Intrinsic Naturalness

February 12, 2010

Posted by Urs Schreiber

$MathML-enabled post (click for more details).$

Lately there have been remarkable developments in higher category theory.

What used to have a touch of alchemy to it – in its mystery, its grand hopes, its plethora of recipes tried out in hard and lonely work in long nights – is becoming chemistry: what used to be hypothesis and conjectures have become theorems; what used to be a philosopher’s stone seemingly out of reach has become a tangible jewel that you can touch, hold against the light, marvel at – and finally use to cut through the glass roof that has been impeding progress for so long: higher category theory.

Or so some think. Discussion with colleagues reveals that the perception of and feelings about what has been achieved – and is being achieved as we speak – varies. While parts of the community are storming ahead with the new technology that has become available, in other parts reservation and scepticism towards this activity is being felt.

Is this really the philosopher’s stone that the search was after? Isn’t its shiny appearance a cheap trick achieved by taking that old pebble called homotopy theory, and polishing it a bit? Is what superficially looks impressive rather intrinsically a kludge geared to serve a purpose and good enough to impress the mundane, but far from being the natural god-given structure that the inner circle of researchers in higher category theory knows – and rightly knows! – is the true goal of the search?

This are, in impressionistic paraphrase, questions being asked behind the scenes. As I have realized in long private discussions recently. And these are good and important questions. If anything as important as a first working implementation of higher category theory is being claimed – explicitly or not – , the claim deserves to be carefully scrutinized.

But careful scrutiny requires an effort to obtain a clear picture of the situation to the same extent that it requires a critical mindset. Imagine the alchemist producing finally the philosopher’s stone – and then discarding it onto the heap of failed attempts for not recognizing it.

Recently I was standing in an alchemist friend’s laboratory and we were looking at that stone. My colleague pointed at it and exclaimed: “Look, it is not natural. This cannot be the answer.” To which I replied: “But wait, you are looking at it as we use it to cut through that glass. Pick it up instead and hold it against the sunlight, so that you see its intrinsic colors, not the reflection of the workbench.” I picked up the stone and held it against the light in three different angles, and we were bathed in its light.

My friend agreed that he hadn’t looked at the stone in this light, and that this did make a difference. He then asked me to share the view from these three angles here on the blog. Which is what I now do.

$MathML-enabled post (click for more details).$

When you are standing at the workbench, it may seem as if the new developments in higher category theory – in $(\infty,1)$ -category theory notably – are based on unnatural kludges from old toolboxes such as notably those tools called model categories and triangulated categories . The impression might arise that the presently active approach is suggesting to fundamentally base higher category theory on model category theory!

But notice that the opposite is true.

Model categories and triangulated categories are toolboxes that have been used for a long time and of which it has always been clear that

they work – they do capture aspects of nature;
they are kludgy and unnatural and at best shadows of a deeper natural structure.

This kind of situation is the killer application for category theory: any field in math full of interesting results but with conceptually awkward foundation. Usually this is a sign that a nice and powerful category theoretic conceptualization is waiting to be uncovered, one that makes all the concepts transparent and makes all the complexity manifestly follow by a systematic process from simple natural basic structures.

And I am claiming that it is this kind of abstract category theory success that we are currently witnessing. If you think that $(\infty,1)$ -category theory is done all with model categories, then you have never tried to read the literature:

after the relation to model category theory is made crystal clear in the theorem that combinatorial model categories are precisely a generators-and-relations presentation of locally presentable $(\infty,1)$ -cartegories in Higher Topos Theory, they are mostly entirely dropped in favor of intrinsic $(\infty,1)$ -category theory. There is hardly a model category model being used as a computational tool in the further development of the theory. Much to the regret of poor souls like me! An explicit model category model of some $(\infty,1)$ -category helps me handle these beasts. It is like choosing a basis for my vector spaces often helps me to handle these. Even though I know that theoretically I should treat them as abstract vector spaces, sometimes an elegant, abstract, intrinsic proof is harder to come by then a kludgy pedestrian component-ridden proof. This fault of a fallible man is not a fault of the theory, though.

I claim it is the other way round: researchers have used and developed model category tools for ages to achieve their goals, thereby producing impressive but – to the eyes of the abstract category theorist or toposopher – unnatural-looking theories. What the present development in higher category theory does achieve is a radical re-conceptualization of these workbench scenarios: it unravels the intrinsic nature underlying these constructions.

Here are three main examples.

$(\infty,1)$ -toposes

If we count Kenneth Brown’s 1973 work as one of the earliest publications where the central idea is put forward, there is now a history of over 35 years of published work on model category structures on simplicial presheaves. Joyal famously suggested this in his letter to Grothendieck as a possible solution to the pursuit of $\infty$ -stacks. Later Jardine intensively developed the theory both in depth and in width. Toën and Vezzosi then generalized the definition to “higher categorical” sites – modeled in turn as model-category sites! The implict claim has all along been that this model structure somehow models higher Grothendieck toposes of $(\infty,1)$ -sheaves/ $\infty$ -stacks. Somehow!

While it could be shown that this was indeed the case in lowest degree, where one could compare with the available intrinsic 2-categorical definition of stack-2-toposes, much beyond that no intrinsic higher category theory was available to check against. And wasn’t it a dubious claim? The first thing most textbooks will tell their readers about stacks is that these are pseudofunctors. But a simplicial presheaf is a strict functor! How on earth can that model an $\infty$ -stack? And how on earth is something as intrinsically intrinsic as topos theory to be incarnated in its higher categorical version in the usual kludgy collection of factorization systems and weak equivalences that the model category structure is based on?

All this is clarified now. Now with HTT an entirely intrinsic, abstract, pure higher category theory definition of $\infty$ -stack has been given: a Grothendieck $(\infty,1)$ -topos of $\infty$ -stacks is a reflective $\infty$ -subcategory of an $(\infty,1)$ -presheaf category

$Sh_{(\infty,1)}(C) \stackrel{\overset{lex}{\leftarrow}}{\hookrightarrow} PSh_{(\infty,1)}(C) \,.$

This is as abstractly category-theoretic and topos-theoretic as it gets. This is as natural and elegant as it gets. If only we can make sense of what all these $(\infty,1)$ -categorical notions mean (adjoint, limit, etc.) then this must be the right and good answer. And now we can make sense of it.

And then there is a theorem: the model category structures on simplicial presheaves that have been guessed for over 35 years, effectively, to model $\infty$ -stack categories do precisely this: every abstractly defined $\infty$ -stack $(\infty,1)$ -topos $Sh_{(\infty,1)}(C)$ has a generators-and-relation presentation by simplicial presheaves on $C$ , and every such presentation presents an $\infty$ -stack $(\infty,1)$ -topos. So what we have now in hand is – on top of a mighty powerful computational technology that has been around all along – an understanding of the entirely natural, abstract and elegant underlying nature of the phenomenon described by this technology.

In 1972 Dan Quillen – famously the inventor of the model category toolset and the advisor of Ken Brown with whom the above story began – had studied homotopy fibers of maps between nerves of categories and remarked:

Someday these ideas will undoubtedly be incorporated into a general homotopy theory for topoi.

This has now become literally true.

For an attempt at an exposition surveying these ideas see
- models for $\infty$ -stack $(\infty,1)$ -toposes.
stable $(\infty,1)$ -categories

If model categories feel bad in their component-ridden non-intrinsicness, then triangulated categories are even worse. While a powerful toolset that has allowed people to venture deep into the heart of homological algebra, dissatisfaction with this toolset has been felt and expressed early on. It was an eduring source of embarrassment how ill-behaved triangulated categories were under various natural operations. Cures for this were put forward, such as notably enhanced pretriangulated dg-categories, but their definition certainly does not manifestly make the concept come closer to the natural intrinsic category theoretic definition that the pure higher category theorist would hope for.

This has changed now. There is a beautiful, simple abstract concept that explains it all: that of a stable $(\infty,1)$ -category.

In every $(\infty,1)$ -category $C$ with a 0-object, there are two god- given operations: the formation of loop space objects and that of suspension objects

$\Omega :C \stackrel{\leftarrow}{\overset{\simeq}{\to}} C : \Sigma \,.$

These are nothing but the pullback and pushforward along itself of the inclusion of the zero object into any other object. An $(\infty,1)$ -category is stable if it is stable under these operations: if $\Omega$ and $\Sigma$ exhibit an auto-equivalence.

That’s a one-line definition. And one whose consideration is compelling. There is no doubt that in the abstract platonic world of pure higher category theory, stable $(\infty,1)$ -categories deserve to be honored for their existence.

And then there is a theorem: when you apply force and crush down a nice and elegant stable $(\infty,1)$ -category to a plain old 1-category by decategorifying it against its will, what you are left with is an ill behaved totally unnatural remnant. All it remembers is a faint shadow of the simple abstract definition above: the shadow of the suspension $\infty$ -functor becomes known as the shift functor, and what are known as the distinguished triangles that give the notion of triangulated categories its name is nothing but a shadow of the long fibration sequences that used to exist in the stable $(\infty,1)$ -category before it was forcefully decategorified.

If you never liked triangulated categories, now is the time to rejoice: you will never ever have to mention them again. Use stable $(\infty,1)$ -category theory instead. This is the right notion. Everything else is a kludge. (Useful as that may be for computations, sometimes.)

For more details on this see the discussion and links given at
- stable $(\infty,1)$ -category.
$(\infty,1)$ -geometry

The new development in higher category theory is notably driven by the desire for a deeper understanding of generalized geometry.

Category theory and topos theory is coming back to its origins here: originally Grothendieck had recognized forcefully that category theory and topos theory provide the right language to speak about the generalized geometry that he was looking at, which happened to be that of generalized spaces modeled on formal duals to commutative rings. This single application remarkably gave rise to a huge development of abstract category theory and topos theory, which in turn provided more tools for these applications.

More recently it became very clear that an entirely higher categorical refinement of all this should not only be thinkable, but is forced upon us by natural examples: the Goerss-Hopkins-Miller theorem shows that the moduli space of derived ellptic curves seems to behave like a higher categorical version of a ringed space, whose structure sheaf consists not of rings of functions, but of $E_\infty$ -rings of functions. That this moduli space wants to be regarded as an object in higher categorical geometry – in derived geometry – becomes quite clear. The big challenge is to make manifest all the required abstract structure that makes this realization not only desireable, but precise and provable.

This is a major motivating example behind derived geometry – higher categorical geometry; higher topos theory.

There have been early attempts to realize this. Notably Toën and Vezzosi constructed an impressive machinery for the description of derived geometry: all entirely built from the model category tool set. They generalize sites to model-sites, sheaves to objects in the model structure on simplicial presheaves on model sites and so on and so forth. An admirable undertaking. But the abstract category theorist who already feels personally hurt by the concept of model category alone, what should he or she think of this? Is such a component-ridden construction supposed to be a realization of intrinsic natural higher category theoretic geometry?

It is a shadow of it. Inspired by these developments, Lurie develops in Structured Spaces the picture of a theory of general abstract higher geometry of charming natural intrinsic beauty. Don’t be fooled by the title of the series: what is being exposed here goes way, way beyond a generalization of the field of algebraic geometry. This is a text with concepts of largely the same intrinsic depth as, say, the Sketches of an Elephant .

The simple central idea is that a concrete generalized space is an $(\infty,1)$ -topos equipped with a structure sheaf $\mathcal{O}$ , modeled as an $\infty$ -stack-valued $\infty$ -copreshaef on the $\infty$ -category $\mathcal{G}$ of spaces on which the geometry is modeled

$\mathcal{O} : \mathcal{G} \to Sh_{(\infty,1)}(X)\,.$

And as shown there, by making concrete choices for that category of test spaces and then turning the crank, the abstract formalism spits out first standard algebraic geoemtry, then algebraic stack theory, then differential geometry, then derived differential geometry, derived $E_\infty$ -spectral geometry and everything else.

An attempt to survey and highlight the central abstract conceptual higher categorical ideas here is at
- notions of space.

These are three examples for the intrinsci naturalness exhibited by recent developments in higher category theory. Instead of being based on kludgy toolsets that have been around for decades, the development shows abstract category theory at its best by unifying all these toolsets into presentations for natural general abstract concepts. And then using the clarity gained this way, the theory goes well beyond what has previously been conceivable.

Or so I think. Hopefully we can discuss this here further.

Posted at February 12, 2010 8:31 PM UTC

TrackBack URL for this Entry: https://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/2169

143 Comments & 3 Trackbacks

Re: Intrinsic naturalness

$MathML-enabled post (click for more details).$

Urs, in your little history of “ $(\infty,1)$ -topos”, I can’t help but think that Dan Quillen should be mentioned somehow. In the introduction to his famous paper “Higher Algebraic K-theory I” (1972), Quillen (who was Ken Brown’s advisor) devotes a paragraph giving an apology for his proofs of Theorems A and B in that paper (these give sufficient conditions for computing the homotopy fiber of a map between nerves of categories). After discussing various techniques which might be used to prove these, he closes with:

“The present approach, based on the Dold-Thom theory of quasi-fibrations, is quite a bit shorter than the others, although it is not as clear as I would have liked, since the main points are in the references. Someday these ideas will undoubtedly be incorporated into a general homotopy theory for topoi.”

The last sentence is now literally true; these theorems are a simple consequence of the “descent” property of spaces, which in fact hold in any $(\infty,1)$ -topos. I’ve often wondered if Quillen actually had something like this in mind when he wrote that.

Posted by: Charles Rezk on February 13, 2010 12:04 AM | Permalink | Reply to this

Re: Intrinsic naturalness

$MathML-enabled post (click for more details).$

Urs, in your little history of “(∞,1)-topos”, I can’t help but think that Dan Quillen should be mentioned somehow. […]

Thanks, Charles, very nice point you are making here.

I have added a brief version of this now to the above entry text. Will add a more detailed paragraph along these lines to the $n$ Lab entry later, when I have a bit of time. Thanks for highlighting this.

Posted by: Urs Schreiber on February 13, 2010 10:45 AM | Permalink | Reply to this

Re: Intrinsic naturalness

Does this imply that in the more mundane applications of quasi-fibrations in ordinary homotopy theory, the
general homotopy theory for topoi will permit more efficient and/or more perspicuous proofs?

Posted by: jim stasheff on February 13, 2010 2:02 PM | Permalink | Reply to this

Re: Intrinsic naturalness

$MathML-enabled post (click for more details).$

I don’t know about “more efficient” proofs, thought I might agree with “perspicuous”. A number of the uses of quasifibrations that I’m aware of are in service of proving things that can be proved in an arbitrary $(\infty,1)$ -topos, and which are more or less straightforward consequences of one of the basic properties of such things (e.g., “descent”). I don’t know all the ways quasifibrations get used, so I wouldn’t want to claim that all uses can be replaced by an appeal to higher topos theory.

And if you’re interested in actual theorems about actual topological spaces, quasifibrations are likely to be useful for the statements of some theorems, even if the proofs appeal to some higher topos tech.

Posted by: Charles Rezk on February 13, 2010 6:41 PM | Permalink | Reply to this

Re: Intrinsic Naturalness

This looks really great, Urs. It just might make me interested enough in (oo,1) things to get beyond the dilettante stage. (In particular, explaining just enough of some wonderful things is a much better to way to get me excited than telling me that some wonderful things exist and that I could learn about them if only I’d put down the wonderful things I’m thinking about and open up some other book.) Thanks! Your effort is much appreciated!

Posted by: James on February 13, 2010 9:08 AM | Permalink | Reply to this

Re: Intrinsic Naturalness

Nice post to read… And Agrippa said unto Paul, ‘In a little thou dost persuade me to become a Christian!’

Posted by: Bruce Bartlett on February 13, 2010 10:54 AM | Permalink | Reply to this

Re: Intrinsic Naturalness

Beautiful! Just what this blog is for.

Can we now begin to tease apart the more and less justified parts of the sceptics’ concerns? Is it possible to agree wholly with what you wrote but still wonder whether these developments are as remarkable as you say? I ask not on the basis of any knowledge - just from things in the air. In fact perhaps the only concrete expression of doubt I’ve read occurs from time to time at the nLab. For example, isn’t Zoran expressing doubt through this discussion.

Posted by: David Corfield on February 13, 2010 11:05 AM | Permalink | Reply to this

Re: Intrinsic Naturalness

$MathML-enabled post (click for more details).$

For example, isn’t Zoran expressing doubt through this discussion.

Just one caveat, if I may: I notice that in the present context there are very many aspects to the situation. Unless and until we are all sure that we all have the same things in mind, I would suggest we try to make carefully explicit what exactly we are thinking of.

Here for instance I am not sure about what exactly you think Zoran thinks is expressing doubt about. Even in the discussion you point to the subject of the discussion changed at least twice from its beginning to where it is now.

One aspect of the discussion is this: Zoran points out that there are questions of $(\infty,1)$ -category theory that are not manifestly modeled by a model category.

This is true, but by itself not in contradiction to anything I meant to say in the above. In fact, it is true that in order to model an $(\infty,1)$ -category by a plain category, the minimal structure necessary is just a category with weak equivalences. By Dwyer-Kan localization every such determines an $(\infty,1)$ -category, and every $(\infty,1)$ -category arises this way, up to equivalence.

All the extra structure provided by a model category is just there to facilitate handling these structures. It’s a tool, not an intrinsic concept.

Posted by: Urs Schreiber on February 13, 2010 2:05 PM | Permalink | Reply to this

Re: Intrinsic Naturalness

$MathML-enabled post (click for more details).$

I don’t understand this stuff well enough, but wasn’t Zoran casting doubt on the power of $(\infinity, 1)$ -categories when he said

You know I would accept your viewpoint that infinity-categories are better than derived functors if they would cover all the cases, but clearly they can not cover the nonsymmetric cases.

and

I know well of about ten of such extremely important cohomologies which do not fit into usual homological or homotopical algebra. People take oldest best understood examples, rephrase them in new language. I appreciate the results, and do not see ANY progress in about a dozen of main examples I was interested in in last 10 years.

and

I know that model categories are just tool to provide the existence of localization at weak equivalences. But in nonsymmetric examples I know it seems that localizing, irrespective of niceties of its description and existence is too rude and can destroy essential nonsymmetric nature of the structure. Talking infinity categories and having all examples in almost symmetric world is self-deceit.

Posted by: David Corfield on February 14, 2010 4:36 PM | Permalink | Reply to this

Re: Intrinsic Naturalness

$MathML-enabled post (click for more details).$

David wrote:

I don’t understand this stuff well enough, but wasn’t Zoran casting doubt on the power of $(\infty,1)$ -categories when he said […].

Do you really think that he was expressing doubt on the power of $(\infty,1)$ -categories? My impression is, that this is more a discussion about certain technichal details here (could anyone explain me, what exactly this quotes by Zoran mean? Non-symmetric case?).

The whole situation reminds me of the reaction when you explain the idea of 1-categories to mathematician who do not use them in their daily work. They usally ask, what can be done with categories, what can not be done without. And it might be very hard to convince people to accept this new perspective, althoug it is very natural and easy….

Posted by: Thomas Nikolaus on February 14, 2010 4:52 PM | Permalink | Reply to this

Re: Intrinsic Naturalness

Sorry if I got things wrong. But still I am intrigued if it is true as Urs said that

While parts of the community are storming ahead with the new technology that has become available, in other parts reservation and scepticism towards this activity is being felt,

that no expression of this reservation seems to have been written down.

Posted by: David Corfield on February 14, 2010 6:22 PM | Permalink | Reply to this

Re: Intrinsic Naturalness

$MathML-enabled post (click for more details).$

I wrote

The whole situation reminds me of the reaction when you explain the idea of 1-categories to mathematician who do not use them in their daily work. They usally ask, what can be done with categories, what can not be done without. And it might be very hard to convince people to accept this new perspective, althoug it is very natural and easy….

Just to clarify David: I didn’t want to blame you for this. Sorry if you got this impression.

It is very good to ask these questions. But I think the algebraic derived category is a very good example where $(\infty,1)$ -categories can clarify the situation. And in fact it is much simpler then the theory of triangulated categories. Maybe we should discuss this in more detail here? Especially the diffrent model structures or the classical derived functor approach? It would be very interesting to understand Zorans point.

Posted by: Thomas Nikolaus on February 14, 2010 6:44 PM | Permalink | Reply to this

Re: Intrinsic Naturalness

$MathML-enabled post (click for more details).$

David, I’m surprised at your apparent surprise. Would you positively expect those with reservations and sceptical feelings towards a particular body of work to publish expressions of those feelings? It seems to me that that’s not really part of the ordinary scholarly process in mathematics. I’m not saying that it shouldn’t be — just that it’s not. Mostly, I think, people are quite conservative in what they write, apart from a few outspoken exceptions.

Or maybe you’re not actually surprised, just intrigued.

Posted by: Tom Leinster on February 14, 2010 9:20 PM | Permalink | Reply to this

Re: Intrinsic Naturalness

I ought not be surprised. In chapter 9 of my book I looked at the debate for and against groupoids. Where one could detect real opposition to the idea that looking beyond groups to groupoids for a more comprehensive understanding of symmetry this was largely through the writings of groupoid promoters, who clearly had met with critical voices.

I’m not intrigued so much as disappointed at the lack of expression of reservation and doubt. You write

It seems to me that that’s not really part of the ordinary scholarly process in mathematics. I’m not saying that it shouldn’t be – just that it’s not.

Yes, it’s not, but it should be. As I wrote early in the life of this blog

‘What is it to pay respect to a research program?’. Often, it translates to a ‘let’s leave each other alone to carry on with our own business’ attitude. To some extent this works. Programmes come to be seen as flourishing or running out of steam. In the latter case, the young don’t sign up, and the old-timers moan to themselves about the passing of the good old days.

This is better than disrespectful rows, but it quite possibly represents a failure to respect other views properly in order to gain fully from them. The theory of rationality I have taken on from Alasdair MacIntyre charges us (p. 16) to make known our assessment of the state of our own programme, not only its successes, but also its current weaknesses and, hopefully temporary, lack of resources in resolving what it takes to be its central problems. These problems may not be taken as central by another group, may barely be recognisable by them, but still it is possible that one of their number may learn this new language and come to see that they have resources to help with these problems. And it is possible that someone coming to learn the rival language as a ‘second first language’ may find resources useful to their own programme of origin.

This exposure of the perceived weaknessess of one’s programme is an uncomfortable activity, just as it is in the project which is our life. And it does not accord well with a climate in which we must scream out how well we are doing to secure resources. But in the right conditions it steers us towards the truth more rapidly. Perhaps we should have a session on this blog of reporting what we take to be the current largest weaknesses of n-category theory.

There are two activities I might separate more clearly there. Exposing what you take to be your own current strengths and weaknesses, and expressing what you take to be the strengths and weaknesses of others. We need to get better, in most walks of life, at discussing possible weaknesses.

Posted by: David Corfield on February 15, 2010 9:40 AM | Permalink | Reply to this

Re: Intrinsic Naturalness

$MathML-enabled post (click for more details).$

David, I’m surprised at your apparent surprise. Would you positively expect those with reservations and sceptical feelings towards a particular body of work to publish expressions of those feelings? It seems to me that that’s not really part of the ordinary scholarly process in mathematics.

I hope there will be some way to voice the concerns. Otherwise we are stuck.

David is trying to understand the reasons for scepticism, in order that they can be addressed, in one way or other. He writes

Can we now begin to tease apart the more and less justified parts of the sceptics’ concerns?

Can you help him answer this question? I’d be very interested, too.

The above entry was meant to address one of the concerns, as far as I understood it. What is the agreement on this concern now? What other concerns are there?

Posted by: Urs Schreiber on February 15, 2010 12:56 AM | Permalink | Reply to this

Re: Intrinsic Naturalness

$MathML-enabled post (click for more details).$

In fact, it is true that in order to model an (∞,1)-category by a plain category, the minimal structure necessary is just a category with weak equivalences. By Dwyer-Kan localization every such determines an (∞,1)-category, and every (∞,1)-category arises this way, up to equivalence.

As Mike kindly points out, Clark Barwick and Dan Kan have produced a detailed account of a

model structure on categories with weak equivalences

that fully formalizes and proves the statement that categories with weak equivalences model the $(\infty,1)$ -category of $(\infty,1)$ -categories.

Posted by: Urs Schreiber on February 15, 2010 12:08 PM | Permalink | Reply to this

Re: Intrinsic naturalness

$MathML-enabled post (click for more details).$

Thanks very much for this, Urs! It’s rather nice to see you explain your personal thoughts about this in the open here. (I was one of those who had a part in the behind-the-scenes discussions with Urs.)

During the discussions Urs was speaking of the intrinsic notion of $(\infty, 1)$ -categories, and I finally interrupted with: what is the intrinsic notion of $(\infty, 1)$ -categories? If you look at the nLab article for instance, not one but a variety of notions is given, so I wasn’t sure what he meant.

Urs’s response: quasi-categories. He went on to remind me that in the old days of higher categories, there was a plethora of definitions of higher categories but not much theory (i.e., theory which develops these structures, not theory that the definitions are built on), whereas now, in the wake of the tremendous activity of Joyal on the theory of quasi-categories (see for example his lecture notes), and of Lurie in his Higher Topos Theory, and of many others, we are at last seeing substantial theoretical progress. Urs of course has been one of the most vigorous promoters of these wonderful developments, here at the Café and elsewhere.

These developments are a strong pragmatic argument for the central importance of quasi-categories, but I was rather interested in hearing more from Urs as to a conceptual reason for why quasi-categories in particular ought to be considered the intrinsic notion. Urs began to say something to this, but I’d like to hear more from him and others here (to what is not a very well-defined question, obviously, but answers would be of interest to me).

Posted by: Todd Trimble on February 13, 2010 2:13 PM | Permalink | Reply to this

Re: Intrinsic naturalness

At least from what I understand of higher category theory, I feel like there probably isn’t an intrinsic or best definition of any kind of higher categorical structure. Even with ordinary algebraic structures, you can have different axiomatizations of the same theory, and it’s not clear that any one set of axiom sets is best. Lawvere theories, and generalizations to other doctrines, do give a categorical structure capturing exactly the intrinsic content of that theory. But I’m not sure that idea can be of much help in giving an intrinsic definition of higher categories, since that invariant categorical structure would itself be a higher dimensional structure.

Even at the level of bicategories, there are many different possible definitions–some algebraic, some not–and it’s not at all clear to me that any one is best.

I am also very interested in what others think about Todd’s question.

Posted by: Patrick Schultz on February 13, 2010 5:24 PM | Permalink | Reply to this

Re: Intrinsic naturalness

$MathML-enabled post (click for more details).$

In order to have a fair discussion we should first name the several models for $(\infty,1)$ -categories. But before I write them down, let me say that they all lead to equivalent concepts of $(\infty,1)$ -categories, but some have technichal advantages/disadvantages. Ant aditionally they are all based on the model of simplicial sets (or better Kan complexes) for $\infty$ -groupoid. You can take this as a definition or assume that the homotopy hypotheses holds and derive it from your model of $\infty$ -groupoids.

1) Simplicially enriched categories

This is a very simple model. It’s just categories enriched over simplicial sets. Better one should say categories enriched in Kan complexes (because these are the fibrant simplicially enriched categories). That means we have objects A,B,C,… and for each two objects a Kan complex $\Hom(A,B)$ . This should be thought of an $\infty$ -groupoid. Thus those are categories strictly enriched in $\infty$ -groupoids.

2) Segal categories

This is more or less the same idea as in 1). It has objects, and for each two objects a simplicial set (Kan complex). But we don’t have a strictly defined composition $\Hom(A,B) \times \Hom(B,C) \to \Hom(A,C)$ but rather a further simplicial set $H(A,B,C)$ with three maps to $\Hom(A,B)$ , $\Hom(B,C)$ and $\Hom(A,C)$ such that the morphism $H(A,B,C) \to \Hom(A,B) \times \Hom(B,C)$ is a weak equivalence. Each choice of invers provides now a composition. Additionally we have furher simplicial sets $\H(A_1,\ldots,A_n)$ which provide the coherence conditions. Thus this should be seen as categories weakly enriched in simplical categories (and the equivalence of 1 and 2 shows that each weakly enriched category is equivalent to a strictly enriched one).

3) Quasi-categories

A simplicial set which admits horn fillers for all inner horns $\Lambda(n,k) \to \Delta(n)$ . The vertices are the objects, the $n$ -simplices are the $n$ -morphisms. The composition of morpisms in this categorie is now just defined by chosing horn fillers (which are unique up to higher cells).

4) Complete Segal Spaces

A simplicial object in simplicial sets with a certain conditon (Segal coniditon). I won’t describe this in full detail here, but it is also easy to understand what this coniditon means. Heuristically one should think about this as follows: The simplicial set $X_0$ is the $\infty$ -groupoid-core of our $(\infty,1)$ -category. The simplicial set $X_1$ is the $\infty$ -groupoid core of the Arrow category and so on.

The advantage of the first two models is that they are very easy to understand. And for example when Dwyer and Kan developed their simplicial localization they realized it as a simplically enriched category. But in those two models it is very hard to describe the functor category between two $(\infty,1)$ -categories and even to talk about weak functors (technically the problem is replace the source category of the functor cofibrantly). In the last two models, though seemingly more complicated, the functor categories arise very naturally. In my opinion it is most easily described in quasi-categories. Because between two quasi-categories $X$ and $Y$ the functor category is just the internal hom $Hom(X,Y)$ in simplicial sets. This Hom-space is then also a quasi-category. A further advantage is, that $\infinity$ -groupoids are naturally a subcategory of quasi-categories.

Posted by: Thomas Nikolaus on February 13, 2010 6:11 PM | Permalink | Reply to this

Re: Intrinsic naturalness

$MathML-enabled post (click for more details).$

Thanks for this response, Thomas. There are a couple of things I’d like to bring out in this discussion. One is what the above-mentioned “equivalence” of these models means.

Let me see if I understand. I hope Urs won’t mind my quoting this snippet from one of our emails:

Kan-enriched categories form a quasicategory. Segal categories form a quasi-category. Quasi-categories form a quasi-category. All these quasi-categories are equivalent as objects in the quasi-category of quasi-categories.

Okay, based on emails with Urs and also what Urs has said here, I think this is what Urs would mean by an intrinsic notion of equivalence between these various modelings. Before making this notion more explicit, I’ll remark that in the Perspectives section of his extensive notes, Joyal exhibits equivalences in terms of model structures on each of these categories (simplicially-enriched categories, Segal categories, Rezk categories, and so on), and indicates that the model structures are all Quillen equivalent. Now, for any model category, there is a corresponding quasi-category, in such a way that Quillen equivalences correspond to equivalences between quasi-categories. (Is it fair to say that this is pretty much state of the art as far as proofs of equivalence go?)

So, let me ask and answer: what is an equivalence between quasi-categories? There is a fundamental category construction (akin to the fundamental groupoid of a simplicial set)

$\tau_1: Set^{\Delta^{op}} \to Cat,$

defined up to isomorphism to be the left adjoint to the nerve functor, and we can regard this as a change-of-base between cartesian closed categories: $\tau_1$ preserves products and hence is strong monoidal. Now $Set^{\Delta^{op}}$ , which is enriched in itself by cartesian closure, becomes enriched in $Cat$ by passing along this change of base. Hence we can regard $Set^{\Delta^{op}}$ as a 2-category (because that’s what $Cat$ -enriched means); the 1-cells in this 2-category are precisely simplicial maps. Now, a quasi-functor between quasi-categories $X$ , $Y$ (which are themselves simplicial sets) is by definition a simplicial map $f: X \to Y$ . We now define a quasi-functor $f: X \to Y$ to be an equivalence if map is an equivalence in $Set^{\Delta^{op}}$ regarded as a 2-category. (If I’m not mistaken, this instantiates the original notion of equivalence between notions of $n$ -category proposed by Baez and Dolan in their original 1995 email, based on equivalence between what Dolan called “homotopy 1-categories”.)

This is all baby stuff to the experts, but it doesn’t hurt to spell it out precisely anyway, since not all of us here are experts.

The other thing I’d like to see brought out in this discussion, whose answer I won’t try to guess for now, is spurred by this snippet from Thomas:

And additionally they are all based on the model of simplicial sets (or better Kan complexes) for $\infty$ -groupoid. You can take this as a definition or assume that the homotopy hypotheses holds and derive it from your model of $\infty$ -groupoids.

The rough idea (see the Idea section here) seems to be that a good notion of $(\infty, 1)$ -category is that of a category (weakly) enriched in “ $\infty$ -groupoids”, provided that the notion of $\infty$ -groupoid is sufficiently good. I’d be interested in a more precise formulation of this rough principle. As for “sufficiently good”, there seems to be a baseline assumption that (at the very least) the homotopy hypothesis is satisfied. What else should one want?

I’m interested in pursuing this because the idea seems to me to be in line with the sort of “egalitarian” sentiments I take to be expressed by Charles here. I mean, besides the technical notion of saying that a good notion of $(\infty, 1)$ -category is that the collection of such putative $(\infty, 1)$ -categories forms (in whatever way) a quasi-category that is quasi-equivalent to the quasi-category of quasi-categories (which seems to be the line Urs is taking), is there some more intuitive properties-based approach one can take?

Posted by: Todd Trimble on February 13, 2010 9:48 PM | Permalink | Reply to this

Re: Intrinsic naturalness

$MathML-enabled post (click for more details).$

Joyal exhibits equivalences in terms of model structures on each of these categories (simplicially-enriched categories, Segal categories, Rezk categories, and so on), and indicates that the model structures are all Quillen equivalent

Yes, that is true. I think Joyal shows that quasicategories are Quillen equivalent to the other three. The full result of exhibiting model structures on all of them and giving direct Quillen equivalences is due to Julie Bergner, as far as I know.

Now, for any model category, there is a corresponding quasi-category, in such a way that Quillen equivalences correspond to equivalences between quasi-categories.

Yes, there is a quasi category for any model category. But I think the more natural construction is the construction of a simplicially enriched category. Either by the Dwyer-Kan-localization (which only depends on the weak equivalences) or by endowing the model category with the extra structure of a simplicially enriched model category and then taking the full subcategory of fibrant-kofibrant objects. Dwyer and Kan showed that those two constructions lead to equivalent function complexes and thus to equivalent $(\infty,1)$ categories.

The quasi-category which Urs means, is now obtained by taking the simplicial nerve of this simplicially enriched category. The simplicial nerve is the right adjoint of the Quillen equivalence between quasi categories and simplicially enriched categories (and obtained as usally by a functor $\Delta \to \text{sCat}$ ).

Furthermore Jacob Lurie gives in HTT a general definition and theory of adjunctions between Quasi-categories. Now the functor described above which associates a quasicategory to a model category, sends Quillen pairs to adjoint funcors between quasi-categories and Quillen equialences to adjoint equivalences. Thus a Quillen equivalence encodes not just an equivalence between $(\infty,1)$ -categoires but rather an adjoint equivalence.

As for “sufficiently good”, there seems to be a baseline assumption that (at the very least) the homotopy hypothesis is satisfied. What else should one want?

First in order to formulate the homotopy hypotheses we need a $(\infty,1)$ -structure on $\infty$ -groupoids. This sounds circular, but the easiest thing would be to have a model structure. Then the homotopy hypotheses would be fullfilled if this model category is Quillen equivalent to simplicial sets (or topological spaces). Furthermore we need a monoidal structure on our model of $\infty$ -groupoids in order to enrich in. But this can not be arbitrary but must be the “same” as the monoidal structure on simplicial Sets. The best think would be to have monoidal model categories such that the Quillen equivalence lifts to a monoidal equivalence. Then we should obtain a Quillen equiovalent notion of $(\infty,1)$ -categories.

As a first example, we could as well as simplicially enriched categories, consider topological enriched categoires and obtain an equivalent model for $(\infty,1)$ -categories…

Posted by: Thomas Nikolaus on February 14, 2010 5:28 PM | Permalink | Reply to this

Re: Intrinsic naturalness

$MathML-enabled post (click for more details).$

Thomas wrote:

Yes, there is a quasi category for any model category. But I think the more natural construction is the construction of a simplicially enriched category. Either by the Dwyer-Kan-localization (which only depends on the weak equivalences) or by endowing the model category with the extra structure of a simplicially enriched model category and then taking the full subcategory of fibrant-kofibrant objects. Dwyer and Kan showed that those two constructions lead to equivalent function complexes and thus to equivalent $(\infty, 1)$ categories.

The quasi-category which Urs means, is now obtained by taking the simplicial nerve of this simplicially enriched category. The simplicial nerve is the right adjoint of the Quillen equivalence between quasi categories and simplicially enriched categories (and obtained as usual by a functor $\Delta \to sCat$ ).

These are excellent technical points to point out – thanks for doing so.

Furthermore Jacob Lurie gives in HTT a general definition and theory of adjunctions between Quasi-categories… a Quillen equivalence encodes not just an equivalence between $(\infty, 1)$ -categories but rather an adjoint equivalence.

Sure.

Posted by: Todd Trimble on February 15, 2010 4:27 PM | Permalink | Reply to this

Re: Intrinsic naturalness

$MathML-enabled post (click for more details).$

Todd wrote:

is there some more intuitive properties-based approach one can take?

I think that’s a great question. Perhaps it’s easier to start by trying to characterize the quasicategory of $(\infty,0)$ -categories, i.e. homotopy types or simplicial sets or $\infty$ -groupoids?

I know of some nice work that characterizes the model category of stable homotopy types, but I don’t know a characterization the model category of plain old homotopy types. Surely someone must have thought about this! Maybe this would hint at some ways forward.

Posted by: John Baez on February 23, 2010 7:00 PM | Permalink | Reply to this

Re: Intrinsic naturalness

$MathML-enabled post (click for more details).$

Todd asked here on the notion of $(\infty,1)$ -category:

is there some more intuitive properties-based approach one can take?

John says:

I think that’s a great question. Perhaps it’s easier to start by trying to characterize the quasicategory of (∞,0)-categories

One way to characterize the quasi-category of $(\infty,0)$ -categories is of course to say that it is the full subcategory on spaces of the homotopy type of a CW-complex. But somehow I feel that this is not the characterization that is being looked for here.

I like this one: $\infty Grpd$ is the terminal $(\infty,1)$ -topos.

It seems to me that concerning Elephant-style fully abstract $\infty$ -topos-theoretic reasoning about $(\infty,1)$ -categories only the surface has been scratched. I am currently entertaining myself here with listing structures provided by $(\infty,1)$ -toposes on purely formal grounds.

Concerning abstract properties of $(\infty,1)Cat$ : not sure what it’s worth, but it is a fact that $(\infty,1)Cat$ has the property that, up to equivalence, the only nontrivial $(\infty,1)$ -functor $(\infty,1)Cat \to (\infty,1)Cat$ that is an equivalence is that which forms opposite $(\infty,1)$ -categories.

These facts, and references for them, are recorded at

Posted by: Urs Schreiber on February 23, 2010 7:52 PM | Permalink | Reply to this

Re: Intrinsic naturalness

$MathML-enabled post (click for more details).$

$\infty Grpd$ is the terminal $(\infty,1)$ -topos

That’s a very nice one, but I think it can be improved a bit. For one thing, it’s not completely intrinsic—it involves quantifying over all other $(\infty,1)$ -toposes. What about “ $\infty Gpd$ is the unique well-pointed $(\infty,1)$ -topos”?

Of course, for this to be a good characterization that doesn’t beg the question, one has to define “ $(\infty,1)$ -topos” using the Giraud axioms, rather than as a lex localization of an $(\infty,1)$ -category of (necessarily $\infty Gpd$ -valued) presheaves.

Posted by: Mike Shulman on February 24, 2010 2:57 AM | Permalink | Reply to this

Re: Intrinsic naturalness

$MathML-enabled post (click for more details).$

Is there a reason why there is a unique well-pointed $(\infty, 1)$ -topos, but not a unique well-pointed topos?

The main point of a well-pointed topos in logic is the equivalence of external and internal properties,

is there an analogue statement for the well-pointed $(\infty, 1)$ -topos?

Posted by: David Corfield on February 24, 2010 9:31 AM | Permalink | Reply to this

Well-pointed topoi

$MathML-enabled post (click for more details).$

Is there a reason why there is a unique well-pointed $(\infty,1)$ -topos, but not a unique well-pointed topos?

There is a unique well-pointed Grothendieck topos, namely Set, so similarly one might expect there to be a unique well-pointed Grothendieck $(\infty,1)$ -topos. I was assuming that in this discussion “topos” meant “Grothendieck topos” since otherwise Urs’ original statement that $\infty Gpd$ (resp. $Set$ ) is the terminal $(\infty,1)$ -topos (resp. 1-topos) is not true either.

As Urs pointed out, no matter what your metalogic, there are lots of different well-pointed elementary topoi. For instance, the category of sets of cardinality $\lt\kappa$ for any strong limit cardinal $\kappa$ . It’s the non-elementary requirements of local smallness and cocompleteness (or, equivalently for a well-pointed topos, Grothendieck-ness) which force a well-pointed topos to be equivalent to $Set$ .

Actually, though, I was also assuming that the metalogic is classical (but no AC required), since in an intuitionistic metalogic there can be well-pointed Grothendieck topoi that are not equivalent to $Set$ (at least, if you define “well-pointed” in the intuitionistically reasonable way; otherwise even $Set$ is not well-pointed intuitionistically). I don’t know whether the intuitionistic $Set$ has a uniquely distinguishing intrinsic non-elementary property.

Posted by: Mike Shulman on February 24, 2010 4:55 PM | Permalink | PGP Sig | Reply to this

Re: Intrinsic naturalness

$MathML-enabled post (click for more details).$

Is there a reason why there is a unique well-pointed (∞,1)-topos, but not a unique well-pointed topos?

I might be wrong about this, Mike will correct me, but my understanding is that the the degree to which you see not a unique ordinary well-pointed topos is precisely the degree to which you bother about subtle foundational issues. Such as whether or not you allow “exlcuded middle in your meta-logic” and things like that. I suppose if you take such subtleties into account, you also see different versions of well-pointed $(\infty,1)$ -toposes.

The main point of a well-pointed topos in logic is the equivalence of external and internal properties,

is there an analogue statement for the well-pointed (∞,1)-topos?

Do we have a notion of “internal $(\infty,1)$ -logic”? There should be one, but I am ignorant of it.

Posted by: Urs Schreiber on February 24, 2010 10:09 AM | Permalink | Reply to this

Re: Intrinsic naturalness

$MathML-enabled post (click for more details).$

Need to clarify this:

my understanding is that the the degree to which you see not a unique ordinary well-pointed topos is precisely the degree to which you bother about subtle foundational issues.

This being so once we also demand that the topos is complete and co-complete.

Otherwsise this fails for trivial reasons. For instance FinSet and $Set_{\kappa}$ (sets of cardinality less than a fixed strong limit cardinal $\kappa$ ) are well-pointed toposes, but do not have all small colimits.

And probably we should add the condition of local smallness, too.

Posted by: Urs Schreiber on February 24, 2010 12:12 PM | Permalink | Reply to this

Re: Intrinsic naturalness

$MathML-enabled post (click for more details).$

Do we have a notion of “internal $(\infty,1)$ -logic”?

I’m fairly sure I know how to do this, but since I haven’t had time yet to carefully write up the internal logic of a 2-topos, I certainly haven’t had time to do the $(\infty,1)$ -version. Some work moving in the right direction can be found in this paper, although this is only the type theory without the logic added yet.

Posted by: Mike Shulman on February 24, 2010 5:03 PM | Permalink | PGP Sig | Reply to this

Re: Intrinsic naturalness

$MathML-enabled post (click for more details).$

Do we have a notion of “internal (∞,1)-logic”?

I’m fairly sure I know how to do this, but since I haven’t had time yet to carefully write up the internal logic of a 2-topos, I certainly haven’t had time to do the (∞,1)-version. Some work moving in the right direction can be found in this paper, although this is only the type theory without the logic added yet.

Okay. I put that into this entry here:

internal logic of an $(\infty,1)$ -topos

If there are further references going in this directoins, let’s collect them there.

Posted by: Urs Schreiber on February 24, 2010 5:53 PM | Permalink | Reply to this

Re: Intrinsic naturalness

What about Voevodsky’s old work on n-types ? Isn’t this a version of higher type theory related to higher fibrations ?

cafe on Voevodsky’s higher lambda calculus

Posted by: Zoran Skoda on February 24, 2010 6:02 PM | Permalink | Reply to this

Re: Intrinsic naturalness

$MathML-enabled post (click for more details).$

Yes, Voevodsky’s work is also in the same direction. I don’t know about “higher fibrations” but he is interested in additional axioms of the type theory that are validated by homotopical models, i.e. in $(\infty,1)$ -categories. I just got to meet him at CMU a couple weeks ago, and talk about this stuff with Steve Awodey, Michael Warren, Peter Lumsdaine, and the other people working on it; I’ve been thinking of blogging about it since then but haven’t had the time yet. There’s a little more recent version of Voevodsky’s ideas on his web page.

Posted by: Mike Shulman on February 24, 2010 8:21 PM | Permalink | Reply to this

Re: Intrinsic naturalness

$MathML-enabled post (click for more details).$

Won’t there be a sense in which this internal logic will have to be interpretable as a ‘logic of space’? If $Set$ is an especially nice topos which being well-pointed allows us to understand 1-logic internally and externally, we might hope that $\infty Gpd$ does the same for $(\infty, 1)$ -logic.

Had I not known anything about logic beyond that it is a language used to formulate statements in a domain and to represent valid reasoning, and had been asked to extract such a thing given the definition of a topos, that would seem to be a tricky task. But give me $Set$ and I can make a start.

I can convey quite a lot about logic with the sets 1, 2, $H$ (the set of humans), and $D$ (the set of dogs). E.g., the predicate ‘French’ is a map $H \to 2$ , the function ‘owner’ is a map $D \to H$ , composing gives me the dog predicate ‘owned by a French person’. Then ‘Fido’ is an arrow $1 \to D$ , ‘True’ is an arrow $1 \to 2$ , as is ‘Fido is owned by a French person’, and so on. I can then work up an account of variables, quantifiers, etc.

So can I not get going with $\infty Gpd$ similarly by picking out a couple of spaces and then analogues of $1$ and $2$ ? We have that object classifiers are supposed to be the analogues of subobject classifiers. And you’re discussing generators below. I just wonder if a simple look at maps between spaces might not suggest what $(\infty, 1)$ -logic is like.

If we have spaces like the surface of the globe and the interval $[0, \infty)$ , and the temperature map between them, won’t $(\infty, 1)$ -logic capture something of our everyday talk about global temperature? [And I promised not to talk about the climate.]

I suppose connectedness will be covered: I can’t reach the US from the UK by land, etc.

[Note: I’ve promoted this comment to a blog post.]

Posted by: David Corfield on February 25, 2010 9:07 AM | Permalink | Reply to this

Re: Intrinsic naturalness

$MathML-enabled post (click for more details).$

it’s not completely intrinsic—it involves quantifying over all other (∞,1)-toposes. What about “∞Gpd is the unique well-pointed (∞,1)-topos”?

Good idea. Here is my (obvious) attempt at making this precise: we might say an $(\infty,1)$ -topos is well-pointed if the global section functor $\Gamma : \mathbf{H} \to \infty Grpd$ is a faithful $(\infty,1)$ -functor.

Is that what you would have in mind? I’d like it. But of course it requires mentioning $\infty Grpd$ itself.

Posted by: Urs Schreiber on February 24, 2010 10:01 AM | Permalink | Reply to this

Re: Intrinsic naturalness

$MathML-enabled post (click for more details).$

we might say an $(\infty,1)$ -topos is well-pointed if the global section functor $\Gamma\colon H\to \infty Grpd$ is a faithful $(\infty,1)$ -functor.

Yes, something like that is what I had in mind. Of course, as you point out, this phrasing already mentions $\infty Grpd$ , but just as in the 1-case it should be rephrasable in an elementary way, the essence being that the terminal object is a suitable sort of generator. Actually, I would want the terminal object to be a strong generator, which is the elementary version of $\Gamma$ being not just faithful but conservative—in the 1-case this comes for free because of subobject classifiers, but in the $n$ -case I think we need to assert it explicitly.

So what is the right notion of “strong generator” in an $(\infty,1)$ -category? Is this in HTT somewhere?

I wonder if there might be subtleties here involving $\infty$ -ness. For instance, in a 1-category with limits, we can say $1$ is a strong generator if any monic $A\to B$ such that every $1\to B$ factors through $A$ is an isomorphism. It then follows that $1$ is a generator, since given $f,g\colon A\to B$ , if $f x = g x$ for all $x\colon 1\to A$ , then by strong generation the equalizer of $f$ and $g$ is iso, so $f=g$ . But in an $(\infty,1)$ -category, “equalizers” are not monic, so maybe we need to be more careful?

Posted by: Mike Shulman on February 24, 2010 8:11 PM | Permalink | Reply to this

Re: Intrinsic naturalness

$MathML-enabled post (click for more details).$

Is this in HTT somewhere?

I don’t think so, at least not in its explicit incarnation.

But in an (∞,1)-category, “equalizers” are not monic, so maybe we need to be more careful?

Right, so we can say instead regular monomorphism.

But we need to refine the argument somehow.

Because we don’t have that two morphisms

$f,g : A \to B$ of

$\infty$ -groupoids are equivalent if for all $x : {*} \to A$ the morphisms

${*} \stackrel{x}{\to} A \stackrel{f}{\to} B$

and

${*} \stackrel{x}{\to} A \stackrel{g}{\to} B$

are equivalent. That just says that $\pi_0(f) = \pi_0(g)$ .

Posted by: Urs Schreiber on February 24, 2010 8:44 PM | Permalink | Reply to this

Re: Intrinsic naturalness

$MathML-enabled post (click for more details).$

Hmm., you’re right, even the correct notion of “generator” is not obvious. The naive notion of “strong generator” is true though: a monomorphism $A\to B$ of $\infty$ -groupoids is an equivalence as soon as every map $∗\to B$ factors through $A$ . But is this property of an $(\infty,1)$ -topos enough to imply that the global-sections functor is an equivalence?

Perhaps what we really want to say is that for every $A$ , the map $\coprod_{∗\to A} ∗ \to A$ is an effective epimorphism? In a 1-topos, that follows once you know that $∗$ is a strong generator as above, since a 1-topos is a regular category and therefore every extremal epimorphism is in fact a regular epimorphism. Is something like that going to be true in an $(\infty,1)$ -topos?

Posted by: Mike Shulman on February 25, 2010 2:49 AM | Permalink | PGP Sig | Reply to this

Re: Intrinsic naturalness

$MathML-enabled post (click for more details).$

I see that HTT uses rather the notion of “generates under small colimits.” I suppose that it would probably suffice to assume that the terminal object generates our $(\infty,1)$ -topos under small colimits and is also “0-compact,” i.e. its corepresentable functor preserves all small colimits. Can we rephrase those in more elementary terms?

Posted by: Mike Shulman on February 25, 2010 3:26 AM | Permalink | Reply to this

Re: Intrinsic naturalness

$MathML-enabled post (click for more details).$

the notion of “generates under small colimits.”

Oh, of course, we have that obvious notion of “generator”.

I suppose that it would probably suffice to assume that the terminal object generates our (∞,1)-topos under small colimits and is also “0-compact,”

Sounds good.

Can we rephrase those in more elementary terms?

I am unsure now what counts as elementary in this game and what not. I might need to hear from Todd which kind of characterization of $\infty Grpd$ he would feel answers his original question. What were you thinking of?

The notion of colimit and preservation of colimit is elementary once we accept that we do have a context for category theory. And it seems we were thinking all along – in this attempt to give an intrinsic characterization of $\infty Grpd$ – of characterizing $(\infty,1)$ -toposes as such via their definition by the $(\infty,1)$ -Giraud axioms. That already involves the clause “Colimits are stable under pullback.” If we accept a clause like that as elementary, then “The terminal object generates the $(\infty,1)$ -topos under colimits.” is elementary, too.

I bet we can give an analogous characterization of $(\infty,1)Cat$ , too. An $(\infty,1)$ -category generated under colimits from its terminal object, and satisfying some more conditions. (Effectively saying that it is a special $(\infty,2)$ -topos.)

I’d be happy with that, but for all this of course we need to assume a woking context of $(\infty,1)$ -category theory, to even make sense of these statements. And I am not sure if the question that started this discussion is taken to assume this or not.

Posted by: Urs Schreiber on February 25, 2010 8:06 AM | Permalink | Reply to this

Re: Intrinsic naturalness

$MathML-enabled post (click for more details).$

Ideally, I’d like “elementary” to mean “expressible in first-order logic.” First-order logic is usually construed as finitary, which means that in a 1-category, only finite limits are elementary in this sense, and for that it’s essential that you can express them as the existence of pullbacks and a terminal object, for instance – quantification over “all finite diagrams” is not finitary first-order. The property of being well-pointed for a 1-topos is also finitary first-order, which is important for the status of ETCS as a foundational theory. Thus, the only non-elementary property in the characterization of Set is the existence of arbitrary (stable and disjoint) coproducts, and that’s precisely the one which you can drop and get a good first-order notion of “structural set theory.”

In an $(\infty,1)$ -category we may need to be more lenient with our notion of “elementary,” since I can see that probably lots of concepts will need at least countable logic in order to be expressed. But I’d like it if the notion of “well-pointed” didn’t require more than countable first-order logic, at least. The notion of “0-compact” is not first-order at all, since it involves quantifying over all small diagrams, and Lurie’s notion of “generates under colimits” is not even legal without some universes, let alone first-order, since it involves quantifying over all subcategories of the $(\infty,1)$ -category in question.

Posted by: Mike Shulman on February 25, 2010 7:27 PM | Permalink | Reply to this

Re: Intrinsic naturalness

$MathML-enabled post (click for more details).$

Ideally, I’d like “elementary” to mean “expressible in first-order logic.”

Okay, I see.

In an (∞,1)-category we may need to be more lenient with our notion of “elementary,”

How about: expressible in first order $(\infty,1)$ -logic ?!

Posted by: Urs Schreiber on February 25, 2010 8:33 PM | Permalink | Reply to this

Re: Intrinsic naturalness

$MathML-enabled post (click for more details).$

How about: expressible in first order $(\infty,1)$ -logic ?!

Hmm, well, maybe. Is (the internal logic of) an ambient $(\infty,1)$ -category the natural context in which to define and study an $(\infty,1)$ -category? Maybe we really want $(\infty,2)$ -logic!

Actually, I think the problem with finitarity arises in $(\infty,1)$ -logic just as much, I guess it’s basically the same problem. So in any case I’m not sure that really answers the question, rather than pushes it to a different level.

Posted by: Mike Shulman on February 25, 2010 10:10 PM | Permalink | Reply to this

Re: Intrinsic naturalness

$MathML-enabled post (click for more details).$

Is (the internal logic of) an ambient (∞,1)-category the natural context in which to define and study an (∞,1)-category? Maybe we really want (∞,2)-logic!

Just by analogy with the ordinary case, where we study 1-toposes using 1-logic, I would have thought the counting continues in this pattern.

Actually, I think the problem with finitarity arises in (∞,1)-logic just as much, I guess it’s basically the same problem.

All right, then I have to take the discussion back to what you said previously, because maybe I am not following now.

Above you wrote:

The property of being well-pointed for a 1-topos is also finitary first-order,

Could you just spell that out in more detail for me? What exactly is finitary here?

Posted by: Urs Schreiber on February 26, 2010 12:13 AM | Permalink | Reply to this

Re: Intrinsic naturalness

$MathML-enabled post (click for more details).$

Just by analogy with the ordinary case, where we study 1-toposes using 1-logic, I would have thought the counting continues in this pattern.

Well, I think that we only study 1-toposes using 1-logic insofar as we study everything using 1-logic. It’s more natural, I think, to study 1-toposes (and any sort of 1-category) using 2-logic. By which I mean, we would formulate the theory of a topos in the sort of logic appropriate to the internal logic of a 2-category. The internal logic of a 1-topos is of course still 1-logic.

What exactly is finitary here?

A well-pointed elementary 1-topos is a model of a first-order theory whose axioms are finite strings of symbols. The types of the theory are “objects” and “arrows.” For instance, the well-pointedness requirement that $1$ is a generator can be stated as “for any objects $X$ and $Y$ and any morphisms $f,g\colon X\to Y$ , if for any morphism $x\colon 1\to X$ we have $f x = g x$ , then $f=g$ .” One could translate that into formal logical symbols like $\forall X,Y:obj, \forall f,g:X\to Y, (\forall x:1\to X, f x = g x) \Rightarrow f =g.$ You can talk about “infinite things” inside this theory, like a natural numbers object, just like in the finitary first-order theory ZFC you can talk about “infinite things”. But there is no infinity happening in the metatheory.

However, if the metatheory is strong enough to include some infinity, then one can talk about logic that includes infinite formulas. This happens with geometric logic, for instance, where you can have formulas that are disjunctions of an infinite set of other formulas. This sort of logic is useful, but many people regard it as not “foundational” since how many formulas there are depends on what sort of axioms relating to infinity you assume in the metatheory.

Finally, one can also consider properties such as completeness and cocompleteness, which cannot even be stated as infinite formulas but belong mostly to the metatheory, since they involve quantifying over “all functors into C” which is not a first-order formula, even an infinite one.

Does that help?

Posted by: Mike Shulman on February 26, 2010 1:48 AM | Permalink | PGP Sig | Reply to this

Re: Intrinsic naturalness

$MathML-enabled post (click for more details).$

You can talk about “infinite things” inside this theory, like a natural numbers object, just like in the finitary first-order theory ZFC you can talk about “infinite things”. But there is no infinity happening in the metatheory.

[…]

Does that help?

Yes, now I understand what is meant by “finitary” here, thanks.

Well, I think that we only study 1-toposes using 1-logic insofar as we study everything using 1-logic. It’s more natural, I think, to study 1-toposes (and any sort of 1-category) using 2-logic.

Could you give me a (simple) example for some 2-logical reasoning applied to a 1-topos? Or any motivating example for 2-logical reasoning? I have the basic idea, but am still lacking a good feeling for what to do with this.

Posted by: Urs Schreiber on February 27, 2010 9:43 AM | Permalink | Reply to this

Re: Intrinsic naturalness

$MathML-enabled post (click for more details).$

Could you give me a (simple) example for some 2-logical reasoning applied to a 1-topos? Or any motivating example for 2-logical reasoning? I have the basic idea, but am still lacking a good feeling for what to do with this.

Reasoning about categories with 2-logic doesn’t really look much different from reasoning about them with 1-logic. The main differences are that

In 2-logic, categories are basic objects, rather than being defined out of sets,
2-logic prevents you from saying evil things, and
2-logic can be interpreted internally in 2-categories.

Here’s a motivating example that I like. Let $S$ be a 1-topos which we want to use as a “mathematical universe”. In this universe we need a notion of “large category,” and the natural thing to choose is a fibered/indexed category over $S$ (which is usually a stack). The point being that if the objects of $S$ are the “sets” then we need a large category $A$ to come with, not just a notion of objects, but a notion of “ $X$ -indexed family of objects of $A$ ” for any $X\in S$ . For example, any geometric morphism $p\colon E\to S$ gives $E$ the structure of an $S$ -indexed 1-topos, in which the category of $X$ -indexed families of objects of $E$ is the slice category $E/p^*X$ . The geometric morphisms $E\to S$ which are “bounded” then play the role of “Grothendieck toposes in the universe of mathematics based on $S$ .”

Now if you want to do category theory with such fibrations or stacks, you need to re-prove all of the basic facts from category theory, but using indexed families and the right “fibrational” notions of limits, colimits, functors, etc. This is tedious work and takes up a lot of Part B of the Elephant. However, what you can do instead is notice that all the usual proofs in category theory can be written in 2-logic (and are constructive), so they can therefore be interpreted internally in the 2-category of fibrations (or stacks) on $S$ . When you translate out what these internal interpretations are saying, they always come out to be equivalent to the more complicated statement written in terms of fibrations, but the internal 2-logic of the 2-category of fibrations does all the work of fibrationalizing things automatically for you. This is just like how the internal logic of a 1-topos makes proving a lot of things much easier, but applied one level up.

Posted by: Mike Shulman on March 1, 2010 10:42 PM | Permalink | PGP Sig | Reply to this

Re: Intrinsic naturalness

$MathML-enabled post (click for more details).$

And to relate that to the previous comment, an important (perhaps essential) fact is that for any 1-topos $S$ (not necessarily well-pointed), the 2-category $St(S)$ of stacks on $S$ contains a particular object $\mathbb{S}$ , the codomain fibration of $S$ , such that the statement “ $\mathbb{S}$ is a well-pointed elementary 1-topos” is true in the internal 2-logic of $St(S)$ . The internal 2-logic of $St(S)$ , particularly when restricted to $\mathbb{S}$ and other stacks “definable” from it, is what I call the stack semantics of $S$ . It’s a natural generalization of the internal logic of $S$ to a language that can deal with unbounded quantifiers (i.e. quantifiers over all sets) and with large categories.

Posted by: Mike Shulman on March 1, 2010 10:52 PM | Permalink | PGP Sig | Reply to this

Re: Intrinsic naturalness

Ah, some interesting slides here of yours, Mike, on stack semantics and other 2-logical things.

I think I once asked you whether we see 2-logic in everyday life, and I seem to recall you saying “No”. I can see that this would make sense if

…just as existing mathematical arguments involving sets can be formalized in the internal logic of a 1-topos, existing mathematical arguments involving categories should be formalizable in the internal logic of a 2-topos.

Showing philosophical colleagues examples of categories outside of mathematics is not so easy, as discussed with regards to the ‘second strand’ here. One tends to resort to the groupoid of the road network or to a poset.

Posted by: David Corfield on March 2, 2010 2:33 PM | Permalink | Reply to this

Re: Intrinsic naturalness

$MathML-enabled post (click for more details).$

I think what I’m saying now about 2-logic (as opposed to what I might have said in the past, or what I might say in the future) is that most proofs in category theory, though they are usually phrased in 1-logic by defining categories as a structure built out of sets, can be more naturally phrased in 2-logic where categories are a basic notion. And that this rephrasing is a valuable thing to do, because there exist semantics for 2-logic in which the categories aren’t built out of the sets in some semantics for 1-logic (such as the stack semantics over a 1-topos, or more generally the internal logic of a 2-category). I also find this rephrasing conceptually helpful; for instance, I think it clarifies the appropriate role (or lack thereof) for the axiom of choice in category theory.

So, right now I think my answer to the question of whether 2-logic appears in everyday life is that it depends on the extent to which we see categories in everyday life. And as you said, those are hard to find.

One tends to resort to the groupoid of the road network or to a poset.

Well, 2-logic has natural fragments called (2,1)-logic and (1,2)-logic, in which groupoids and posets, respectively, are the basic notions. And of course (2,1)-logic generalizes to $(\infty,1)$ -logic as we were discussing in the other thread. Do you think those would find more natural examples in everyday life?

Posted by: Mike Shulman on March 2, 2010 7:16 PM | Permalink | PGP Sig | Reply to this

Re: Intrinsic naturalness

Do you think those would find more natural examples in everyday life?

I’d imagine so. There must be reasoning we carry out on the groupoid of routes between towns, or on the poset of ancestorship which could be well captured.

Posted by: David Corfield on March 2, 2010 9:05 PM | Permalink | Reply to this

Re: Intrinsic naturalness

$MathML-enabled post (click for more details).$

the extent to which we see categories in everyday life. And as you said, those are hard to find.

Hey, you should read the works of a famous guy called John Baez, who has been busy for years emphasizing how categories are everywhere in physical and computer processes. So as you type, there is a wealth of category theory going on.

Groupoid theory is the science of static spaces . Category theory is the science of processes – of directed spaces.

As you know. But I needed to intervene here with David and you happily agreeing on the $n$ -category Café that there are no categories around us.

Maybe categories are a bit like air: they are so ubiquitous that it becomes hard to see them.

Posted by: Urs Schreiber on March 2, 2010 7:35 PM | Permalink | Reply to this

Re: Intrinsic naturalness

$MathML-enabled post (click for more details).$

Well yes, but there’s no very clear manifestation of categories in our ordinary language. There are all sorts of higher-dimensional spaces around, but they were so obscure to our forefathers that Poincaré needed to justify the study of them to fellow mathematicians as late as the 1890s. They hadn’t, and still haven’t, made their presence felt in everyday language.

The founders of analytic philosophy were very excited by the idea that their new logic was going to resolve a heap of philosophical problems. E.g., what should I make of the statement “The king of France is bald.” as uttered in 1900? Many took it to be meaningless, but Russell with his new logic could translate it as

$\exists x. (x is the king of France \& x is bald).$

Then the statement can be seen to be meaningful, but false.

I’m sure many of my colleagues would become much more interested in category theory if you could show them an ordinary sentence and render it in 2-logic in an illuminating way.

Posted by: David Corfield on March 2, 2010 8:07 PM | Permalink | Reply to this

Re: Intrinsic naturalness

$MathML-enabled post (click for more details).$

but there’s no very clear manifestation of categories in our ordinary language.

Hm, I am a bit lost now. This sounds like saying to me that there is no clear manifestation of processes in our ordinary language.

Lawvere with his Categorical dynamics , Bob Coecke with his monoidal category quantum mechanics, the people who invented directed spaces to describe concurrent programming, they are all saying: when we talk about physical and computational processes, we are talking about categories.

Posted by: Urs Schreiber on March 2, 2010 9:18 PM | Permalink | Reply to this

Re: Intrinsic naturalness

$MathML-enabled post (click for more details).$

I mean explicit manifestations of categories and 2-logic in ordinary language are lacking, at least it seems so.

Imagine we are living in 1800 and you say to me “Every Mozart piano concerto has been listened to by some person.” I reply “So someone has heard every Mozart piano concerto.” You say “No, that doesn’t follow.” We can probably resolve our differences, but not by the official logic of the day – the Aristotelian syllogism – considered by some to be the final form of logic.

That logic has already, from the earliest times, proceeded upon this sure path [of a science] is evidenced by the fact that since Aristotle it has not required to retrace a single step, unless, indeed, we care to count as improvements the removal of certain needless subtleties or the clearer exposition of its recognised teaching, features which concern the elegance rather than the certainty of the science. It is remarkable also that to the present day this logic has not been able to advance a single step, and is thus to all appearance a closed and completed body of doctrine. If some of the moderns have thought to enlarge it by introducing psychological chapters on the different faculties of knowledge (imagination, wit, etc.), metaphysical chapters on the origin of knowledge or on the different kinds of certainty according to difference in the objects (idealism, scepticism, etc.), or anthropological chapters on prejudices, their causes and remedies, this could only arise from their ignorance of the peculiar nature of logical science. We do not enlarge but disfigure sciences, if we allow them to trespass upon one another’s territory. The sphere of logic is quite precisely delimited; its sole concern is to give an exhaustive exposition and a strict proof of the formal rules of all thought, whether it be a priori or empirical, whatever be its origin or its object, and whatever hindrances, accidental or natural, it may encounter in our minds. (Critique of Pure Reason, Bviii)

From today’s perspective we see our 1800 counterparts as desperately needing quantifiers, and all the apparatus of 1-logic, to disentangle $\exists \forall$ and $\forall \exists$ . My wonder then is whether anyone in 2200 will look back at ordinary people in the street (not computer scientists or mathematical physicists) in 2000 and say “What they needed was 2-logic.”

Posted by: David Corfield on March 3, 2010 9:59 AM | Permalink | Reply to this

Re: Intrinsic naturalness

[Will] anyone in 2200 will look back at ordinary people in the street in 2000 and say “What they needed was 2-logic.”[?]

As I’m always reminded when the Amtrak conductor cautions that “all doors will not open”, ordinary people still need 1-logic.

Posted by: John Armstrong on March 3, 2010 4:25 PM | Permalink | Reply to this

Re: Intrinsic naturalness

$MathML-enabled post (click for more details).$

David, let me ask you this: have philosophers fully assimilated the use of dependent types in their study of logic and ordinary language? If so, then I don’t think there’s much new that (what I mean by) 2-logic has to offer them, at least not more than category theory in general has to offer, since what I mean by 2-logic is really just a particular sort of dependent type theory which is adapted to describing categories and excluding evil statements.

There may, of course, be other more exciting kinds of “2-logic” waiting to be discovered.

Posted by: Mike Shulman on March 3, 2010 6:10 AM | Permalink | PGP Sig | Reply to this

Re: Intrinsic naturalness

$MathML-enabled post (click for more details).$

Mike asks

have philosophers fully assimilated the use of dependent types in their study of logic and ordinary language?

I would say hardly at all, aside from a small amount of interest in Martin-Löf type theory. Don’t get misled by Steve Awodey working in a philosophy department. I once spoke with him about the gap between mathematical and philosophical logic and neither of us could see the chasm closing soon. Princeton philosopher Hans Halvorson’s sabbatical with Ieke Moerdijk at Utrecht may help category theoretic logic gain recognition.

Looking at the Stanford Encyclopedia, I see that dependent types gets a brief mention in the page on type theory, but note this is written by a computer scientist Thierry Coquand.

Just to get things straight, dependent type theory only counts as 2-logic for you if it is of a particular sort? Plain dependent type theory is part of 1-logic, as shown by the claim

The internal logic of locally cartesian closed categories is dependent type theory?

Dependent type theory seems to me to be a good case of something beyond the ordinary (typed) predicate calculus which captures everyday language. Think of variants:

Every month there’s a day on which some team has all its players fit.
In all teams there is a player who is fit on every day of some month.

$\forall m \in M. \exists d \in D(m). \exists t \in T. \forall p \in P(t). F(p, d)$ .
$\forall t \in T. \exists p \in P(t). \exists m \in M. \forall d \in D(m). F(p, d)$ .

It’s horribly clunky to have to write for the first of these

$\forall m \in M. \exists d \in D. \exists t \in T. \forall p \in P. (Belongs(d, m) \& Belongs(p, t) \& F(p, d))$ ,

with two different Belongs relations.

Posted by: David Corfield on March 3, 2010 11:34 AM | Permalink | Reply to this

Re: Intrinsic naturalness

$MathML-enabled post (click for more details).$

I am thinking of logic and dependent types as a general framework which includes both various kinds of “1-logic” (i.e. the natural internal logics of 1-categories) and “2-logic” (the natural internal logics of 2-categories) as special cases. Many people are familiar only with the parts of logic/type-theory that are some kind of 1-logic, but in addition to 2-logic and even $(\infty,1)$ -logic, there are other kinds of logic. Most of the internal languages of categories are characterized by “extensionality” and “exactness” properties, but in Martin-Löf’s original intensional type theory neither of these are assumed. And intensional dependent type theory is not just some wacky constructivist thing; one can argue that it’s the most natural logical framework in which to prove correctness of computer programs.

Posted by: Mike Shulman on March 3, 2010 3:33 PM | Permalink | Reply to this

Re: Intrinsic naturalness

$MathML-enabled post (click for more details).$

I think maybe we mean different things by “everyday life.” I don’t think I talk a whole lot about such “processes” in my everyday life. I think of physical and computer processes as part of a mathematical description of something, and I certainly agree that categories are everywhere in the mathematical descriptions of things.

Posted by: Mike Shulman on March 3, 2010 3:24 PM | Permalink | Reply to this

Re: Intrinsic naturalness

$MathML-enabled post (click for more details).$

Mike wrote:

I think maybe we mean different things by “everyday life”.

If everyday life means small talk interactions, then I don’t need the natural numbers either, I don’t think I’ll ever need a number bigger than, say, $10^100$ .

But David seems to mean the “everyday live” of philosophers, and “ordinary people” = “philosophers in academia”. (@David: Sorry for putting words in your mouth, I hope they are at least the right ones :-)

An author who tries to make the point that categories are useful in software engineering would be José L. Fiadeiro (amazon) (but I’m not convinced, since the examples in his book do not really relate to software engineering). One aspect of software engineering is to model real life processes like the interaction of people in an organization in order to get something done, and translate a part of these processes into a software system. So, if categories are of use here, you could say that they are used to express “everyday life processess” from the viewpoint of the responsible persons. Judging from the organizations that I know, that is very much not “everyday life” from the viewpoint of most other people, but I guess you know what I mean.

Posted by: Tim van Beek on March 3, 2010 5:54 PM | Permalink | Reply to this

Re: Intrinsic naturalness

$MathML-enabled post (click for more details).$

I think maybe we mean different things by “everyday life.” I don’t think I talk a whole lot about such “processes” in my everyday life.

I may be missing what you are actually after when you write

[..] whether 2-logic appears in everyday life […] depends on the extent to which we see categories in everyday life.

I think we see categories in our everyday perception the way we see clouds in the sky and pebbles on the floor: we may not talk much about this over dinner, but it is still part of our immediate experience.

For instance there is a category whose objects are points in spacetimes, whose morphisms are timelike curves in spacetime, and composition is concatenation of curves.

This category is the grand category of processes in which your and my life are sub-processes. This is everyday experience.

And people talk about this process. Of course they won’t use the language of categories and 2-logic. It is not their job to do so. It is the job of the 2-logician to look at how people talk about this and then point out how this is really talking about categories.

You said one of three main difference between 1-logic and 2-logic is

2-logic prevents you from saying evil things,

There are people who spend lots of energy on trying to make non-evil statements about that grand category of spacetime and timelike paths that I mentioned. They call it “coordinate free language”, but it means precisely this: non-evil statements about a category.

They are writing thick books about this and are talking about it on conferences. Many of them have never heard of categories and might seriously benefit from a bit of 2-logic. It should be the job of the 2-logician to see what they are currently doing and recognize the good application of 2-logic, in as far as there is one.

But maybe I am missing your whole point and maybe we should not continue this discussion but come back to something more technical.

I would like to get back to understanding $(\infty,1)$ -type theory and logic. I want to get a better grasp of what it means that an $(\infty,1)$ -topos has essentially ordinary logic but non-standard type theory.

Can you give me an example statement or construction inside an $(\infty,1)$ -topos that would illuminate this interplay between these two aspects, and the difference to the ordinary situation?

I just need to get my foot on the ground here somewhere. We keep talking about $(\infty,1)$ -logic and so on, and still don’t feel I could explain to any bystander properly what it’s all about.

Posted by: Urs Schreiber on March 3, 2010 6:23 PM | Permalink | Reply to this

Re: Intrinsic naturalness

$MathML-enabled post (click for more details).$

I really don’t think that 2-logic has much to offer to everyday language and to philosophers that isn’t just coming from dependent types, which are more general than 2-logic anyway. The only things that make 2-logic special, among dependent type theories, are that everything is a category, that you can’t say evil things, and that it can be interpreted in lots of other 2-categories. Of these, the first is false in everyday language, and I’m skeptical of the usefulness of the other two. In particular:

There are people who spend lots of energy on trying to make non-evil statements about that grand category of spacetime and timelike paths that I mentioned. They call it “coordinate free language”, but it means precisely this: non-evil statements about a category.

I don’t think that’s quite the same thing. The way you described that category, there are no nonidentity isomorphisms, so evil is irrelevant. (In fact, many people would argue that there are no nonidentity endomorphisms.) Coordinate-free language certainly has something to do with isomorphism-invariance—in particular, it seems to me like Einstein’s “hole paradox” is just a misunderstanding of what it means for something to be determined up to isomorphism—but I don’t think it’s that simple.

I would like to get back to understanding (∞,1)-type theory and logic.

Great, let’s do it in the other thread.

Posted by: Mike Shulman on March 5, 2010 9:22 PM | Permalink | PGP Sig | Reply to this

Re: Intrinsic naturalness

$MathML-enabled post (click for more details).$

The way you described that category, there are no nonidentity isomorphisms, so evil is irrelevant.

This category is introduced in the physics textbooks in the equivalent incarnation of its Cech cover replacement, for the maximal atlas: objects are spacetime points in a fixed coordinate patch, morphisms are generated from paths and changes of coordinate system.

I really don’t think that 2-logic has much to offer to everyday language

If you say so, I believe you, I don’t know about 2-logic. What I started objecting to and still am is the assertion that peple don’t perceive categories as such much in their everyday life, as in David’s statement

Showing philosophical colleagues examples of categories outside of mathematics is not so easy, […]. One tends to resort to the groupoid of the road network or to a poset.

But okay, we have exchanged the arguments, let’s move on to something more concrete.

Posted by: Urs Schreiber on March 6, 2010 9:41 AM | Permalink | Reply to this

Re: Intrinsic naturalness

$MathML-enabled post (click for more details).$

I’m not sure I see how my statement relates to the assertion “people don’t perceive categories as such much in their everyday life” – they could perfectly well perceive categories without it being easy to convince philosophers of their presence. Indeed, this had better be the case if you are right, because as a matter of fact convincing philosophers of the importance of categories has been a hard task.

There’s something group theoretic going on in our perceptual apparatus, allowing us to recognise objects as retaining their size even as their image shrinks on our retina, but I don’t see that group theory manifests itself in our everyday vision speech.

Anyway let’s move on. Looking back at older 2-logic discussions, I see John making the Baez-Dolan point that:

The great thing about 2-logic is that it can be completely described in one sentence. A theory in 2-logic has a concrete 2-groupoid of models, and any concrete 2-groupoid is the 2-groupoid of models of an essentially unique theory in 2-logic.

This seems a different approach to Mike’s idea of 2-logic as the logic internal to 2-categories. You can see why we thought modal logic ought to come into the mix. Given that models for a modal theory assign a space/groupoid for the collection of worlds and then something sheaf-like to capture the non-modal part, the collection of models would seem to be a 2-category.

The Baez-Dolan approach links up with their property, structure, stuff story. Think of what the highest level syntax gets assigned.

In 0-logic (i.e., propositional), the propositional letters are assigned truth values ( $-1$ -groupoids).
In 1-logic (i.e, typed predicate), the types are assigned sets (0-groupoids), the stuff. Functions and relations provide the structure, and axioms the properties.
In 2-logic, the metatypes need to be assigned groupoids, the 2-stuff.

Posted by: David Corfield on March 7, 2010 9:10 AM | Permalink | Reply to this

Re: Intrinsic naturalness

“I don’t see that group theory manifests itself in our everyday vision speech.”

But visual symmetry gives us one of the easiest examples of groups.

Posted by: jim stasheff on March 7, 2010 1:51 PM | Permalink | Reply to this

Re: Intrinsic naturalness

$MathML-enabled post (click for more details).$

This seems a different approach to Mike’s idea of 2-logic as the logic internal to 2-categories.

Yes, for a while I tried to avoid saying “2-logic” and said “2-categorical logic” instead, to make clear exactly what I was talking about and try not to step on anyone else’s toes. But after a while I seem to have given that up.

However, I don’t think what John is talking about there is that different from the 2-categorical logic I’m working on. Later on he refers to the appendix of Baez-Shulman, which helped in the genesis of my current ideas.

In 2-categorical logic, just like in 1-categorical logic, a “theory” can have models in many different (2-)categories. If $T$ is an $n$ -theory and $C$ is an $n$ -category with sufficient structure to interpret $T$ , then there is an $n$ -category $Mod(T,C)$ of models of $T$ in $C$ . Moreover, the construction $Mod(-,-)$ is the common homset of an adjunction between theories and (structured) categories, so we have

$Theories( T, Th(C)) \simeq Mod(T,C) \simeq Categories(Syn(T), C)$

where $Th(C)$ is the canonical “theory of the category $C$ ,” which is sometimes called its “internal logic,” and $Syn(T)$ is the “syntactic category” of the theory $T$ .

In particular, any $n$ -categorical theory $T$ has an $n$ -category $Mod(T,(n-1) Cat)$ of models in the prototypical $n$ -category $(n-1) Cat$ . That is, any 1-categorical theory has a 1-category of $Set$ -models, and any 2-categorical theory has a 2-category of $Cat$ -models. You can of course forget the noninvertible morphisms in both cases to get an $n$ -groupoid, and I think that’s what John is referring to by

A theory in 2-logic has a concrete 2-groupoid of models.

However, I don’t understand what he means by

any concrete 2-groupoid is the 2-groupoid of models of an essentially unique theory in 2-logic.

I don’t even understand this in the 1-case. I don’t think I know any definition of “theory” for which it’s non-tautologously true that every concrete 1-groupoid is the 1-groupoid of models of an essentially unique theory.

Posted by: Mike Shulman on March 8, 2010 3:18 AM | Permalink | PGP Sig | Reply to this

Re: Intrinsic naturalness

I think the 2-groupoid of models idea is a categorification of things contained in Todd’s posts here and here.

Posted by: David Corfield on March 9, 2010 10:46 PM | Permalink | Reply to this

Re: Intrinsic naturalness

$MathML-enabled post (click for more details).$

Thanks for the links, David, that helps. At those posts Todd describes how for some set $X$ , there is an equivalence between

subgroups of $Aut(X)$ , and
subtheories of $P(X^*)$ .

Here $P(X^*)$ is “the complete theory of the set $X$ ,” including all the operations on $X$ of all different arities. Todd describes it in the hybrid style of Lawvere theories, but it could be given a truly syntactic presentation as well. So I gather that perhaps a subgroup of $Aut(X)$ is what is meant by a “concrete group,” with a “concrete groupoid” being a multi-object generalization of this? I guess giving a set $X$ and a subgroup of $Aut(X)$ is the same as giving a group together with a faithful functor to $Set$ , which is the usual definition of “concrete category.”

The notion of “theory” here is, however, I think, not the same as what I would use that word for. At those posts, anyway, Todd is mostly talking about “complete” theories, whereas the theories I find interesting, like the theory of groups or the 2-theory of monoidal categories, are almost never complete. Perhaps the incomplete theories are where the groups become groupoids (there being then multiple models)? But the theories of groups and monoidal categories actually have proper-class many models, so the groupoids should be large — but I would expect a theory to have only a set of sorts and operations, so I don’t see how every large concrete groupoid could be the groupoid of models of some theory. Maybe Todd or John or someone else can help me make the connection between those posts and John’s statement?

(I’m also interested in more general kinds of theories, like theories in geometric logic or in constructive first-order logic, for which there can be inequivalent theories that have exactly the same models in $Set$ . In particular, there are many different, consistent, theories which have no models in $Set$ at all.)

(I’m also worried about how the correspondence works in 2 dimensions even in the classical case, since the presentation of theories used in those posts seems to lean rather heavily on relations, whereas in 2 dimensions most functors are not recoverable from their graph as a relation—you need to use their graph as a profunctor.)

Posted by: Mike Shulman on March 12, 2010 3:03 AM | Permalink | Reply to this

Re: Intrinsic naturalness

I was trying to push on to the 2 dimensional case here and here. Tim Porter had a go here.

If only we could get you, John, Todd and Jim together to talk about 2-logic, I’m sure something good would emerge.

Posted by: David Corfield on March 12, 2010 8:50 AM | Permalink | Reply to this

Re: Intrinsic naturalness

$MathML-enabled post (click for more details).$

Those approaches seem to be along the lines of replacing truth values with sets. My feeling is that in 2-dimensional logic, we still need ordinary truth values, in addition to sets and categories. I wrote a bit about my feelings here. But there is surely a connection waiting to be explored.

If only we could get you, John, Todd and Jim together to talk about 2-logic, I’m sure something good would emerge.

Yes! Unfortunately I wasn’t around and thinking about this stuff yet back when you all were. John, Todd, Jim, are any of you listening?

Posted by: Mike Shulman on March 12, 2010 5:39 PM | Permalink | Reply to this

Re: Intrinsic naturalness

$MathML-enabled post (click for more details).$

We now define a quasi-functor $f : X \to Y$ to be an equivalence if map is an equivalence in $Set^{\Delta^{op}}$ regarded as a 2-category.

Yes. Maybe just for emphasis, some comments:

The full $sSet$ -subcategory of $sSet$ on those simplicial sets that are quasi-categories is a quasi-category enriched category: the $(\infty,2)$ -category of quasi-categories. This may be presented by the $sSet_{Joyal}$ -enriched model category $sSet_{Joyal}$ .

Two qasi-categories are equivalent if they are equivalent as objects in that $(\infty,2)$ -category. As you say, one may determine this by applying the homotopy-category functor hom-wise and checking equivalence in the resulting $2$ -category of quasi-categories.

Alternatively, to determine this, it is sufficient to first look at the maximal $(\infty,1)$ -category inside $(\infty,1)Cat$ obtained by retaining in each hom- $(\infty,1)$ -category only the maximal $(\infty,0)$ -category.

The result is the $(\infty,1)$ -category of $(\infty,1)$ -categories (HTT, def. 3.0.0.1). One can check equivalence here by forming the homotopy category of this $(\infty,1)$ -category by applying the functor $\pi_0$ hom-wise. Two quasi-categories are equivalent if they are isomorphic in the resulting category.

Equivalently, this $(\infty,1)$ -category of $(\infty,1)$ -categories is presented by Lurie’s “Cartesian model structure” (prop 3.1.3.7; with $S = pt$ ). As plain model categories (i.e. without the enrichment) this is equivalent to the Joyal model structure (theorem 3.1.5.1 (A0)) but the Cartesian model structure is indeed $sSet_{Quillen}$ -enriched instead of $sSet_{Joyal}$ -enriched (and has other nice properties, such as supporting a presentation of the $\infty$ -Grothendieck construction). So two quasi-categories are equivalent if they are equivalent in the corresponding homotopy categories of these model categories, which are the same as the result of applying $\pi_0$ hom-wise to the $(\infty,1)$ -category of $(\infty,1)$ -categories.

Posted by: Urs Schreiber on February 14, 2010 2:05 PM | Permalink | Reply to this

Re: Intrinsic naturalness

$MathML-enabled post (click for more details).$

I have expanded the entry

$(\infty,1)$ -category of $(\infty,1)$ -categories

a bit more.

Posted by: Urs Schreiber on February 15, 2010 12:32 PM | Permalink | Reply to this

Re: Intrinsic naturalness

$MathML-enabled post (click for more details).$

Are quasicategories really the “instrinsic” notion of $(\infty,1)$ -category? I don’t really see what justifies this claim. Sure, quasicategories may be a convenient notion of $(\infty,1)$ -category. And at this date, it is certainly the best developed version. But I find it hard to see why it is the “instrinsically” correct notion. (For one thing, as far as I know there’s as yet no easy generalization of quasi-category to $(\infty,n)$ -categories; at best there’s Verity’s model for $(\infty,2)$ .)

I kind of doubt that there is an “intrinsic” notion to be found. There are simply various models, which each have their own advantages and disadvantanges. The collection of all such models, together with the ways of passing between them, is all there is. If there’s a way to single out a best version, I don’t see it.

For instance, we have topological spaces and simplicial sets. These are both models for the homotopy theory of spaces (or, if you like, the $(\infty,1)$ -category of $(\infty,0)$ -groupoids.) But I don’t see a reason to pick one over the other as intrinsic. Likewise, among the various models for categories of spectra, I don’t see an intrinsic model. Of course, if you step back, you get to say that these are just certain $(\infty,1)$ -categories, which are each characterized by a suitable universal property. But saying this involves a deliberate act of abstraction away from particular models.

Posted by: Charles Rezk on February 13, 2010 5:46 PM | Permalink | Reply to this

Re: Intrinsic naturalness

$MathML-enabled post (click for more details).$

These developments are a strong pragmatic argument for the central importance of quasi-categories, but I was rather interested in hearing more from Urs as to a conceptual reason for why quasi-categories in particular ought to be considered the intrinsic notion.

I’ll try to say some things in an attempt to clarify my point of view.

To begin with, let us consider a truncated part of the question on which there should be common and long-time agreement: bicategories:

An intrinsically category-theoretic point of view on bicategories is that these are categories weakly enriched in categories.
It turns out that up to equivalence every bicategory is one even strictly enriched in categories, so that to some extent this is still an intrinsic notion.
Bicategories also have a simplicial nerve and may be characterized entirely by properties of this simplicial nerve. While the simplicial nerve looks different when you write it to paper, form a conceptual point of view it is a pretty obvious repackaging of the same data without, and that’s important, involving any unnecessary choices.

So I regard the first and the third perspective as being intrinsic and category theoretic. And with due care taken, also the second one.

I suppose that in some way or other we agree on this situation? Please let me know.

Because from there on develops my notion of intrinsic category theoretic formulation of $(\infty,1)$ -category:

The right analog of the first point above should be: an $(\infty,1)$ -category is a category weakly enriched in $\infty$ -groupoids. A very natural-looking formulation of what “category weakly enriched in $\infty$ -groupoids” might mean is the notion of a “complete Segal space” or “Segal category”. While unfortunately these technical terms don’t quite reflect the category theoretic relevance that they do have, this I regard as a nicely intrinsic category theoretic definition of $(\infty,1)$ -category.
One may also semi-strictify this and consider strictly $\infty$ -groupoid enriched categories. With due care taken that this involves a little bit of choices this will still be okay.
There is an obvious simplicial nerve of these structures and they are fully characterized by the fact that this nerve is a what Joyal called a quasi-category.

A lot of work has shown that these four notions (complete Segal space, Segal category, Kan-complex enriched category and quasicategory) all form a structure of this kind (for instance the Kan-complex enriched category of quasi-categories is just the subcategory of the $sSet$ -ategory $sSet$ on quasi-categories and taking the maximal Kan complexes in each hom-simplicial set) and are all equivalent as objects of one of these.

So four notions that a-priori look as if they should naturally and intrinsically encode the notion of $(\infty,1)$ -category are all equivalent in all of their four perspectives.

This gives a nice web of self-consistency checks on these notions and reinforces my belief that together this captures the notion of $(\infty,1)$ -category suitably.

That I think of as a concpetual foundation of $(\infty,1)$ -category theory. The pragmatic aspect that you mention serves to reinforce the observation that, indeed, this theoretical construct does what the right notion is expetced to do: it supports a category theory up to coherent $\infty$ -equivalence, in which each and every of the esteemed constructions and theorems from ordinary category theory find their evident counterpart.

This together makes me think: yes, evidently what has been found here is the right intrinsic notion of $(\infty,1)$ -category theory.

That this intrinsic notion itself comes in different incarntions does not worry me any more than it did for the example of bicategories that we started with. On the contrary, with the well-established experience from category theory, which teaches us that many concepts may come to us in very different guises but may still form equivalent categories and hence be equivalent as abstract concepts, precisely this situation is to be expected:

whatever notion of $(\infty,1)$ -category that surfaces should be expected to have incarnations in many different ways that all are equivalent as objects of $(\infty,1)$ -categories in any of the incarnations.

And at this point I am drwing on the important apsect of $(\infty,1)$ -categories which is not present for bicatgories or tricategories: $(\infty,1)$ -categories do reflect on themselves and on everything else in fact in that for determining if any two things are equivalent, it is fully sufficient to know the $(\infty,1)$ -category in which they jointly live. Even if both these things really live even in an $(\infty,n)$ -category: for determuining equivalence the $(\infty,1)$ -category truncation of that is fully sufficient.

Given all this, what I perceive has been established is precisely what one would have hoped can be and would have expected will eventually be established. A web of notions that in a perfectly self-consistent manner reflect on each other and thereby present to me one abstract intrinsic notion, which I refer togeneraly as the notion of $(\infty,1)$ -category.

I find it hard to imagine arguments that would suggest that the notion of $(\infty,1)$ -category should be something not reflected by this web of notions.

What I can easily imagine, though, is that the web of realizations can be increased quite a lot. Certainly it increases when we pas from $(\infty,1)$ to $(\infty,n)$ , whith for instance Charles Rezk’s Theta-spaces joining the picture. I mention these particularly to come to one important point: in Rezk’s realization we see an interplay between globular and simplicial methods: a Theta-space is a structure, roughly, that has a simplicial set of globular $n$ -cells, for each $n$ . If we take the prefix “ $\infty$ -” to be an indication that a simplicial method is being used and that of “ $\omega$ -” that a globular method is being used, then this provides us with the notion of an $\infty-\omega-category$ in a useful sense.

You see whaat I am getting at now. i would for instance be very surprised if the following could not be shown to be true:

Conjecture There is a notion of simplicial nerve of a Trimble $\omega$ -category, and Trimble $\omega$ -categories in which all except possibly the 1-morphisms are invertible are precisely characterized by the fact that their simplicial nerve is a quasi-category.

I do expect that with the required work invested one can construct the Trimble $\omega$ -category of all Trimble $\omega$ -categories with this property, then find the maximal Trimble $\omega$ -category inside this in which all 2-morphisms are invertible and then demonstrate that under simplicial nerve this is the quasi-category of all quasi-categeories.

The same comment applies to Batanin’s definition of $\omega$ -categories. Also I think Verity’s definition will fit nicely into a further extension of the web of interrelated idea. You can see that Lurie hints at that already on the bottom of p. 5 of $(\infty,2)$ -categories and the Goodwillie calculus .

All of these realizations of the notion of $(\infty,1)$ -category I would regard as intrinsic. All of them may be presented by model categories, which are a generators-and-relations presentation more than an intrinsic notion.

Posted by: Urs Schreiber on February 13, 2010 6:05 PM | Permalink | Reply to this

Re: Intrinsic naturalness

$MathML-enabled post (click for more details).$

Urs, thanks for this detailed comment. I actually see nothing wrong with what you are saying here. In particular, so far I see absolutely nothing wrong with the idea that the collection of all (for whatever given notion of) “ $(\infty, 1)$ -categories” ought to form a quasi-category. Given that, I think I also agree (I think I had better agree!) that for any such notion of $(\infty, 1)$ -category worthy of the name, this quasi-category had better be quasi-equivalent to the quasi-category $QCat$ . That’s certainly a necessary condition, because otherwise we are talking about fundamentally different notions! And I think it’s sufficient too; it falls in line with that Baez-Dolan email.

The rest is details as they say… I like the general spirit of optimism expressed in your comment (as applied to Batanin or Trimble-style definitions, for instance). You asked me once in email about a model category structure on the Trimble-things. I didn’t then and don’t now have an answer though.

I hope you weren’t identifying me as one of the pessimists! That said, comparing different notions of higher categories is, generally, hard work – even in the case of 2-dimensional structures!

The use of model category theory does seem to be a wonderful apparatus for making comparisons, and it gives me much food for thought. Being behind the times, I don’t know what is the status of inverstigating model category structures on the more algebraic notions of $(\infty, 1)$ -categories (e.g., the one you’d get by appropriate truncation of Batanin $\infty$ -categories). Can anyone enlighten me?

Posted by: Todd Trimble on February 13, 2010 10:50 PM | Permalink | Reply to this

Re: Intrinsic naturalness

$MathML-enabled post (click for more details).$

Urs, thanks for this detailed comment. I actually see nothing wrong with what you are saying here.

Okay, thanks.

I hope you weren’t identifying me as one of the pessimists!

Do you mean pessimists or sceptics? If asked I would have said maybe: while it seems clear that in the end algebraic and geometric definitions of higher categories should be two sides of the same coin, my impression is that some people who invested lots of work on the algebraic side see the emphasis of success of the geometric side sometimes implicitly as a threat that the algebraic approach is claimed to be less worthwhile and are sceptical about this. My feeling is that this is unwarranted: the algebraic and the geometric definitions will in the end be seen to nicely harmonize and be complementary to each other, both useful for different purposes, both aspects of the same underlying reality.

That said, comparing different notions of higher categories is, generally, hard work – even in the case of 2-dimensional structures!

Yes, indeed.

I don’t know what is the status of investigating model category structures on the more algebraic notions of (∞,1)-categories (e.g., the one you’d get by appropriate truncation of Batanin ∞-categories). Can anyone enlighten me?

All I know in this direction are the Brown-Golasinski model structure on strict globular $\omega$ -groupoids and the Lafont-Metayer-Worytkewicz “folk”-model structure on strict globular $\omega$ -categories. (And it seems that the first is indeed the obvious restriction of the latter.)

In these cases, as probably in all algebraic formulations, all objects are fibrant and the crucial point is to get the cofibrant objects under control. This are the “computad”-type objects, those that are freely generated in some sense.

Posted by: Urs Schreiber on February 14, 2010 2:25 PM | Permalink | Reply to this

Re: Intrinsic naturalness

$MathML-enabled post (click for more details).$

Urs wrote:

If asked I would have said maybe: while it seems clear that in the end algebraic and geometric definitions of higher categories should be two sides of the same coin, my impression is that some people who invested lots of work on the algebraic side see the emphasis of success of the geometric side sometimes implicitly as a threat that the algebraic approach is claimed to be less worthwhile and are sceptical about this. My feeling is that this is unwarranted: the algebraic and the geometric definitions will in the end be seen to nicely harmonize and be complementary to each other, both useful for different purposes, both aspects of the same underlying reality.

That seems to be carefully and cautiously worded sentiment, and I personally believe there’s a good chance that this impression is correct. But then the object of the skepticism as expressed here seems to me different from that of the blog entry: “While parts of the community are storming ahead with the new technology that has become available, in other parts reservation and scepticism towards this activity is being felt. [Directly followed by] Is this really the philosopher’s stone that the search was after? Isn’t its shiny appearance a cheap trick achieved by taking that old pebble called homotopy theory, and polishing it a bit?” I’m still not sure the latter description accurately describes the feelings of unnamed colleagues, and (as one of those who was engaged in private discussions with you) I would certainly reject it if applied to me.

In truth, the more I learn about the world of $(\infty, 1)$ -category theory, the more enchanted I become: tremendous vistas are opening up!

Posted by: Todd Trimble on February 15, 2010 4:57 PM | Permalink | Reply to this

Re: Intrinsic naturalness

$MathML-enabled post (click for more details).$

Urs helpfully remarked

[After I asked]

I don’t know what is the status of investigating model category structures on the more algebraic notions of $(\infty, 1)$ -categories (e.g., the one you’d get by appropriate truncation of Batanin $\infty$ -categories). Can anyone enlighten me?

All I know in this direction are the Brown-Golasinski model structure on strict globular $\omega$ -groupoids and the Lafont-Metayer-Worytkewicz “folk”-model structure on strict globular $\omega$ -categories. (And it seems that the first is indeed the obvious restriction of the latter.)

In these cases, as probably in all algebraic formulations, all objects are fibrant and the crucial point is to get the cofibrant objects under control. This are the “computad”-type objects, those that are freely generated in some sense.

Thanks for this! It gives me much food for thought.

Posted by: Todd Trimble on February 15, 2010 5:14 PM | Permalink | Reply to this

Re: Intrinsic naturalness

$MathML-enabled post (click for more details).$

I’m still not sure the latter description accurately describes the feelings of unnamed colleagues

Yes, it’s a dilemma: me neither! I am not even sure I know who all the unnamed colleagues are. What I wrote in the above entry is, even though based on very long discussion (as you know) maybe more a response to what I manage to deduce these feelings are, than to actual feelings. As I mentioned in a previous comment here.

I still believe that an open expression of these feelings will allow for constructive discussion that is bound to make everybody much happier.

In truth, the more I learn about the world of (∞,1)-category theory, the more enchanted I become: tremendous vistas are opening up!

I am glad about this!

Posted by: Urs Schreiber on February 15, 2010 5:15 PM | Permalink | Reply to this

Re: Intrinsic naturalness

$MathML-enabled post (click for more details).$

Urs, I wonder why you speak here of $(\infty, 1)$ -categories rather than $\infty$ -groupoids? —

And at this point I am drawing on the important aspect of $(\infty,1)$ -categories which is not present for bicategories or tricategories: $(\infty,1)$ -categories do reflect on themselves and on everything else in fact in that for determining if any two things are equivalent, it is fully sufficient to know the $(\infty,1)$ -category in which they jointly live. Even if both these things really live even in an $(\infty,n)$ -category: for determining equivalence the $(\infty,1)$ -category truncation of that is fully sufficient.

As far as I can see, you could replace ‘ $(\infty,1)$ -category’ by ‘ $\infty$ -groupoid’ throughout this passage and it would remain true. And it would seem to make the statement more natural:

for determining if any two things are equivalent, it is fully sufficient to know the $\infty$ -groupoid in which they jointly live.

After all, we’re using the observation that equivalences only ever mention invertible cells. So why apply the observation only in dimensions $\gt 1$ ?

Posted by: Tom Leinster on February 14, 2010 2:52 AM | Permalink | Reply to this

Re: Intrinsic naturalness

$MathML-enabled post (click for more details).$

As far as I can see, you could replace ‘(∞,1)-category’ by ‘∞-groupoid’ throughout this passage

Yes, that’s true, or with “ $(\infty,n)$ -category” for any $n \geq 0$ , the important point being that equivalences are understood in all dimensions.

Posted by: Urs Schreiber on February 14, 2010 9:36 AM | Permalink | Reply to this

Re: Intrinsic naturalness

$MathML-enabled post (click for more details).$

Urs wrote:

If we take the prefix “ $\infty$ -” to be an indication that a simplicial method is being used

That was just something you were doing for the purposes of that sentence, right…?

Posted by: Tom Leinster on February 14, 2010 3:00 AM | Permalink | Reply to this

Re: Intrinsic naturalness

$MathML-enabled post (click for more details).$

That was just something you were doing for the purposes of that sentence, right…?

Yes, that’s right. I was just trying to amplify that adopting the $(\infty,n)$ -perspective does not have to mean that simplicial methods are necessarily singled out over non-simplicial methods. One “simplicially-globular” model is already given by Theta-spaces.

Other globular methods are alread lurking here and there:

for instance one way to write the $\infty$ -groupoid/Kan-complex $Hom_C(x,y)$ of morphisms between objects $x$ and $y$ in a quasi-category $C$ is to let it in degree $n$ be given by those maps $f : \Delta^n \to C$ such that

$f|_{\{0,1,2, \cdots, n\}} = x$
$f|_{\{n+1\}} = y$ .

(For instance section 1.2.2 in HTT).

This picks, in a way, the “globular” simplices between $x$ and $y$ . For instance for $n=2$ such an $f$ is a bigon in $C$ between $x$ and $y$ .

Posted by: Urs Schreiber on February 14, 2010 10:04 AM | Permalink | Reply to this

Re: Intrinsic naturalness

$MathML-enabled post (click for more details).$

I took me so long to type my above reply that now I see Patrick Schultz and Charles Rezk have posted comments that I hadn’t seen yet when I had started typing.

Reading what they write I get the impression that there is essential agreement with what I wrote, so I’ll leave it at that for the moment.

Posted by: Urs Schreiber on February 13, 2010 6:09 PM | Permalink | Reply to this

Re: Intrinsic Naturalness

Very interesting! It might be useful for some to point out that HTT is Lurie’s Higher Topos Theory. It took me some googling to find out. (All those false hits containing http!)

Posted by: Maarten Bergvelt on February 13, 2010 5:46 PM | Permalink | Reply to this

Re: Intrinsic Naturalness

$MathML-enabled post (click for more details).$

It might be useful for some to point out that HTT is Lurie’s Higher Topos Theory.

Okay, I have added the link. I tried to reduce the number of hyperlinks in the entry to a minimum, because past experience with entries where I provided all keywords with hyperlinks have shown that readers follow not a single one of them, once there are too many.

So here I thought I’d provide precisely three links that lead to in-depth discussion of what the entry is about. Any one of them would quickly lead to HTT itself.

But maybe that was too minimalistic. I have added now also in the link to $(\infty,1)$ -category.

Posted by: Urs Schreiber on February 13, 2010 6:19 PM | Permalink | Reply to this

Re: Intrinsic Naturalness

Urs wrote:

> All it remembers is a faint shadow of the simple abstract definition above: the shadow of the suspension ∞-functor becomes known as the shift functor, and what are known as the distinguished triangles that give the notion of triangulated categories

It has been well known since 1960s that triangulated categories were not the right thing to do, e.g. the nonfunctoriality of cones was ever a problem, and that instead one should define a gentler derived category by dividing all complexes by acyclic complexes in a milder way. This has been accomplished by DG-enhancements of Kapranov and Bondal introduced in 1988-1989 with immediate applications to algebraic geometry and with the construction of B. Keller and Drinfel’d of the DG-quotient of DG-categories for which the enhanced derived category as a special case. DG-categories were introduced at the end of 1950-s and they are very simple; the definition at least is simpler to work with than simplicially defined infinity categories. The property of being pretriangulated, hence having a triangulated category as a homology takes a bit effort to understand, as much as does the proof that every stable category has a triangulated “shadow”, reagrdless if we have an earlier stable model category, Segal stable category of French school or stabel quasi-categories of Joyal and of Lurie. It is totally parallel and equally complicated or simple. What I want to emphasise is that in homological algebra the construction has been expected since 1960s and the development inside it has presumably lead to constructions of Kapranov, Bondal, Keller, Drinfel’d and finally more general A-infinity version of Lyubashenko and Manzyuk.

Though from the point of view of functoriality etc. the triangulated category is worse than the stable infinity category, in practice it is often not so. For the most striking example, the most relevant category for mathematical physics, the category of D-branes in B-model, is a special case of a derived category of coherent sheaves on a quasiprojective variety over complex numbers. Recent strong result of Lunts and Orlov has shown that this category has a unique dg-enhancement! Hence for a reasonable variety, the derived category says it all, with philosophical consequences.

I have been at Oberwolfach Arbeitsgemeinschaft few years ago when somebody suggested derived algebraic geometry as a topic of a next AG. In the scope of the discussion it appeared that the guys who suggested the topic did not have strong arguments that the topic is important and I, who also did not know much at the time (even in comparison to now, what is still not much) about the topic, felt compelled to say few arguments which I knew. I was relatively successful and got enough topic sympathy that the topic got overall place 2 among about 10 topics considered. Gerd Faltings looked at us suspiciously and after a short discussion asked “are there any theorems there ?”. Of course he knew that there is much difficult mathematics in derived geometry but expressed his scepticism on weather all this machinery is justified by real hardcore applications, to previously unsolved but recognizable problems, and especially those outside other similar fields of suspicious standing like superstring theory, TQFT and noncommutative geometry.

I feel compelled to push myself and others around me to address these external questions of really new fields of applications, rather than continuing multiplying the formalisms within the field in which the things are already known to work with improvement of clarity and in some specialistic hands also of effectiveness.

Posted by: Zoran Skoda on February 15, 2010 6:06 PM | Permalink | Reply to this

Re: Intrinsic Naturalness

$MathML-enabled post (click for more details).$

Thanks, Zoran, for this nice comment! You should put some of that into the entry

enhanded triangulated category

which seems to be waiting for somebody to brush it up a bit more, given its importance.

In my above entry I wrote:

Cures for this were put forward, such as notably enhanced pretriangulated dg-categories, but their definition certainly does not manifestly make the concept come closer to the natural intrinsic category theoretic definition that the pure higher category theorist would hope for.

As I had tried to indicate, despite the impression the entry might make, I hold no grudge against either model categories or enhanced triangulated categories and in fact typically feel glad to have them around as power tools to help me get around. In the entry I tried to disregard all practical matters and emphasize the plain abstract nonsense, for the sake of it. As far as that goes, stable $(\infty,1)$ -categories are certainly the right abstract notion. But of course that does not mean that in practice we don’t gladly fall back to presenting them with dg-enhanced triangulated categories as soon as there is anything to compute.

But I hold that as soon as we are done with computing and want to figure out what deserves to be computed next, it pays to go back to the abstract picture in order to navigate the global structure of the problem. For instance for understading

what happens under stabilization in a given situation;
or for deciding structural questions as what the right notion of module, derivation and Kähler differentials for a chosen notion of derived site are;
or for studying general questions of geometric $\infty$ -function theory

and other general abstract structural questions like this, it seems crucially important to lift from dg-enhanced triangulated categories to the intrinsic notion of stable $(\infty,1)$ -categories. Once you have figured out what exactly to compute next using this abstract notion, for instance which derived integral transform to compute next, it’s likely convenient to fall back to the models given by dg-categories.

That’s how I see work in mathematics: we always pass back and forth between the perspective of general abstract reasoning that tells us what to do and concrete models that tell us how to do it.

Posted by: Urs Schreiber on February 15, 2010 6:57 PM | Permalink | Reply to this

Re: Intrinsic Naturalness

Urs, you attached notions like more abstract, intrinsic etc. to stable infinity categories as opposed to pretriangulated dg-categories. I do not see what you actually mean. Do you mean

1. that quasicategories are more intrinsic than dg-categories or

2. you like talk infinity without specilizing to which of the conjecturally equivalent approaches to infinity categories you refer to.

If 2. then the only really proved thing is that some approaches lead to Quillen equivalent model strutures. But using Quillen functors (not only equivalences) to change gears and prove various things is the normal toolbox in mathematics since Quillen’s work. Thus I do not see anything new in this, except that we have few new concrete and important Quillen equivalences at our disposal.

If 1. then indeed dg-categories are more alike the simplicially enriched approach with the need for strictification choices after some homotopical constructions; what may be sometimes technically more cumbersome. It is better to work with dg categories with A-infinity functors instead of just dg-functors to improve the flexibility without need to zig-zags. Quasi-categories have their own negative imperfections like the non-uniqueness of fillers. These are all features, not a principal difference.

In my understanding, the stabilization which you mentioned has its analogue in dg-category world (pretriangulatization functor defined in 1990). Finally for the integral transform and business of pushforwards and alike the technology is far more developed in algebraic geometry and in algebraic setups and with another intuitive geometric language which is pretty flexible. It depends on personal background if you want to do next paper in this direction in dg or quasi-category language. I do not see what is intrinsical in such a personal choice. If it is so intrinsic can you then guess an easy strategy to define quasi-category analogue of the Fukaya A-infinity category for a symplectic manifold, to mention the most important example ?

In close future, the difference will remain the difference between algebraic topology and algebraic geometry community. Algebraic topologists will embrace quasicategories as they already talk mainly simplicial, Quillen and homotopy llimits. Geometers will stay closer to dg/A-infty-categories and the computations with twisted complexes, adapted classes and alike, plus the geometric language on top of it (like in Kontsevich’s work). For most mathematicians far from both disciplines (and their main applications like geometric representation theory and very formal aspects of mathematical physics) and in foundations, this will stay for long out of scope anyway. Thus I consider most urgent to widen the frontier of applications. E.g. the very important formalism of perverse sheaves is a typical child of triangulated formalism, is there an advantage to possibly enhance it to a stable infinity version of the theory, to throw an example ?

Posted by: Zoran Skoda on February 15, 2010 10:40 PM | Permalink | Reply to this

Re: Intrinsic Naturalness

$MathML-enabled post (click for more details).$

It’s probably worth mentioning that, over a field of characteristic zero, pre-triangulated dg-categories and stable

(\infty,1)

categories are the same thing. But, you are asking why is the

(\infty,1)

description the “right” way of thinking, particularly for the Fukaya category. From the point of view of the physics, the derived category of coherent sheaves and the Fukaya category arise from topological A- and B-model string theories respectively (this isn’t quite correct, but it’s close enough). But now we have Lurie’s version of the cobordism hypothesis, and we know that these are classified by (roughly) Calabi-Yau

(\infty,1)

categories. (Over a field of characteristic zero, you get Costello’s result which relates this to Calabi-Yau A

_\infty

categories.) So, in that sense, at least the

(\infty,1)

category is the natural object.

Posted by: Aaron Bergman on February 16, 2010 3:05 AM | Permalink | Reply to this

Re: Intrinsic Naturalness

Aaron, I attended a talk by Kontsevich in Paris in June 2004 which was dedicated to the recipe how to get a QFT from Calabi-Yau A-infinity category. While from some SCFT models he obtained the A-infinity more than ten years earlier leading to the homological mirror symmetry and the notion of D-branes preceding the dicovery of their role in physics community, he said that for 10 years he did not know how precisely to invert the process. Costello’s result few months later is more or less equivalent to Kontsevich’s Spring 2004 solution. He said that the recipe while of theoretical importance, iss in his view not expected to be ever practical to perform in any real-world cases. It is of course of importance weather the passage is possible to explicitly perform: It is not clear to me if the decription which Lurie’s theorem provides can make the inverse connection betwee the categorical/homological (in any picture, being it quasicategory or A-infinity flavour) and a SCFT formulation of the “B-model so to speak” practically accessible in contrary to pessimistic Konstevich’s prognosis.

Posted by: Zoran Skoda on February 17, 2010 1:12 PM | Permalink | Reply to this

Re: Intrinsic Naturalness

$MathML-enabled post (click for more details).$

I’m definitely not claiming that these things are going to be easy (or even feasible) to do calculations with, just that they do arise from the physics. And even when physicists sit down to do calculations, we end up with A

_\infty

stuff.

Posted by: Aaron Bergman on February 17, 2010 2:00 PM | Permalink | Reply to this

Re: Intrinsic Naturalness

$MathML-enabled post (click for more details).$

Zoran,

Of course stable $\infty$ -categories and pretriangulated dg or $A_\infty$ categories are simply the same thing, so there’s not much point arguing over the difference. eg there’s no more problem defining Fukaya categories in an $\infty$ -context than an $A_\infty$ one (in other words, it’s equally poorly understood in general!), and likewise for perverse sheaves. I personally like thinking in the dg language, and the $A_\infty$ language is natural in symplectic geometry, but one can easily pass between all the languages.

Where a deep and substantial issue arises is in the unstable theory. Most directly relevant to this discussion, you can ask how to treat the collection of dg/ $A_\infty$ /stable $\infty$ categories, and this is where I think $\infty$ -categorical machinery is undoubtedly the way to go. (In particular there isn’t a truly unstable analogue of dg categories – there are dg categories where you haven’t yet thrown in all the cones you need, but that’s quite a different issue.) For example if you want to define notions of monoidal or symmetric monoidal dg categories, of descent theory or deformation theory, of internal homs and tensor products of dg categories etc, there are clear and simple advantages to $\infty$ categories, and many concrete and geometric applications. For example one can apply this to study categories of perverse sheaves as you ask (in that particular case I can cite my most recent paper with Nadler, though we certainly have only begun to see the impact $\infty$ -categorical ideas will have in the subject - and of course what we do pales in comparison with what Jacob’s doing, the most spectacular of which IMHO is the work on TFT that Aaron cites..)

Posted by: David Ben-Zvi on February 16, 2010 4:12 AM | Permalink | Reply to this

Re: Intrinsic Naturalness

$MathML-enabled post (click for more details).$

Urs, you attached notions like more abstract, intrinsic etc. to stable infinity categories as opposed to pretriangulated dg-categories. I do not see what you actually mean.

Maybe I am expressing myself badly. I would have thought that I am making an evident point that I would have expected is uncontroversial:

saying “an $(\infty,1)$ -category with 0-object for which looping and suspension is an auto-equivalence” is more conceptual than writing out the definition of a pretriangulated dg-category.

Wouldn’t you agree with that?

Posted by: Urs Schreiber on February 16, 2010 10:42 AM | Permalink | Reply to this

Re: Intrinsic Naturalness

$MathML-enabled post (click for more details).$

Urs – I haven’t thought this through carefully but I think you can say pretriangulated dg category much more easily – after all dg category just means category enriched in chain complexes. Then we have a closure operation on these which involves adding cones of morphisms (or if you like, adding twisted complexes), and we are simply looking for the dg categories closed under this.. is this right? something of this kind certainly should be. I think saying the homotopy category is triangulated is fine too if you already accept triangulated categories (which I don’t particularly like) - but it’s probably a lazy way of saying intrinsically the dg analog of the characterization of stable categories. In any case I don’t think the stable $\infty$ -version is simpler, it’s just more naturally a part of a much more flexible language (ie we know what an $\infty$ -category is which isn’t stable).

Posted by: David Ben-Zvi on February 16, 2010 3:16 PM | Permalink | Reply to this

Re: Intrinsic Naturalness

$MathML-enabled post (click for more details).$

David, thanks for your comments. I am bit busy with some other things at the moment that prevent me from putting much energy into your very sensible suggestion that we

[…] can say pretriangulated dg category much more easily – after all dg category just means category enriched in chain complexes. Then we have a closure operation on these which involves adding cones of morphisms (or if you like, adding twisted complexes), and we are simply looking for the dg categories closed under this.. is this right?

but with you and Zoran now entering the discussion here, I feel I need to remind everyone of where this discussion came from:

for whatever it’s worth, the starting point of the entry and discussion here was an attempt to point out to the purely pure category theorist how $(\infty,1)$ -category theory has a nice pure conceptual category theoretic formulation that does not rely on concrete models, but just uses these when convenient.

Assuming that pure category theorist has accepted the abstract notion of $(\infty,1)$ -category itself, there is then hardly anything simpler – for the pure category theorist’s mind, remember! – than the statement:

an $(\infty,1)$ -category with a 0-object such that pullback of the 0-object-inclusion along itself is an equivalence.

I mean, this discussion here was about how nice and abstract the concepts of $(\infty,1)$ -category theory are. There is hardly anything simpler, from this abstract point of view, than the definition of a stable $(\infty,1)$ -category. Here “simple” means “follows immediately and naturally from abstract reasoning”. It doesn’t have to mean: “is easy to unwind in full detail in components”. Instead, the emphasis was on naturality .

Remember the characterization of the abstract category theorists the way somebody used to characterize them, jokingly, as those who say: “Please don’t give me an example, otherwise I won’t understand the definition anymore.” ;-) It’s a different mind-set, where one moves around just in the realm of abstract structures and tries to see nice patterns there.

You see, on the one hand I am talking to these “unnamed colleagues” here who feel that things like dg-categories, twisted complexes, triangulated structures and so on are a sign that concepts have been messed up, telling them how everything can be done very conceptually using just abstract nonsense and no models, and now Zoran and you jump in and shout out how everything is nicely done in terms of concrete models! :-)

That’s fine with me, and I am glad this is being discussed, right now it just makes me feel a bit like a guy in one of these cartoon scenes, with a little angel sitting on his left shoulder and a little devil on the right, whispering opposite truths in oposite ears at the same time.

Posted by: Urs Schreiber on February 16, 2010 3:47 PM | Permalink | Reply to this

Re: Intrinsic Naturalness

$MathML-enabled post (click for more details).$

Urs, as far as the difference between quasicategory and dg-category approach to stability, the difference is a matter of a choice of the definition. In the world of dg or A-infty categories there are several equivalent definitions of stability and if you like you can equally choose a closer looking condition. You can say that the A-infinity category $C$ is stable (Seidel says triangulated) if

1) every morphism in $H^0(C)$ can be completed to an exact triangle (this means that the shift/suspension is defined!)

2) that this shift from 1) is invertible in the weak sense: for every object $X$ there is some $Y$ such that $S(Y)$ is equivalent to $X$ in $H^0(C)$

While for loop objects etc. in quasi-category you need things like homotopy pullbacks you here need exact triangles what is about the same kind of device.

things like dg-categories, twisted complexes, triangulated structures and so on are a sign that concepts have been messed up

How can you put correct concept like dg-category on the same footing like approximate like triangulated catgeory ? Twisted complexes have a role similar to anafunctors and various cocycles between complexes: this is seen most readily in the early papers in gluing complexes in complex geometry (Toledo and others). Now if you often use words like cocycle etc. that does not mean that the formulas are less complicated. And I am sure that it is formulas which scared your friend, not the fact that this is ultimately not any less or more intutive than the anafunctor picture for cocycles. For the Maurer-Cartan kind of equations you can use the geometric language on top as Kontsevich does, and I am sure he knows what are wrong and what are messed up structures here.

Posted by: Zoran Skoda on February 16, 2010 5:15 PM | Permalink | Reply to this

Re: Intrinsic Naturalness

$MathML-enabled post (click for more details).$

I think it’s misleading to talk about a dg category approach to stability – dg categories are all already essentially stable objects - in particular they’re already enriched in spectra. The issue of passing to pretriangulated envelopes is one of cocompletion, not of stability - you’re adding some diagrams that you had forgotten to add, but you’re not stabilizing (or “linearizing”) the category, since it already was so. That’s an enormous difference with the $\infty$ world, in which there’s an honest unstable setting. Of course you can talk about $\infty$ -categories that are not stable but are enriched in spectra, and these would be precise analogs of dg categories that aren’t pretriangulated, but I think that misses Urs’ main points.

Posted by: David Ben-Zvi on February 17, 2010 9:26 PM | Permalink | Reply to this

Re: Intrinsic Naturalness

$MathML-enabled post (click for more details).$

While for loop objects etc. in quasi-category you need things like homotopy pullbacks you here need exact triangles what is about the same kind of device.

Sure, but, see, this thread effectively started with me playing the following game:

Game rules. Say as much as possible about constructions and results in $(\infty,1)$ -category theory using only intrinsic category theoretic languge, including notions such as

limit, colimit, pullback, terminal object, initial object, zero object, adjoint functor, fully faithful functor, left exact functor, presheaf, sheaf, topos, ect.

Just this abstract stuff that exists in any context in which there is a notion of category theory exists.

Since “homotopy pullback” is just “pullback” in this context, this is allowed by the rules of the game. But things like “twisted complex” or “dg-category” are not allowed.

We don’t need to continue playing this game. But this was the game I was playing. Because I had reason to think that viewing this game being played would be pleasing to some readers. I understand well that the game seems pointless to other readers.

In any case, since I agree entirely with all technical points you make, and since I seem to be the only active contributor caring about the above game, maybe we should just leave it at that.

Posted by: Urs Schreiber on February 16, 2010 5:57 PM | Permalink | Reply to this

Re: Intrinsic Naturalness

> this thread effectively started with me playing the following game

Well Urs, the entry started with the claim that recently something big happened in higher category theory (I agree) at the conceptual level (well agree a bit less, the technology being largely advanced, systematized and multiplied, the concepts just made gradually more precise for 2-3 decades…). Main example you quote – triangulated categories – are naturally corrected. I confirm, yes, but in essence a while ago, with e.g. dg-categories for which it is less usual, but possible as I quoted above to define stability as the existence of suspension functor and its weak invertibility, what was the main point of “internal simplicity” of the concept. Having in the same approach also other (nontrivially) equivalent ways to say the same definition is a nice feature; so e.g. the fact that one can say the idea with and without twisted complexes should be greeted as a good thing.

Now you made me aware that you meant more of something what is undoubtfully a beautiful idea – to use just 1-categorical language in its infinity-reincarnation; the infinity categorification in its linguistic version. This is not again recent, this “game” is played in connection to some deep applications in mind still, e.g. in Grothendieck’s 1983 letter. There is an important, valuable, substantial, though limited in its expressivity, part of the pool of statements you can say about inf-categories in this language. In particular you can hardly use any concrete infinity category in the construction as it would be often very difficult to define it with this language. A specialist will like to play with such clean language, and a working mathematician will be at least equally so interested with how to connect and shape things from real world into the shapes ready for fruitful higher categorical treatment.

To Thomas: by nonsymmetric case I mean the following. In generalizing homological algebra, one looks at the categories with some additional structure, more or less dealing with exactness. E.g. in Quillen exact categories one has a choice of admissible short exact sequences, effectively admissible epis and admissible monos. Now such a setup is self-dual: in the dual category we replace admissible monos by admissible epis and we get the same axioms hold. The definition of left derived functors can be defined hence one can equally in such a framework define right derived functors. Now, if one wants to be able to define just left derived but not right derived or viceversa, a weaker structure on the category is sufficient. Say just admissible epis and few axioms on them. I am not talking about existence of left or right derived functors, I am talking about expressibility of what they mean at all in a meaningful way. To say it different to express what a notion of a derived functor of a functor from category A to category C is there is a need for some additional structure on the category A. Homotopical frameworks offer such a structure which provides both notions simultaneously (again: I am not taking existence but expressivity). There are examples of structures which make sense just of left derived functors only but not right. See e.g. for a sample framework with only intermediate generality Rosenberg (pdf).

Posted by: Zoran Skoda on February 16, 2010 11:04 PM | Permalink | Reply to this

Re: Intrinsic Naturalness

$MathML-enabled post (click for more details).$

Thanks for your last message, Zoran, we are painting a picture together now:

the entry started with the claim that recently something big happened in higher category theory (I agree) at the conceptual level (well agree a bit less, the technology being largely advanced, systematized and multiplied, the concepts just made gradually more precise for 2-3 decades…).

Right, for 3 decades or so ideas have been materializing.

quasi-categories were conceived in 1973… only that their full impact as a carrier of full-blown higher category theory took a long time to become fully apparent;
$\infty$ -stacks were conceived in 1973… only that their full impact as a carrier of full-blown higher topos theory took a long time to become fully apparent;
pretriangulated dg-categories were conceived in the (what?) late 1980s (?)… but I think their full impact as a carrier of full-blown abstract stable homotopy theory wasn’t fully manifest until rather recently, just think of Lurie’s re-telling of the Goodwillie-calculus story.

So I think for the majority of concepts in the “recently something big”-event it is true that essentially they have been “in the air”, probably. But I think it is crucial that on top of all the technological progress there is now considerble abstract conceptual progress: not only does all this technology exist, but we can point to these constructions now and call them by their natural abstract name. And thereby see a much larger connnected picture.

Posted by: Urs Schreiber on February 17, 2010 12:25 AM | Permalink | Reply to this

Re: Intrinsic Naturalness

$MathML-enabled post (click for more details).$

It seems to me that the theory of pretriangulated dg categories or

A_{\infty}

categories is a special case of the theory of stable

\infty

-categories, which is strictly more general. For instance, the category of spectra doesn’t have an algebraic model, I think. In the context of

\infty

-categories, I would guess that pretriangulated dg categories should correspond to stable

\infty

-categories endowed with the structure of being enriched and tensored over

Hk

-modules for

Hk

the Eilenberg-MacLane spectrum of the ground ring

k

. Thus if you are interested in extending algebraic geometry and homological algebra to more general, homotopical contexts, stable

\infty

-categories really are necessary and dg categories are insufficient.

Posted by: Chris Brav on February 17, 2010 4:55 AM | Permalink | Reply to this

Re: Intrinsic Naturalness

Right. But as far as the solution for correcting the concept of triangulated categories, and the derived category in particular, the pretriangulatedness condition for dg-categories was discovered before the business of stable quasicategories, I think. The fact that they were immediately applied to hard problems in geometry shows that they have had incorporated devices with good motivation. General case of dg-categories is from a bit before the general enriched category business around 1960 (Kelly 1965, somewhat earlier Eilenberg, but I think there is also one refernce from 1958, do not recall right). Somebody else could teach us when the stable model categories were considered in connection with homological algebra first; Lurie in his work says that the theory, has been well known to experts by the time he wrote his variant of stability treatment.

Lurie’s treatment of Goodwillie calculus which you mention is a big breakthrough however (as are many other things in his work, but not so much the treatment of enhancing the derived catgeories, at least not the special cases he made available) and it will be interesting to see more of it soon!

Chris: Stable inf-categories may be more general in characteristics zero. But, in positive (and even mixed) characteristic dg and A-infinity business is another story and of undeniable importance for algebraic geometry. What is the appropriate generalization at stable inf-side I’d like to hear (I think it is not known so far).

Posted by: Zoran Skoda on February 17, 2010 10:40 AM | Permalink | Reply to this

Re: Intrinsic Naturalness

$MathML-enabled post (click for more details).$

Stable infinity categories are not more general in char zero - k-linear stable infinity categories are precisely the same as k-linear dg categories in any characteristic (zero, positive, mixed). What Chris is helpfully pointing out is that stable categories make sense over k for k any commutative ring spectrum (and thus GIVE the correct def of what a k-linear dg category say should be in that generality). Where the difference does become crucial in positive characteristic is in considering symmetric monoidal structures, where asking for a dg model is clearly the wrong thing.

Posted by: David Ben-Zvi on February 17, 2010 9:31 PM | Permalink | Reply to this

Re: Intrinsic Naturalness

$MathML-enabled post (click for more details).$

the pretriangulatedness condition for dg-categories was discovered before the business of stable quasicategories

Sure. History rarely discovers concepts in their platonic logical order. On the contrary, history usually goes the opposite direction, as here, too.

By the way, you keep saying “stable quasi-category” as if it were important that we refer to quasi-categories.

But my point is that we have an abstract intrinsic category-theoretic definition that does not depend on any choice of realization:

all we need to say “stable $(\infty,1)$ -category” is a context of $(\infty,1)$ -category theory that allows us to say the words “terminal object”, “initial object”, “pullback”, “pushout”, “equivalence”.

With that and only that abstract input, we can say “stable $(\infty,1)$ -category”.

Then if we concretely incarnate an $(\infty,1)$ -categories by a quasi-category, we can work out in detail what this means in that incarnation. But the concept is independent of that incarnation and much “simpler” than any concrete incarnation will be, in as far as economy of concepts goes.

Instead of quasi-categories I can take for instance sSet-categories as incarnations of $(\infty,1)$ -categories. Since I know what “pullback” etc. means in an sSet-category incarnation of an $(\infty,1)$ -category, I immediately get a definition of “stable $sSet$ -category”.

Similarly, as soon as I understand how, under Dold-Kan, dg-categories are special $sSet$ -categories, hence also incarnations of certain $(\infty,1)$ -categories, I can then check what the abstract prescription: “form for each object the pullback of the zero-object along itself and check if that produces an equivalence” means in this concrete case. And I will discover that it means precisely that my dg-category is pre-triangulated.

But it is hardly a god-given demand that I incarnate my stable $(\infty,1)$ -category on a structure that involves the notion of chain complexes.

Posted by: Urs Schreiber on February 17, 2010 12:34 PM | Permalink | Reply to this

Re: Intrinsic Naturalness

$MathML-enabled post (click for more details).$

Is it clear from the point of view of the definitions and theorems in current usage in quasi-category approach if the collection of stable quasi-categories is monadic ? The pretriangulation endofunctor can be extended to a monad (Bondal-Kapranov pdf) in dg-categories and the pretriangulated categories are precisely the algebras over that monad. Stabilization functor is defined for presentable quasi-categories; so maybe monadicity over that collection ?

Posted by: Zoran Skoda on February 17, 2010 1:30 PM | Permalink | Reply to this

Re: Intrinsic Naturalness

$MathML-enabled post (click for more details).$

It seems to me that the theory of pretriangulated dg categories or $A_\infty$ categories is a special case of the theory of stable ∞-categories, which is strictly more general.

That should be right. Extra assumptions needed here were briefly mentioned at the very beginning of this discussion on dg-categoriess here. But we should (have) discuss(ed) this in more detail.

Posted by: Urs Schreiber on February 17, 2010 12:38 PM | Permalink | Reply to this

Re: Intrinsic Naturalness

$MathML-enabled post (click for more details).$

For what it’s worth, I like this game. But I’m not sure that dg-categories should be excluded from it. I would include “enriched categories” in my list of “intrinsic category theory language,” and of course dg-categories are just categories enriched in chain complexes. The enrichment is strict, of course, but we can just regard that as giving one particular model for the “real” $(\infty,1)$ version of enrichment, just as strictly simplicially enriched categories are a good enough model for $(\infty,1)$ -categories. $A_\infty$ -categories (in the algebraic sense) would be another non-strict model.

The interesting thing is then that for an $(\infty,1)$ -category to be dg-enriched, it suffices for it to satisfy the stability condition. I think of this as an $(\infty,1)$ -improvement of the classical fact that a category is enriched in commutative monoids as soon as it has finite biproducts.

Posted by: Mike Shulman on February 17, 2010 2:08 AM | Permalink | PGP Sig | Reply to this

Re: Intrinsic Naturalness

$MathML-enabled post (click for more details).$

Thanks, Urs! This is a really important thing to be discussing, and I also feel that everyone’s reservations need to be brought out into the open and discussed honestly. I think we should be able to do that while still acknowledging the amazing successes that have been achieved, and thus without feeling as though we’re devaluing or bad-mouthing someone’s work.

Considering myself as a category theorist, I tend to fall more on the side of using and advocating for ideas like model categories and connections with homotopy theory. But I have to admit that one of the things I don’t like, aesthetically, about the theory of $(\infty,1)$ -categories, is the plethora of different equivalent models. At a rational level, I realize that this is ridiculous—one of the big ideas of category theory is that equivalent things can freely be substituted. Even if you have very different-looking definitions of some pair of categories, if they define equivalent categories, you can freely pass back and forth between them. (My favorite example is affine schemes, as either certain locally ringed spaces or $Rings^{op}$ .) So I shouldn’t feel bad about having very different-looking definitions of $(\infty,1)$ -category as long as they produce equivalent $(\infty,2)$ -categories. But somehow the fact that you have to choose a notion of $(\infty,2)$ -category in order to express this equivalence (even if you’d get an equivalent equivalence for any other choice) is aesthetically unpleasing. Of course this is already present in 1-category theory: there are lots of equivalent definitions of a category, all of which produce equivalent (2-)categories $Cat$ , but of course you have to choose a notion of (2-)category in order to say what the equivalence between versions of $Cat$ means. But psychologically, that choice is easier to ignore, somehow.

One concrete thing I don’t like is that so many “naturally occurring” $(\infty,1)$ -categories are not “naturally occurring” in the useful form of a quasicategory or a complete Segal space—rather those forms of them have to be constructed in some generators-and-relations way from a model category or a simplicially enriched category (which is usually the fibrant+cofibrant objects of some model category). This is already true even for the most basic $(\infty,1)$ -category, namely $\infty Gpd$ , whose quasicategory incarnation is generally defined as the coherent nerve of the simplicially enriched category of Kan complexes. So even to understand what $\infty Gpd$ looks like, as a quasicategory, I need to understand the coherent nerve, which is combinatorially imposing. It also, I think, partly undercuts the argument that intrinsic $(\infty,1)$ -category theory is replacing model category theory.

Josh Nichols-Barrer once proposed a more “intrinsic” definition of the quasicategory $\infty Gpd$ , and also of the quasicategory of $(\infty,1)$ -categories itself. It went like this: an $n$ -simplex in the simplicial set $\infty Gpd$ is a left fibration over $\Delta^n$ . Since left fibrations are the quasi-categorical analogue of Grothendieck opfibrations in groupoids, such a fibration is a natural model for a functor $\Delta^n \to \infty Gpd$ , which is what the $n$ -simplices in $\infty Gpd$ should be. Similarly, an $n$ -simplex in the simplicial set $(\infty,1)Cat$ is a coCartesian fibration over $\Delta^n$ . I liked this idea very much, but the problem is that Josh hadn’t been able to prove that these simplicial sets actually were quasicategories. (One would of course then want to show that they are equivalent to the usual definitions of these quasicategories.) Has anyone else thought about a definition like this? That was several years ago, so perhaps such an equivalence is now to be found in the literature somewhere?

Posted by: Mike Shulman on February 15, 2010 8:08 PM | Permalink | PGP Sig | Reply to this

Re: Intrinsic Naturalness

$MathML-enabled post (click for more details).$

Thanks for your message, Mike. Good that you are joining in!

Josh Nichols-Barrer once proposed a more “intrinsic” definition of the quasicategory ∞Gpd, and also of the quasicategory of (∞,1)-categories itself. It went like this: an $n$ -simplex in the simplicial set ∞Gpd is a left fibration over $Delta^n$ . […] Similarly, an n-simplex in the simplicial set (∞,1)Cat is a coCartesian fibration over Δ n.

That sounds like a good idea.

but the problem is that Josh hadn’t been able to prove that these simplicial sets actually were quasicategories. (One would of course then want to show that they are equivalent to the usual definitions of these quasicategories.) Has anyone else thought about a definition like this?

Let me see, probably this pretty directly gives not the quasi-category but the comple Segal space incarnation of $(\infty,1)Cat$ and $\infty Grpd$ .

The crucuial input is the $(\infty,1)$ -Grothendieck construction based on the universal fibration of $(\infty,1)$ -categories.

This gives an equivalence of the $(\infty,1)$ -categories $CartFib(\Delta^n)$ and $RFib(\Delta^n)$ on the one hand and the $(\infty,1)$ -categories $Func(\Delta^n,N((\infty,1)Cat))$ and $Func(\Delta^n,N(\infty Grpd))$ , respectively on the other.

For Josh Nichols-Barrer’s suggestion one would have to see if this restricts to a bijection on 0-cells, which seems an unreasonable demand. But it makes sense to restrict to the cores and pick the maximal $\infty$ -groupoids inside these $(\infty,1)$ -categories.

Doing that – in view of the Grothendieck construction – is constructing precisely the complete Segal space version of $(\infty,1)Cat$ and $\infty Grpd$ , along the lines of the discussion on p. 27 here (and in a minute this will also be at complete Segal space…)

Posted by: Urs Schreiber on February 15, 2010 9:55 PM | Permalink | Reply to this

Re: Intrinsic Naturalness

$MathML-enabled post (click for more details).$

For Josh Nichols-Barrer’s suggestion one would have to see if this restricts to a bijection on 0-cells, which seems an unreasonable demand.

Why a bijection? Josh wasn’t proposing that his construction gives something isomorphic to the other definition, only equivalent to it.

Posted by: Mike Shulman on February 15, 2010 11:24 PM | Permalink | Reply to this

Re: Intrinsic Naturalness

$MathML-enabled post (click for more details).$

Why a bijection?

Hm, right, that wasn’t well said.

I believe I meant to express that by truncating these $(\infty,1)$ -categories of fibrations to their 0-cells, one loses control over how they relate to $\infty$ -functors.

But let’s see, we can probably play around a little with the various Quillen equivalences between complete Segal spaces and quasi-categories (there are two different ones). I am currently looking at

Joyal, Tierney, Quasi-categories vs Segal spaces

One functor that takes a quasi-category to a Segal space, is the functor $t^!$ on p. 13. I was suggesting that the simplicial space

$[n] \mapsto Core(CartFib(\Delta^n))$

is, under the Grothendieck construction, manifestly

$t^!(N((\infty,1)Cat))$

which is the bisimplicial set defined as

$t^!(N((\infty,1)Cat))_{m n} = Hom_{sSet}(\Delta[m] \times Ex^\infty(\Delta[n]), N((\infty,1)Cat))$

But then there is another Quillen equivalence back from bisimplicial sets to simplicial sets, which simply evaluates at $n=0$ . That’s the functor denoted $i^*_1$ in that article.

Unless it’s already too late for me to think straight (that’s actually likely), this would mean that we have an equivalence of quasi-categories between

$i^*_1([m] \mapsto Core(CartFib(\Delta[m]))) = ([m] \mapsto CartFib(\Delta[m])_0)$

and

$N((\infty,1)Cat)$ .

Which would be the desired result. But let me check this…

Posted by: Urs Schreiber on February 15, 2010 11:48 PM | Permalink | Reply to this

Re: Intrinsic Naturalness

$MathML-enabled post (click for more details).$

I wrote:

Which would be the desired result. But let me check this…

Okay, now I am more awake, that helps. I’ll try to say again what I said above, but now in a more ordered fashion. All notation and facts that I use are collected at model structure for complete Segal spaces.

Write $N(\infty Grpd) \subset N((\infty,1)Cat)$ for the quasi-category of $\infty$ -groupoids inside the quasi-category of quasi-categories. For $n \in \mathbb{N}$ write $LFib(\Delta[n]) \subset coCartFib(\Delta[n])$ for the $(\infty,1)$ -category of left fibrations inside that of coCartesian fibrations over the $n$ -simplex.

Claim. There is a canonical equivalences of quasi-categories

$([n] \mapsto coCart(\Delta[n])_0) \stackrel{\simeq}{\to} N((\infty,1)Cat)$

that restricts to an equivalence of quasi-categories

$([n] \mapsto LFib(\Delta[n])_0) \stackrel{\simeq}{\to} N(\infty Grpd) \,.$

Idea of Proof. By applying the $(\infty,1)$ -Grothendieck construction columnwise we have a morphism of bisimplicial sets

$([n] \mapsto Core(coCart(\Delta[n]))) \stackrel{\simeq}{\to} ([n] \mapsto Core(Func(\Delta[n],N(\infty,1)Cat)))$

which is columnwise a weak equivalence in $SSet_{Quillen}$ , hence a weak equivalence in the vertical model structure on bisimplicial sets, hence a weak equivalence in the Rezk model structure $BiSSet_{Rezk}$ for complete Seagl spaces. The object on the right is manifestly the image under $t^! : SSet \to BiSSet$ of $N((\infty,1)Cat)$ , so that we have a weak equivalence

$([n] \mapsto Core(LFib(\Delta[n]))) \stackrel{\simeq}{\to} t^!(N((\infty,1)Cat))$

The object on the right is fibrant and that on the left is cofibrant. So as a side remark, from $(t_! \dashv t^!) : BiSSet_{Rezk} \leftrightarrow SSet_{Joyal}$ being a Quillen equivalence we have a weak equivalence of quasi-categories

$t_!([n] \mapsto Core(coCart(\Delta[n]))) \stackrel{\simeq}{\to} N((\infty,1)Cat) \,.$

Here $t_!(X_{\bullet,\bullet}) = \int^{[k],[l]} X_{k,l} \cdot \Delta[k] \times Ex^\infty(\Delta[l])$ . That might be of interest for something, but here we want something else.

To get that, I need that $([n] \mapsto Core(coCart(\Delta[n])))$ is fibrant in $BiSSet_{Rezk}$ , hence is a complete Segal space. That looks like it should be true but I am not sure yet how to make this precise. This is a gap in the argument.

But if I allow myself to assume that this object is a complete Segal space, then I can apply the right adjoint of the other Quillen equivalence

$(p_1^* \dashv i_1^*) : SSet_{Joyal} \leftrightarrow BiSSet_{Rezk}$

which will preserve this weak equivalence between fibrant objects and produce a weak equivalence of quasi-categories

$i_1^*([n] \mapsto Core(coCart(\Delta[n]))) \stackrel{\simeq}{\to} i_1^* t^!(N((\infty,1)Cat)) \,.$

This $i_1^*$ simply evaluates the first row of a bisimplicial set and we know that

$i_1^* t^! : SSet \to SSet$

is isomorphic to the identity functor. Therefore this is the desired equivalence of quasi-categories

$([n] \mapsto coCart(\Delta[n])_0) \stackrel{\simeq}{\to} N((\infty,1)Cat) \,.$

Posted by: Urs Schreiber on February 16, 2010 10:11 AM | Permalink | Reply to this

Re: Intrinsic Naturalness

$MathML-enabled post (click for more details).$

Thanks for trying to answer this! I’m working through your argument, which seems very promising. A couple of questions.

The object on the right is manifestly the image under $t^! : SSet \to BiSSet$ of $N((\infty,1)Cat)$

That isn’t clear to me, since the definition of $t^!$ here includes an extra $Ex^\infty$ . It looks to me from that definition like $t^! N((\infty,1)Cat)$ should be

$[n] \mapsto Func(Ex^\infty \Delta[n], (\infty,1)Cat)$

which doesn’t look quite the same as $[n] \mapsto Core(Func(\Delta[n],N(\infty,1)Cat))$ .

Also, when you wrote

$([n] \mapsto Core(LFib(\Delta[n]))) \stackrel{\simeq}{\to} t^!(N((\infty,1)Cat))$

Did you by any chance mean $coCart$ instead of $LFib$ ?

Posted by: Mike Shulman on February 17, 2010 2:21 AM | Permalink | PGP Sig | Reply to this

Re: Intrinsic Naturalness

$MathML-enabled post (click for more details).$

Thanks for the reply!

It looks to me from that definition like $t^! N((\infty,1)Cat)$ should be $[n] \mapsto Func(Ex^\infty \Delta[n], N((\infty,1)Cat))$ which doesn’t quite look quite the same as $[n] \mapsto Core(Func(\Delta[n], N((\infty,1)Cat))$ .

I agree, that this isn’t the same, but also the first expression doesn’t seem to me what $t^!$ yields.

Let’s go the other way round and unwrap the second expression:

for any quasi-category $A$ a $k$ -cell in $Func(\Delta[n], A)$ is a map

$\Delta[n] \times \Delta[k] \to A \,.$

And in $Core(Func(\Delta[n]), A)$ we retain only those $k$ -cells $\Delta[k] \to Func(\Delta[n],A)$ that are invertible. Which is the same as maps out of (the nerve of) the free groupoid $\Delta'[k]$ generated from $[k]$ , so a map $\Delta'[k] \to Func(\Delta[n],A)$ hence the same as morphisms

$\Delta[n] \times \Delta'[k] \to A \,.$

This is the way $t^!$ is written on p. 13 in Joyal/Tierney.

I thought I’d just change to more explicit notation and write equivalently $Ex^\infty \Delta[k]$ for $\Delta'[k]$ .

Did you by any chance mean coCart instead of LFib?

Yes! :-) I mixed that up a bit. Also when producing the comment I kept changing my mind as to whether I write a comment on just $\infty Grpd$ or $(\infty,1)Cat$ or both.

Posted by: Urs Schreiber on February 17, 2010 9:21 AM | Permalink | Reply to this

Re: Intrinsic Naturalness

$MathML-enabled post (click for more details).$

Hang on, what’s the definition of the core of a quasicategory? I thought it meant the maximal sub-Kan-complex. If so, then isn’t a $k$ -simplex in $Core(X)$ just a $k$ -simplex in $X$ all of whose 1-simplices are invertible? That doesn’t seem to me to be the same as maps out of the free groupoid $\Delta'[k]$ generated by $[k]$ . Even its 1-simplices are not the same, since a given 1-simplex $\Delta[1] \to X$ might extend to multiple different maps from $\Delta'[1]$ (i.e. a 1-simplex in a quasicategory may have more than one homotopy inverse).

Also, I really don’t think that $\Delta'[k]$ is the same as $Ex^\infty \Delta[k]$ , although they are equivalent Kan complexes (both being contractible).

Posted by: Mike Shulman on February 17, 2010 5:03 PM | Permalink | PGP Sig | Reply to this

Re: Intrinsic Naturalness

$MathML-enabled post (click for more details).$

Hang on, what’s the definition of the core of a quasicategory? I thought it meant the maximal sub-Kan-complex. If so, then isn’t a k-simplex in Core(X) just a k-simplex in X all of whose 1-simplices are invertible?

Yes.

That doesn’t seem to me to be the same as maps out of the free groupoid $\Delta'[k]$ generated by $[k]$ .

It is equivalent: for $C$ a quasi-category the canonical morphism

$([n] \mapsto Hom(\Delta' [n], C)) \;\; \to \;\; Core(C)$

is an acyclic fibration in $sSet_{Quillen}$ .

I have now typed some details on this into this section here:

Model structure for quasi-categories – relation to model structure for Kan complexes

You write:

Also, I really don’t think that $\Delta'[k]$ is the same as $Ex^\infty \Delta[k]$ , although they are equivalent Kan complexes (both being contractible).

I’d think that the construction won’t depend on which fibrant replacement functor we use. But let me think about this a bit more. For the time being read all the $Ex^\infty \Delta[k]$ in the above comments as $\Delta'[k]$ . That was just notation and didn’t enter the argument.

Posted by: Urs Schreiber on February 17, 2010 8:44 PM | Permalink | Reply to this

Re: Intrinsic Naturalness

$MathML-enabled post (click for more details).$

I believe I can now complete my proof of Josh Nichols-Barrer’s suggestion that the simplicial set

$([n] \mapsto coCart(\Delta[n])_0)$

is a quasi-category equivalent to the quasi-category of all small quasi-categories, $N((\infty,1)Cat)$ .

Above I had given an argument that constructed an equivalence of quasi-categories

$([n] \mapsto coCart(\Delta[n])_0) \;\; \stackrel{\simeq}{\to} \;\; N((\infty,1)Cat)$

under the assumption that the bisimplicial set

$([n] \mapsto Core(coCart(\Delta[n])))$

is a complete Segal space, where now $Core(coCart(\Delta[n]))$ is the Kan complex of coCartesian fibrations of quasi-categories over $\Delta[n]$ .

Here now I want to prove the validity of this assumption, thereby completing the proof.

To check that our bisimplicial set is a Segal space, it is sufficient to show that for all $k,l \in \mathbb{N}$ the diagram

$\array{ coCart(\Delta[k+l]) &\to& coCart(\Delta[l]) \\ \downarrow && \downarrow^{s} \\ coCart(\Delta[k]) &\stackrel{t}{\to}& coCart(\Delta[0]) }$

is a homotopy pullback diagram of quasi-categories.

Now, $coCart(\Delta[k])$ is the homotopy coherent nerve

$coCart(\Delta[k]) = N( (sSet^+/\Delta[k])^\circ)$

of the simplicial category of fibrant-cofibrant objects in the simplicial model category of marked simplicial sets over $\Delta[k]$ .

This and related facts from HTT that I am using are collected and referenced now at $(\infty,1)$ -Grothendieck construction.

So consider instead for a moment the corresponding pullback diagram of simplicial categories

$\array{ && (sSet^+/\Delta[l])^\circ \\ && \downarrow^{s} \\ (sSet^+/\Delta[k])^\circ &\stackrel{t}{\to}& (sSet^+/\Delta[0])^\circ } \,.$

I want now to

take a fibrant replacement of this diagram,
then compute its homotopy limit as an ordinary limit,
then apply the homotopy coherent nerve to the diagram to get the desired homotopy pullback of quasi-categories and
then show that this is equivalent to the diagram we started with.

By the fact that all model categories appearing here are $SSet_{Quillen}$ -enriched, all simplicial categories in the diagram are already fibrant in the model structure on $sSet$ -categories. So to get a diagram that is fibrant in the diagram category we need to make at least one of the morphisms into a fibration.

For that, I use the facts about (co)Cartesian fibrations over simplices from HTT, section 3.2.2.

Prop. 3.2.2.7 there implies that I can find equivalent SSet-categories of coCartesian fibrations by replacing all coCartesian fibrations over $\Delta[n]$ with mapping simplices (see there for the definition): specially adapted coCartesian fibrations classified by sequences of $(\infty,1)$ -functors $C_0 \to C_2 \to \cdots \to C_n$ .

So I pass to a weakly equivalent pullback diagram

$\array{ && (sSet^+/\Delta[l])^\circ_{sk} \\ && \downarrow^{s} \\ (sSet^+/\Delta[k])^\circ_{sk} &\stackrel{t}{\to}& (sSet^+/\Delta[0])^\circ_{sk} }$

where my simplicial categories are partially skeletatized this way by retaining for each coCartesian fibration the corresponding mapping simplex. The point is that then remark 3.2.2.5 in HTT implies that the two morphisms in the pullback diagram are fibrations of sSet-categories. I think. I won’t spell this part out here in full detail. But for instance the fact that homotopy equivalences lift through the morphisms (one of the conditions on fibrations in $SSet Cat$ ) boils now down to the fact that given a sequence of $(\infty,1)$ -functors $A_0 \to \cdots \to A_n$ , from a (homotopy-)equivalence $A_n \stackrel{\simeq}{\to} A_n$ of the last $(\infty,1)$ -category we obtain homotopy equivalence of sequences of $(\infty,1)$ -functors

$\array{ A_0 &\to& A_1 &\to& \cdots &\to& A_n \\ \downarrow^{\mathrlap{Id}} && \downarrow^{\mathrlap{Id}} && && \downarrow^{\mathrlap{\simeq}} \\ A_0 &\to& A_1 &\to& \cdots &\to& A'_n } \,.$

By similar reasoning, I think, one finds that the ordinary pullback of my partially skeletized simplicial categories is the partial skeletization of $(sSet/\Delta[k+l])^\circ$ : meaning that

$\array{ (sSet^+/\Delta[k+l])^\circ_{sk} &\to& (sSet^+/\Delta[l])^\circ_{sk} \\ \downarrow && \downarrow^{s} \\ (sSet^+/\Delta[k])^\circ_{sk} &\stackrel{t}{\to}& (sSet^+/\Delta[0])^\circ_{sk} }$

is an ordinary limit over a fibrant pullback diagram. Since the homotopy coherent nerve is right Quillen, it follows then that

$\array{ N((sSet^+/\Delta[k+l])^\circ_{sk}) &\to& N((sSet^+/\Delta[l])^\circ_{sk}) \\ \downarrow && \downarrow^{s} \\ N((sSet^+/\Delta[k])^\circ_{sk}) &\stackrel{t}{\to}& N((sSet^+/\Delta[0])^\circ_{sk}) }$

is a homotopy pullback of quasi-categories. And since $N$ preserves weak equivalences between fibrant objects, this is weakly equivalent to the pullback diagram that we wanted to produce.

Posted by: Urs Schreiber on February 16, 2010 10:03 PM | Permalink | Reply to this

Re: Intrinsic Naturalness

$MathML-enabled post (click for more details).$

This is also looking promising. I’m not sure exactly what you meant by “retaining for each coCartesian fibration the corresponding mapping simplex;” did you mean to look at the full sub-simplicial-categories whose objects are mapping simplices? It seems to me that in order to get the honest pullback to be what you want, you should look instead at the (still equivalent) simplicial-category whose objects are $k$ -tuples (or $l$ -tuples, etc.) of composable maps, with hom-spaces induced from the corresponding mapping simplices—since it seems to me that a priori a given coCartesian fibration over a simplex might be a mapping simplex of more than one different composable tuple. Maybe that’s what you intended?

The one thing I don’t see how to do is the part that you “didn’t spell out in full detail” about why functoriality implies that those maps are fibrations. I see that homotopy equivalences lift, but it seems that for it to be a fibration on hom-spaces you also need some sort of functoriality on homotopies?

Posted by: Mike Shulman on February 17, 2010 4:08 AM | Permalink | PGP Sig | Reply to this

Re: Intrinsic Naturalness

$MathML-enabled post (click for more details).$

It seems to me that in order to get the honest pullback to be what you want, you should look instead at the (still equivalent) simplicial-category whose objects are k-tuples (or l-tuples, etc.) of composable maps, with hom-spaces induced from the corresponding mapping simplices—since it seems to me that a priori a given coCartesian fibration over a simplex might be a mapping simplex of more than one different composable tuple. Maybe that’s what you intended?

Thanks, that’s a good point. Yes, I think that’s the right way to say it.

The one thing I don’t see how to do is the part that you “didn’t spell out in full detail” about why functoriality implies that those maps are fibrations. I see that homotopy equivalences lift, but it seems that for it to be a fibration on hom-spaces you also need some sort of functoriality on homotopies?

True. Probably I should try to spell that out in full detail.

Posted by: Urs Schreiber on February 17, 2010 9:30 AM | Permalink | Reply to this

Re: Intrinsic Naturalness

$MathML-enabled post (click for more details).$

Also, I think a (complete) Segal space additionally has to be Reedy fibrant. This was missing from the nLab page, but I just added it.

Posted by: Mike Shulman on February 17, 2010 4:50 PM | Permalink | Reply to this

Re: Intrinsic Naturalness

$MathML-enabled post (click for more details).$

Also, I think a (complete) Segal space additionally has to be Reedy fibrant. This was missing from the nLab page, but I just added it.

Thanks, Mike, true.

So there is a bit left to show to prove that $Core(coCart(\Delta[-])$ is a complete Segal space. For the moment I am running a bit out of steam, with other questions waiting for my attention. But it would be nice if we could clarify this eventually.

Posted by: Urs Schreiber on February 18, 2010 9:06 AM | Permalink | Reply to this

Re: Intrinsic Naturalness

$MathML-enabled post (click for more details).$

This discussion is going way over my head! Just to make sure I understand what’s going on, can I propose a simpler version of this? Suppose I build a simplicial set $K$ by setting $K([n])$ to be the set of all Kan fibrations $E\to \Delta[n]$ . We would want to claim that $K$ is a model for the $\infty$ -groupoid of $\infty$ -groupoids, right?

There are two things I could mean by this: (1) $K$ is a simplicial set which is weakly equivalent to $N(\infty,1)Cat$ , (2) $K$ is a Kan complex which is weakly equivalent to $N(\infty,1)Cat$ .

Do we want to claim (2), or just (1)? I’m pretty sure (1) is true, but not at all convinced (2) is. I have a map $f:X\to \Lambda^i[2]\subset \Delta[2]$ , which restricts to a Kan fibration over each non-degenerate edge in $\Lambda^i[2]$ , and I want to construct a Kan fibration $g: Y\to \Delta[2]$ whose restriction to $\Lambda^i[2]$ is $f$ . How do I do this?

There’s a variant construction, where instead of using Kan fibrations, I’ll use “sharp” maps. A map $f:E\to B$ of simplicial sets is sharp if for every map $g\colon \Delta[n]\to B$ , the pullback of $E\to B\leftarrow \Delta[n]$ is weakly equivalent to the homotopy pullback of the same diagram. Then I define a simplicial set $S$ so that $S([n])$ is the set of sharp maps $S\to \Delta[n]$ .

The guy $S$ is a bit nicer than $K$ , because the definition of sharp map turns out to be “local”: a map $E\to B$ is sharp if and only if it’s pullback along any map $\Delta[n]\to B$ is sharp. So every sharp map $E\to B$ is “classified” by a map $B\to S$ .

Anyway, you can prove the analogue of (1) for $S$ . But you can also show that that $S$ is not a Kan complex.

Posted by: Charles Rezk on February 19, 2010 12:14 AM | Permalink | Reply to this

Re: Intrinsic Naturalness

$MathML-enabled post (click for more details).$

The original hope was that (2) would also be true. Of course, what you point out is the problem involved in showing that it is true. But my intuition suggests to me that it might be true, since a Kan fibration over $\Delta^1$ is just a “nonalgebraic” notion of an equivalence between two Kan complexes (the fibers over the vertices), and two equivalences can be composed. Intuitively, given $X\to \Lambda^2_1$ which is Kan over each $\Delta^1$ , one wants to do a sort of fibrant replacement over $\Delta^2$ which doesn’t change the fibers over $\Lambda^2_1$ .

The approach that Urs sketched using complete Segal spaces seems promising to me, because it’s much clearer that given such an $X$ we can find a Kan fibration over $\Delta^1$ whose restriction to $\Lambda^2_1$ is equivalent to $X$ . So we should start out by keeping around the equivalences, giving ourselves a simplicial space rather than a simplicial set. The problem then becomes showing that this simplicial space is in fact a CSS. Of course, maybe it isn’t, but if not that would be interesting to know too. Maybe it’ll at least have some easily-describable fibrant replacement that could supply a good “intrinsic” notion of the quasicategory of quasicategories.

Posted by: Mike Shulman on February 19, 2010 3:41 PM | Permalink | PGP Sig | Reply to this

Re: Intrinsic Naturalness

$MathML-enabled post (click for more details).$

Suppose I build a simplicial set $K$ by setting $K([n])$ to be the set of all Kan fibrations $E \to \Delta[n]$ . We would want to claim that $K$ is a model for the ∞-groupoid of ∞-groupoids, right?

Yes. That’s the kind of statement we have in mind.

Or, as another variant (just for emphasis) let $L([n])$ be the set of all left fibrations $E \to \Delta[n]$ . Then we want to claim that $L$ is a model for the $(\infty,1)$ -category of $\infty$ -groupoids.

There are two things I could mean by this: (1) $K$ is a simplicial set which is weakly equivalent to $N(\infty,1)Cat$ , (2) $K$ is a Kan complex which is weakly equivalent to $N(\infty,1)Cat$ .

Given your previous sentence I suppose instead of $N((\infty,1)Cat)$ you here mean $N(\infty Grpd)$ , or in fact the maximal Kan complex $Core(N(\infty Grpd))$ inside it?

Do we want to claim (2), or just (1)?

we want to claim (1). Weakly equivalent in the model structure $sSet_{Joyal}$ .

I’m pretty sure (1) is true

Okay, good! Is there any chance we could look in your cards that tell you this? Does your argument generalize if we generalize Kan fibrations to left fibrations, and further to coCartesian fibrations?

Posted by: Urs Schreiber on February 19, 2010 12:36 AM | Permalink | Reply to this

Re: Intrinsic Naturalness

$MathML-enabled post (click for more details).$

Yes, $N(\infty,1)Cat$ is a typo, I meant $N(\infty Gpd)$ .

I’ll need to think about what the argument actually was, but it’s something like this. In the case of $S$ , it amounts to the fact that $S$ “represents” sharp maps. Remember that a map $f:\Delta[n]\to S$ amounts to a sharp map $p_f:E_f\to S$ , and this leads to the construction of a “tautological” sharp map $p:ES\to S$ , so that the pullback of $p$ along $f$ is isomorphic to $p_f$ .

I can’t figure out how to draw diagrams here, so this is a little awkward. Let $p_1:E_1\to B_1$ and $p_2:E_2\to B_2$ be sharp maps. Let $g: p_1 \Rightarrow p_2$ represent a pullback square, whose left and right sides are $p_1$ and $p_2$ , and suppose that $g: B_1\to B_2$ is a cofibration. Let $f_1: p_1 \Rightarrow p$ be a pullback square, where $p:ES\to S$ as above. Then it is not hard to show that there exists a pullback square $f_2:p_2\Rightarrow p$ such that $f_2g=f_1$ .

Then you show that any $p:ES\to S$ with the above property must be such that $S$ is a model for $N(\infty Gpd)$ . I think you can also carry out this argument with $K$ , instead of $S$ , using Kan fibrations instead of sharp maps. In my previous post, I said that $S$ is “local”, but $K$ also has this property.

I don’t know anything about left fibrations, or coCartesian fibrations.

Posted by: Charles Rezk on February 19, 2010 6:51 PM | Permalink | Reply to this

Re: Intrinsic Naturalness

Quote: there are lots of equivalent definitions of a category

but one of those definitions predominates and leads readily to the definition of an A_\infty cat

are we anywhere near such a consensus for this discussion?

Posted by: jim stasheff on February 16, 2010 1:06 PM | Permalink | Reply to this

Re: Intrinsic Naturalness

$MathML-enabled post (click for more details).$

Quote: there are lots of equivalent definitions of a category

but one of those definitions predominates and leads readily to the definition of an $A_\infty$ cat

are we anywhere near such a consensus for this discussion?

As far as I am concerned, I might need more hints concerning which consensus precisely you mean.

Just one caveat: when people like David Ben-Zvi say “ $A_\infty$ -category” as in a comment above they mean the linear version, i.e. categories weakly enriched in chain complexes, each of which is $A_\infty$ -equivalent to a dg-category, i.e. to a category enriched strictly in chain complexes.

Last time we had a big between-the-lines fight on whether or not this usage of the term should be regarded as the standard usage. But fact is that some people do use the term differently and more generally. An attempt to sort this out is at

$A_\infty$ -category.

Posted by: Urs Schreiber on February 16, 2010 1:51 PM | Permalink | Reply to this

Re: Intrinsic Naturalness

I hesitate to mess with your A-infty cat page but take
exception to:

1.for all X,Y in Ob(C) the Hom-sets Hom C(X,Y) are finite dimensional chain complexes of matbfZ-graded modules

Why fin dim?

2.for all objects X 1,…,X n in Ob(C) there is a family of linear composition maps (the higher compositions) m n:Hom C(X 0,X 1)⊗Hom C(X 1,X 2)⊗⋯⊗Hom C(X n−1,X n)→Hom C(X 0,X n) of degree n−2 (homological grading convention is used) for n≥1

3. m 1 is the differential on the chain complex Hom C(X,Y)

4. m n satisfy the quadratic A ∞-associativity equation for all n≥0.

m 1 and m 2 will be chain maps

Strange to call m_1 a chain map

but the compositions m i of higher order are not chain maps, nevertheless they are Massey products.

No, although thory are related to Massey products

Posted by: jim stasheff on February 17, 2010 1:35 PM | Permalink | Reply to this

Re: Intrinsic Naturalness

$MathML-enabled post (click for more details).$

I hesitate to mess with your A-infty cat page but take exception to: […]

Thanks!

That page is probably also more generally in need of a kind soul to take care of it and brush it up a bit. But that person won’t be me right now, as I have to take care of something else. But you all know where the edit-button is…

Posted by: Urs Schreiber on February 17, 2010 1:47 PM | Permalink | Reply to this

Re: Intrinsic Naturalness

$MathML-enabled post (click for more details).$

So although in some respects matters are a little clearer, we still don’t really understand the reservations some category theorists have about the push towards $(\infty, 1)$ -categories. Are there reservations from the other party to the marriage? Are some homotopy theorists complaining about the quasi-category take-over?

Posted by: David Corfield on February 17, 2010 12:39 PM | Permalink | Reply to this

Re: Intrinsic Naturalness

Do the current notions of space shed any light on Cartier’s merger of Grothendieck’s and Connes’s versions of space, and on Connes’s objection to the merger?

Posted by: David Corfield on February 18, 2010 2:20 PM | Permalink | Reply to this

Re: Intrinsic Naturalness

$MathML-enabled post (click for more details).$

Do the current notions of space shed any light on Cartier’s merger of Grothendieck’s and Connes’s versions of space, and on Connes’s objection to the merger?

Good question. We have talked about this a bit here and there in past entries.

Here is what might be one way to understand the relation:

It turns out that $(\infty,1)$ -toposes are – while crucially different – surprisingly similar to stable $(\infty,1)$ -categories:

every (Grothendieck-) $(\infty,1)$ -tops is the left exact localization of an $(\infty,1)$ -category of $(\infty,1)$ -presheaves;
every stable $(\infty,1)$ -category is the left exact localization of an $(\infty,1)$ -category of stabilized $(\infty,1)$ -presheaves, i.e. of spectrum-valued presheaves.

Now, following Grothendieck we maythink of an $(\infty,1)$ -topos as the formal dual to a generalized concrete space. This is what happens in the discussion at notions of space.

But since stable $(\infty,1)$ -categories are conceptually (while different) not entirely unsimilar, we might wonder what would happen if we regard stable $(\infty,1)$ -categories as formal duals of generalized spaces, too.

And this is what is happening in “noncommutative geometry”. To see this clearly, you have to go beyond Connes’ traditional way of presenting the theory to Kontsevich’s “derived” version, where:

associative algebras are generalized to general $dg$ -algebras/ $A_\infty$ -algebras; (thought of as functions on a noncommutative derived space)
one shifts attention from the algebra itself ot its (linear) $A_\infty$ -category of modules (thought of as the category of quasicoherent sheaves on that noncommutative space)…
…and then goes one step further and regards every $A_\infty$ -category as a category of quasicoherent sheaves on a generalized space, hence as a formal dual to a generalized space.

This is the program that Kontsevich and others have been developing for some time now. Effectively they do with $A_\infty$ -catgeories everything that people also do with $(\infty,1)$ -toposes: for instance there is a notion of geometric morphism between $A_\infty$ -catgeories coming from an adjoint pair of functors that describes a morphisms of the formally dual would-be spaces, and so on.

But now recall from the discussion we are having further above, that a (linear) $A_\infty$ -category is nothing but a kind of stable $(\infty,1)$ -category!

So from this perspective one can see that much of what happens in noncommutative geometry is analogous to the description of generalized spaces as formal duals to $(\infty,1)$ -toposes, only that instead of those one uses stable $(\infty,1)$ -categories.

We had a little bit of debate here and on other blogs to which extent this picture is correct, but to me it looks like a good conceptual organization.

Particularly interesting is then the aspect that there is a canonical map that takes an $(\infty,1)$ -topos to a stable $(\infty,1)$ -category, by “stabilizing” or “linearizing” it. This suggests that everything one does in “nonlinear” $(\infty,1)$ -topos theory has “linear approximations” in stable $(\infty,1)$ -cat theory and hence in “derived noncommutative geometry”. The study of these “linearizaitons” is called Goodwillie calculus.

Posted by: Urs Schreiber on February 18, 2010 2:58 PM | Permalink | Reply to this

Re: Intrinsic Naturalness

As you know I absolutely agree with you and with Kontsevich that there is no substantial formal difference between commutative and noncommutative derived algebraic geometry. The unsubstantial differences are in traditionally taken set of sources of examples, local models and focuses of research.

However, this is not so in non-derived picture: the abelian categories (rather than stable-infinity categories) of quasicoherent sheaves, deal with a generalization of sheaf theory which does not lead to topoi or infinity topoi.

At the very moment derived noncommutative geometry is more vigorously advancing, while the progress on non-derived side of

noncommutative algebraic geometry

is, in my opinion, slow, due to general lack of interest in the foundations of this subject.

Posted by: Zoran Skoda on February 18, 2010 4:33 PM | Permalink | Reply to this

Re: Intrinsic Naturalness

$MathML-enabled post (click for more details).$

If on the non-derived side of things, you look to abelian categories in the noncommutative case and to $(\infty, 1)$ -toposes in the commutative case, does Freyd’s results on the comparison between abelian categories and toposes hold any interest here?

Posted by: David Corfield on February 22, 2010 10:51 AM | Permalink | Reply to this

Re: Intrinsic Naturalness

$MathML-enabled post (click for more details).$

The link to Freyd does not work. It is of interest, but I only heard of it from Baez’s comment. Anyway, it could be just a start of the unification; the exactness properties for sheafification functors and alike are much weaker in noncommutative case and these things are serious issue.

Posted by: Zoran Skoda on February 22, 2010 1:34 PM | Permalink | Reply to this

Re: Intrinsic Naturalness

Link fixed. We also talked about Freyd’s result back here.

Posted by: David Corfield on February 22, 2010 1:52 PM | Permalink | Reply to this

Re: Intrinsic Naturalness

Hi David,

Sorry for being completly off topic, but the link on the main page right beside your portrait seems to be outdated, (that would be “http://dcorfield.pwp.blueyonder.co.uk/”).

Posted by: Tim van Beek on February 23, 2010 11:26 AM | Permalink | Reply to this

Re: Intrinsic Naturalness

Thanks. I’ll have it changed.

Posted by: David Corfield on February 23, 2010 4:33 PM | Permalink | Reply to this

Re: Intrinsic Naturalness

$MathML-enabled post (click for more details).$

In addition to the sheaf condition for quasicoherent sheaves on lattices of Gabriel localizations (Fred van Oystaeyen, Willaert, A. Verschoren, A. Rosenberg and others) there are also attempts to do do the functor point of view as well in noncommutative geometry, e.g. this preprint from 1999: ps. Rosenberg also devised a notion of sheaves of sets in and on Q-categories. In any case we talk about some sort of sheaf condition on NAff which is opposite to the category of (noncommutative) rings. The natural candidates for Grothendieck topologies do not satisfy the stability under pullbacks axiom. In a way intersecting a cover with another localization does not make its cover in general.

Noncommutative Zariski sheaves, sheafification for Gabriel localization and formal smoothness are some special cases of this machinery. Here we talk sheaves of sets, hence no passage to abelian case, and still the sheaves on such “sites” do not make Grothendieck topoi. On NAff there are some genuine Grothendieck topologies though, like noncommutative smooth topology, good for many things. But localizations and noncommutative schemes are very useful when we have their structure (explicit nice covers for example), and most experts would not dispense from them.

I know of further cases needing derived descent (when the localizations are having weaker flatness conditions, like Cohn localization), but passing to stable (infinity,1)-categories would loose some essential information as far as I can see.

So much is known about hints where to go, but no complete solutions exist. I emphasise again: because of general lack of interests into foundations of nonderived noncommutative algebraic geometry.

Posted by: Zoran Skoda on February 22, 2010 6:43 PM | Permalink | Reply to this

Re: Intrinsic Naturalness

Urs said

> thought of as the category of quasicoherent sheaves on that noncommutative space

A-infinity category presents just (unique enhancement of) the derived category of qcoh sheaves. While from the abelian category of qcoh sheaves we can reconstruct the space, outside of few special classes of nice varieties, one can NOT fully reconstruct the space from the derived category (or its stable enhancement); some tipically finite amount of infromation is lost when passing from the abelian category to the stable category. As an extension of this, the derived noncommutative geometry is just a part of the story of the true noncommutative algebraic geometry (abelian side), which is harder and where the noncommutativity plays a more crucial role than in derived geometry in which there is no essential difference.

It is a pity that you are watering down this truly beuatiful and elegant picture of the derived geometry which you are teaching us, by making a repeated false claim that there is no difference between derived and nonderived noncommutative algebraic geometry, by identifying them in interpretation and behaviour.

Posted by: Zoran Skoda on March 3, 2010 2:13 PM | Permalink | Reply to this

Re: Intrinsic Naturalness

Oh sorry, I missed the comment which I was reading and I thought that Urs has posted a new comment, as I thought I did spell out clearly before precise statement about the distinction. Now I see it is an old comment from Feb 18. I hope(d) we agreed about the derived/nonderived distinction by now.

Posted by: Zoran Skoda on March 3, 2010 2:23 PM | Permalink | Reply to this

Read the post (Infinity, 1)-logic
Weblog: The n-Category Café
Excerpt: In search of the internal logic of a $(\infty, 1)$-topos
Tracked: February 25, 2010 9:39 AM

Read the post In Praise of Dependent Types
Weblog: The n-Category Café
Excerpt: Dependent types should be everyone's friend!
Tracked: March 3, 2010 6:40 PM

Read the post Confessions of a Higher Category Theorist
Weblog: The n-Category Café
Excerpt: Why might we continue to search for more models for higher categories, aside from the simplicial ones that have proven so useful?
Tracked: April 29, 2010 3:04 AM

The n-Category Café

Skip to the Main Content

February 12, 2010