New Hire at UCR

February 27, 2008

Posted by John Baez

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Yay!

Julia Bergner has accepted a tenure-track position in the math department here at U. C. Riverside. She’s just finishing her third year as a postdoc at the University of Kansas. She’s done a lot of important work on the ‘homotopy theory of homotopy theories’.

So, life here at UCR just got a lot more interesting for people who like $n$ -categories.

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What Julia Bergner wisely calls the ‘homotopy theory of homotopy theories’, I prefer to call the $(\infty,1)$ -category of $(\infty,1)$ -categories.

Quite roughly, an $(\infty,1)$ -category is an $\infty$ -category where all the $j$ -morphisms with $j \gt 1$ are invertible. They’re a step in the direction of understanding fully general $\infty$ -categories, but they’re a lot easier to understand, in part because they’re related to topology.

It’s a wonderful fact, not yet fully understood, that $\infty$ -categories where all the $j$ -morphisms are weakly invertible are the same as topological spaces — at least as far as homotopy theory is concerned. So, you can think of an $(\infty,1)$ as an $\infty$ -category such that between any pair of objects we have a space of morphisms. Such things are incredibly common. The most famous is the category of topological spaces! Another is the category of chain complexes. Indeed, in any situation where we can do something like homotopy theory, there’s probably an $(\infty,1)$ -category lurking in the background. That’s why Julie wisely calls an $(\infty,1)$ -category a ‘homotopy theory’.

Julie has been studying ways to make these pretty words precise. There are at least as many ways to define $(\infty,1)$ -categories as to define $\infty$ -categories. In fact, right now there might be more! Here are a few:

Topologically enriched categories. These are categories that have a topological space of morphisms between any two objects.
Simplicially enriched categories. These are categories that have a simplicial set of morphisms between any two objects.
Model categories. These are a standard framework for studying general ‘homotopy theories’. Any model category gives a simplicially enriched category using an old trick called the Dwyer–Kan simplicial localization.
$A_\infty$ -categories. These are things like topologically or simplicially enriched categories, but where composition of morphisms is only associative ‘up to coherent homotopy’.
Quasicategories. These are simplicial sets satisfying a weakened version of the ‘horn-filling condition’ that defines Kan complexes, which themselves describe $\infty$ -categories with all $j$ -morphisms invertible.
Segal categories. A ‘Segal precategory’ is a simplicial space whose space of 0-simplices is discrete. A nice one of these is a Segal category.
Complete Segal spaces. Your eyes are glazing over, so I won’t even attempt to explain these.

We need to relate all these concepts, to keep from going insane! But how?

How could we relate definitions of $\infty$ -category, for starters?

If Fred has his definition of $\infty$ -category, and Ned has his, how can we tell if their definitions are equivalent? We summon a neutral arbiter, say Ted, who has his own definition of $\infty$ -category. Ted must also have defined an $\infty$ -category of his $\infty$ -categories, say $TED$ .

Then we set Ted to work! He must set up a Ted- $\infty$ -category of Fred- $\infty$ -categories, say $FRED$ . He also must set up a Ted- $\infty$ -category of Ned- $\infty$ -categories, $NED$ .

Then,Ted needs to see if $FRED$ is equivalent to $NED$ , viewed as objects in $TED$ .

This is clearly a lot of work, and subject to various pitfalls. But, this is what Julia has successfully been doing — not for $\infty$ -categories, but for $(\infty,1)$ -categories!

She started by making the category of simplicially enriched categories into a model category. Like any model category, this gives a simplicially enriched category. So, there is a (large) simplicially enriched category of (small) simplicially enriched categories — the ‘homotopy theory of homotopy theories’.

Then, she obtained model categories starting from some other definitions of $(\infty,1)$ -category, and showed these were all ‘Quillen equivalent’. Joyal and Tierney have also been doing this sort of thing.

As a result, we now know there are simplicially enriched categories of ‘Segal categories’, ‘complete Segal spaces’, and ‘quasicategories’, all equivalent to the simplicially enriched category of simplicially enriched categories.

She gave a shockingly clear introduction to all this mind-boggling work at the Fields Institute last winter:

You can hear it and read the writeup here:

Julia Bergner, A survey of $(\infty,1)$ -categories. Audio version: Model categories Quillen equivalent to the quasi-category model structure.

She’s also written lots of other interesting papers.

So, all UCR math grad students will soon be experts on $(\infty,1)$ -categories… and we’ll take over the world.

Posted at February 27, 2008 2:52 PM UTC

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34 Comments & 2 Trackbacks

Re: New Hire at UCR

I thought I recognized that name. I saw her talk on “Thirteen ways of looking at a topological group” at the JMM last year in New Orleans. An excellent speaker.

Posted by: John Armstrong on February 27, 2008 4:18 PM | Permalink | Reply to this

Re: New Hire at UCR

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This is all really nice. I’ve always been interested in $\infty$ -categories from afar, but didn’t know it was possible to think about them in any sort of meaningful way without getting covered in opetopes and globules.

I think I recently read somewhere on this blog that, for any $k$ , it’s the same for all $j$ -morphisms to have inverses for $j \gt k$ , as it is for all $j$ -morphisms to have duals for all $j \gt k$ . Is this right? So can $(\infty,1)$ -categories equally be described as $\infty$ -categories with duals for all $j$ -morphisms with $j \gt 1$ ?

I really like this idea of a correspondence between topological spaces and $(\infty,1)$ -categories. Given an $(\infty,1)$ -category, how exactly do you construct a topological space out of it? I should probably just click through some of the links in the post and find out myself… but asking people is easier!

Posted by: Jamie Vicary on February 27, 2008 6:47 PM | Permalink | Reply to this

Re: New Hire at UCR

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Jamie wrote:

I really like this idea of a correspondence between topological spaces and $(\infty,1)$ -categories.

No, no, no! As I said, the correspondence is between topological spaces and $\infty$ -categories where all $j$ -morphisms are invertible, not just those with $j \gt 1$ . These could be called $(\infty,0)$ -categories, but people just call ‘em $\infty$ -groupoids.

Don’t worry, I’m not really upset: I just want this point to be very clear.

So, let’s fix this next question:

Given an $(\infty,0)$ -category, or $\infty$ -groupoid, how exactly do you construct a topological space out of it?

It depends, of course, which setup you used to describe your $\infty$ -groupoid in the first place. The easiest one is the simplicial framework, where an $\infty$ -groupoid is described as a specially nice simplicial set called a ‘Kan complex’. Then we just do a process called ‘geometric realization’, which amounts to staring at a simplicial set and seeing it as a space made of simplices.

For a bit more info, try this. For more detail, try this. For even more detail, try Peter May’s Simplicial Objects in Algebraic Topology.

Posted by: John Baez on February 27, 2008 8:06 PM | Permalink | Reply to this

Re: New Hire at UCR

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I just want this point to be very clear.

It’s now seared into my mind!

Thanks for those links about turning $\infty$ -groupoids into topological spaces. Something that’s a bit weird about this is that we ALREADY have a correspondence (modulo some technicalities) between topological spaces and 1-categories, thanks to the nerve construction! So there should be some sort of correspondence between 1-categories and $\infty$ -groupoids, which is a bit surprising to me. I wonder if there is a nice way to see how this works, at least for simple cases.

Am I right in getting the impression that, especially since Julia Bergner’s recent work, there is now no doubt about what the right notion of $(\infty,1)$ -category is?

Posted by: Jamie Vicary on February 27, 2008 9:49 PM | Permalink | Reply to this

Re: New Hire at UCR

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…we ALREADY have a correspondence (modulo some technicalities) between topological spaces and 1-categories, thanks to the nerve construction! So there should be some sort of correspondence between 1-categories and $\infty$ -groupoids, which is a bit surprising to me.

Here’s how I like to think about it. If I have an $n$ -category, there’s a canonical way to turn it into an $n$ -groupoid: you add inverses to all the morphisms. Of course, when I say “inverses” I mean in the appropriate $n$ -categorical sense: inverses up to equivalence. That means that if I have an $n$ -category, and I regard it as an $m$ -category with only identity $k$ -cells for $n$ < $k\le m$ , and then make it into an $m$ -groupoid, I get a different $m$ -groupoid than I would by first making my $n$ -category into an $n$ -groupoid and then regarding it as an $m$ -groupoid, since the sense of “inverse” is different.

In particular, consider $n=1$ and $m=\infty$ . Taking the nerve of a category should be thought of (I think) as first regarding it as an $\infty$ -category, and then making it into an $\infty$ -groupoid, by adding inverses-up-to-homotopy to all the arrows. Of course what you get is different from the 1-groupoid you’d get by adding strict inverses to all the arrows.

Now it turns out that all $\infty$ -groupoids (that is, spaces) can be constructed from 1-categories in this way. This is the only nontrivial (or at least non-formal) part, and it may or may not be surprising; I’m not sure. But I think this way of thinking about it makes it more clear what exactly the correspondence you refer to is doing.

Posted by: Mike Shulman on February 28, 2008 5:48 AM | Permalink | Reply to this

Re: New Hire at UCR

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Dear Mike,

Thanks very much for that perspective. I certainly am surprised… you seem to be saying that every $\infty$ -groupoid can be freely generated in an appropriate sense from a 1-category! Surely I’ve misunderstood you.

Posted by: Jamie Vicary on February 28, 2008 9:48 AM | Permalink | Reply to this

Re: New Hire at UCR

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Actually, I am saying that. In fact, I’m saying that every $\infty$ -groupoid can be freely generated from a 0-category—by which I mean a poset. The point is that $n$ -categories contain a whole lot more information than $n$ -groupoids—so much so that a 0-category can contain all the information in an $\infty$ -groupoid. For example, a connected 1-groupoid is determined by just a single group, but nothing of the sort can be said for a 1-category.

Maybe it would be fun and illuminating to look in a little more detail about how we can present an $\infty$ -groupoid by a 1-category. The basic idea is that if you have a strictly commuting square of morphisms in an $n$ -category, each of which has an inverse equivalence, then the corresponding square of inverses only commutes up to equivalence. Therefore, if we have a commuting square of 1-cells in a 1-category, when we freely add inverses, a 2-cell equivalence magically pops into being inside the resulting square of inverses. And similarly for higher cells. The trick is then to cook up a 1-category such that when we $\infty$ -groupoidify it, the higher cells that pop into being are just right to generate some given $infty$ -groupoid we started with.

Homotopy theorists have been doing this for a long time, although of course they didn’t use words like “ $\infty$ -groupoid”, and they came up with a clever way to do this all at once. First we make our $\infty$ -groupoid $X$ into a simplicial set. Its 0-simplices are objects of $X$ , its 1-simplices are 1-cells of $X$ , its 2-simplices are quadruples $(f,g,h,\alpha)$ where $\alpha$ is a 2-cell $g f\to h$ , and so on. Then we consider the category of simplices of that simplicial set; call it $S X$ . An object of $S X$ is an $n$ -simplex in $X$ , for some value of $n$ , and a morphism from $\sigma$ to $\tau$ says essentially that $\sigma$ is a face of $\tau$ .

(Note that in a simplicial set, one simplex $x$ can be a face of another simplex $y$ in more than one way; thus $S X$ is really a category rather than a poset. You can massage it some more to make it into a poset, but let’s not bother with that for now. Let’s also not worry about degeneracies.)

Now, we have a (pseudo $\infty$ -)functor from $S X$ to $X$ , which sends each simplex to its last object. Thus it sends a 1-simplex to its target, and a 2-simplex $(f,g,h,\alpha)$ to the common target of $g$ and $h$ . If $f: x\to y$ is a 1-cell in $X$ , then the inclusion $x\to f$ gets sent to $f$ itself, while the inclusion $y\to f$ gets sent to the identity on $y$ . And so on.

So far all of this worked if $X$ were an arbitrary $\infty$ -category, but now suppose it’s an $\infty$ -groupoid. Then the above functor $S X\to X$ factors through the $\infty$ -groupoid reflection of $S X$ , call it $G S X$ , and the claim is that the induced functor $G S X\to X$ is an equivalence.

Now, a functor between $\infty$ -groupoids is an equivalence iff it is essentially surjective on $k$ -cells for all $k$ . Clearly $G S X\to X$ is surjective on 0-cells. For the rest, let’s just look at 1-cells and 2-cells, and just at an easy special case; I think that’s enough to get the idea. First note that if $f: x\to y$ is a 1-cell in $X$ , then we have the zigzag $x \to f \leftarrow y$ in $S X$ . Since $f\leftarrow y$ gets sent to the identity of $y$ , when we invert it in $G S X$ , the composite of $x\to f$ with the inverse of $f\leftarrow y$ will end up mapping to $f$ up to equivalence.

Now consider a 2-cell $\alpha: f \to g : x\to y$ in $X$ . This is the first not-so-obvious case, since $S X$ has no 2-cells itself. $f$ and $g$ are represented in $S X$ by the zigzags $x\to f \leftarrow y$ and $x\to g \leftarrow y$ . Now we have a 2-simplex $\sigma = (f, 1, g, \alpha)$ in $S X$ , with inclusions $f\to \sigma$ and $g\to \sigma$ . Now $x\to f \to \sigma$ is equal to $x\to g \to\sigma$ , and while $y\to f\to \sigma$ isn’t equal to $y\to g\to\sigma$ , they get identified in $G S X$ since they are related by a degeneracy. If you think about that long enough, this implies that when we pass to $G S X$ , a 2-cell appears relating the zigzag $x\to f \leftarrow y$ to $x\to g\leftarrow y$ , and this 2-cell will map onto $\alpha$ in $X$ .

Thus, we can model higher cells in an $\infty$ -groupoid with zigzags and morphisms between zigzags in a 1-category. By the way, if you’ve seen something called the hammock localization, this should look sort of familiar. I’m not sure whether or not that’s helpful, but there it is.

Posted by: Mike Shulman on February 28, 2008 7:19 PM | Permalink | Reply to this

Re: New Hire at UCR

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Thanks for this interesting recipe. One point I’m having trouble with is where you say

Now, a functor between $\infty$ -groupoids is an equivalence iff it is essentially surjective on k-cells for all k. Clearly GSX $\rightarrow$ X is surjective on 0-cells.

OK, clearly it’s surjective on 0-cells, but it seems very far from faithful on equivalence classes of 0-cells. No two objects in GSX will be isomorphic, but it seems to me that different objects can get sent to the same object in X.

I can convince myself that the “fully faithful” bit of the condition for equivalence can be omitted for $n$ -cells for $n \ge 1$ … but I can’t understand how you can omit it for $n=0$ .

Posted by: Jamie Vicary on February 28, 2008 9:32 PM | Permalink | Reply to this

Re: New Hire at UCR

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Actually, one way to define an equivalence of $n$ -categories is as an $n$ -functor that is essentially surjective on $k$ -cells for all $k$ . You just have to remember that $k$ -cells in an $n$ -category for $k=n+1$ are the same as equalities. The point is that injective is the same as full on equalities: if there is an equality $f(x)=f(y)$ , then there is an equality $x=y$ mapping onto it.

For example, a 1-functor is an equivalence iff it is essentially surjective on 0-cells, (essentially) surjective on 1-cells (aka full) and (essentially) surjective on 2-cells (aka full on equalities between 1-cells, aka faithful). More trivially, a function between sets (0-groupoids) is an equivalence (aka bijection) iff it is surjective on objects and surjective on equalities (aka injective).

In general, fullness on $k$ -cells for $k>n$ is what gives you essential injectivity on $n$ -cells. A full and faithful functor is essentially injective on isomorphism classes of objects, and so on.

In this particular case, it’s true that no two objects in $S X$ will be isomorphic (or equivalent), but many of them become equivalent when we pass to $G S X$ , since when we do that every morphism in $S X$ gets turned into an equivalence.

Posted by: Mike Shulman on February 29, 2008 12:31 AM | Permalink | Reply to this

Re: New Hire at UCR

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Jamie wrote:

It’s now seared into my mind!

Good — I can smell the charred brain cells from here.

Something that’s a bit weird about this is that we ALREADY have a correspondence (modulo some technicalities) between topological spaces and 1-categories, thanks to the nerve construction! So there should be some sort of correspondence between 1-categories and $\infty$ -groupoids, which is a bit surprising to me.

Right, I’ve often puzzled over this.

For me the correspondence between spaces and $\infty$ -groupoids, while technically tricky, is morally very simple, since we get spaces with vanishing homotopy groups above the $n$ th level from $n$ -groupoids, and there’s a beautiful way to build up $n$ -groupoids layer by layer which mimics the theory of Postnikov towers — a similar way of building up spaces.

What’s shocking, from this point of view, is that you can get all spaces in one go from mere 1-categories! What’s left for the $n$ -categories to do?

I wonder if there is a nice way to see how this works, at least for simple cases.

Here’s one way to see a bit of what’s going on. Build a space from a bunch of simplices stuck together — technically, a simplicial set.

On the one hand, we can turn this into an $\infty$ -groupoid — technically a Kan complex — by throwing in higher-dimensional simplices ad infinitum, but in a very bland way. A Kan complex is a simplicial set where every ‘horn’ has a ‘filler’. A ‘horn’ looks like a simplex missing one face and also its interior; a ‘filler’ is the obvious way of filling a horn and getting a simplex. So, we just keep filling every horn we see, recursively, until we get a Kan complex. Viewed as a space, this Kan complex is homotopy equivalent to the simplicial set we started with.

On the other hand, we can take the partially ordered set of simplices in our simplicial set, where the order relation is ‘ $x$ is a face of $y$ ’. This partially ordered set is a category, and the nerve of this category is a space. This space is also homotopy equivalent to the space we started with!

Both these constructions can be visualized, with a little practice… and some enlightenment is supposed to ensue.

By the way, this also means that every space is already the nerve of a mere partially ordered set!

Posted by: John Baez on February 28, 2008 12:38 AM | Permalink | Reply to this

Re: New Hire at UCR

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John pointed out that

every space is already the nerve of a mere partially ordered set!

And I bet there are non-equivalent posets which have nerves that are isomorphic as topological spaces, right?

You ask What’s left for the n-categories to do? — but you could wonder this equally for the 1-categories!

Also, I would’ve thought that nonisomorphic topological spaces can give rise to the same $\infty$ -groupoid, since knowing the homotopy groups isn’t enough to define the space — am I right? Or does knowing how they ‘fit together’ into the $\infty$ -groupoid give you enough information?

Posted by: Jamie Vicary on February 28, 2008 10:22 AM | Permalink | Reply to this

Re: New Hire at UCR

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Also, I would’ve thought that nonisomorphic topological spaces can give rise to the same $\infty$ -groupoid, since knowing the homotopy groups isn’t enough to define the space — am I right? Or does knowing how they ‘fit together’ into the $\infty$ -groupoid give you enough information?

You are right. But when we’re doing homotopy theory, which is what we’re doing when we compare spaces to $\infty$ -groupoids, we only consider our spaces up to homotopy equivalence rather than isomorphism. And in fact, we really only consider them up to weak homotopy equivalence—or equivalently, by “space” we really mean (say) “CW complex”.

Posted by: Mike Shulman on February 28, 2008 3:52 PM | Permalink | Reply to this

Re: New Hire at UCR

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Jamie wrote:

And I bet there are non-equivalent posets which have nerves that are isomorphic as topological spaces, right?

Yes:

$\bullet \to \bullet$

and

$\bullet \to \bullet \leftarrow \bullet$

Or: take any piecewise-linear manifold, triangulate it any way you like, take the poset of faces, take its nerve… and you get your manifold back up to homeomorphism.

Also, I would’ve thought that nonisomorphic topological spaces can give rise to the same $\infty$ -groupoid…

That’s true, but…

In everything I’ve been writing in this thread I’m talking about homotopy theory. So, the suitable notion of ‘sameness’ for topological spaces is not homeomorphism but homotopy equivalence. (I’ll assume we’re restricting attention to ‘nice’ spaces like CW complexes; otherwise we need to use weak homotopy equivalence, which I’d rather not discuss now).

This makes perfect sense: we’ve got $(\infty,1)$ -category of nice spaces, continuous maps, homotopies between continuous maps, etcetera. So, we should say two objects in here are ‘the same’ not if they’re isomorphic, but if they’re equivalent.

Similarly, we’ve got an $(\infty,1)$ -category of $\infty$ -groupoids.

And, when I say something like:

It’s a wonderful fact, not yet fully understood, that $\infty$ -categories where all the $j$ -morphisms are weakly invertible are the same as topological spaces — at least as far as homotopy theory is concerned.

I’m asserting the equivalence of these two $(\infty,1)$ -categories: the $(\infty,1)$ -category of nice spaces, and the $(\infty,1)$ -category of $\infty$ -groupoids.

And, this is not just a grand dream: it’s a theorem, if we use the simplicial approach to $\infty$ -groupoids and $(\infty,1)$ -categories.

since knowing the homotopy groups isn’t enough to define the space — am I right?

That’s true but it’s not really relevant to what we’re talking about, which is “in what sense are spaces the same as $\infty$ -groupoids?”

You see, there’s vastly more information in an $\infty$ -groupoid than a mere list of homotopy groups! There’s also all the Postnikov data saying how these groups are stuck together. This information is enough to determine the corresponding nice space up to homotopy equivalence.

(By the way, if you’d prefer to avoid the word ‘nice’, then we need to talk about weak homotopy equivalence, which is what model categories are designed to handle. I’d rather not talk about that stuff now, but in a full version of the story it’s important).

Posted by: John Baez on February 28, 2008 6:01 PM | Permalink | Reply to this

Re: New Hire at UCR

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Dear John and Mike,

Thank you for those comments, I hadn’t realised that it was weak homotopy equivalence and CW complexes that are the buzzwords here, rather than homeomorphism and topological space.

We’ve talked about getting topological spaces from three types of categories: posets, arbitrary 1-categories and $\infty$ -groupoids. But it seems that only the last of these works cleanly enough to get a proper equivalence to a category of topological spaces (in this case, the category of CW complexes and weak homotopy equivalence — or in fact, by Whitehead’s theorem, full homotopy equivalence.)

Posted by: Jamie Vicary on February 28, 2008 9:34 PM | Permalink | Reply to this

Re: New Hire at UCR

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We can actually get a full equivalence of $(\infty,1)$ -categories from 1-categories too, if we set ourselves up correctly. The basic idea is that if I have a 1-category $C$ and a class $W\subset Arr(C)$ of “weak equivalences”, then I can make $C$ into an $(\infty,1)$ -category by making all the maps in $W$ into equivalences. If $W= Arr(C)$ we get the $\infty$ -groupoid I was talking about above. Now, what’s being asserted is that the following $(\infty,1)$ -categories are all equivalent.

$\infty$ -groupoids, $\infty$ -functors, $\infty$ -natural transformations, etc.
CW complexes, continuous maps, homotopies, homotopies between homotopies, etc.
The result of starting with the 1-category of CW complexes and creating an $(\infty,1)$ -category by making the homotopy equivalences into $(\infty,1)$ -categorical equivalences.
The result of starting with the 1-category of all topological spaces and creating an $(\infty,1)$ -category by making the weak homotopy equivalences into $(\infty,1)$ -categorical equivalences.
The result of starting with the 1-category of small 1-categories and creating an $(\infty,1)$ -category by making the functors whose nerves are homotopy equivalences into $(\infty,1)$ -categorical equivalences.

It seems to me that there ought to be a class of maps between posets that you could invert to get the same result too, but as far as I know no one has written this down. Probably the best formal way we have to say all this, as John alluded to, is with the language of model categories.

Posted by: Mike Shulman on February 29, 2008 1:26 AM | Permalink | Reply to this

Re: New Hire at UCR

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An in addition to getting an $\infty$ -groupoid out of a 1-category (say by taking the nerve and applying Kan’s fibrant replacement functor $Ex^\infty$ , giving a Kan complex AKA an $\infty$ -groupoid), given a Kan complex, we can form its 1-category of simplices. This category has its classifying space (weakly) homotopy equivalent to the geometric realisation of the Kan complex we started with!

I think this construction first appeared here:

Chen, Yuh-ching,
Stacks, costacks and axiomatic homology
Trans. Amer. Math. Soc. 145 (1969), pp 105-116

But the proof of the above statements is more likely found here:

Lee, Ming Jung
Homotopy for functors
Proc. Amer. Math. Soc. 36 (1972), pp 571-577

There is also a paper by an R. Fritsch and D. Latch which talks about this category of simplices construction as being a homotopy inverse for the nerve funtor.

(Disclaimer: these are the easiest, earliest references Google Scholar/MathSciNet would give me after minimal effort)

A more up to date reference is linked to below.

Note that the category of simplices of the simplicial set $\Delta$ is the category of simplices $\mathbf{\Delta}$

Posted by: David Roberts on February 29, 2008 12:41 AM | Permalink | Reply to this

Re: New Hire at UCR

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$\infty$ -categories where all the $j$ -morphisms are weakly invertible are the same as topological spaces

Could you remind me what happens to this statement when we restict to $\omega$ -Categories = strict globular infinity-categories?

I would also like to better understand how $\omega Cat$ -enriched categories (using the extended Gray-like tensor product on $\omega Cat$ ) fit into the big picture. Anything known about that?

Maybe to start with: is it known if $\omega Cat$ (the (monoidal) 1-category of strict infinity-categories) supports a model category structure?

Posted by: Urs Schreiber on February 27, 2008 7:08 PM | Permalink | Reply to this

Re: New Hire at UCR

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…is it known if $\omega$ Cat (the (monoidal) 1-category of strict infinity-categories) supports a model category structure?

Are you looking for this?

Posted by: Mike Shulman on February 28, 2008 5:27 AM | Permalink | Reply to this

Re: New Hire at UCR

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…is it known if $\omega Cat$ (the (monoidal) 1-category of strict infinity-categories) supports a model category structure?

Are you looking for this?

Thanks! Cool.

So doesn’t that mean that we are entitled to expect that we can add also the following item to the list of models for $(\infty,1)$ -categories that John gave above (or is it $(\infty,2)$ then?):

$\;\;\;\bullet$ $(\omega Cat, \otimes)$ -enriched categories

(where $\otimes$ is that tensor product on $\omega Cat$ which extends the Gray tensor product on $2 Cat$ ).

That would be some kind of “strictification” for $(\infty,1)$ -categories. For instance if we truncate at $n=3$ we know something like this is right: every tricategory is equivalent to a Gray category, which is a special case of an $(\omega Cat, \otimes)$ -enriched category (essentially, I think).

Well, what I am really after is the answer to this question:

In what I am doing it happens that it would be particularly convenient if I could restrict attention to $(\omega Cat,\otimes)$ -enriched categories. What I want to know is: am I on the safe side when adopting that restriction? Meaning: will that restriction mean that I am losing lots of information that would be available had I allowed another model for $\infty$ -categories?

Sorry for the vague kind of question. But I guess it should be clear what I mean.

Posted by: Urs Schreiber on February 28, 2008 10:15 AM | Permalink | Reply to this

Re: New Hire at UCR

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Well, first of all, to get $(\infty,1)$ -categories you probably want to look at categories enriched in $\infty$ -groupoids, rather than $\infty$ -categories. And I think the answer is that you’ll be losing a lot of information. It is true that weak 3-categories can be strictified to categories enriched over 2-categories with the Gray tensor product (aka Gray-categories), but one of the first things you do in proving that is apply `homwise’ the lower-dimensional fact that weak 2-categories can be strictified to fully strict ones. If you try to do that for weak 4-categories, the most you can expect is to get a category enriched over `semistrict’ 3-categories (that is, Gray-categories), not a category enriched over strict 3-categories. So I see no reason to expect that you could get everything with categories enriched in strict $\infty$ -categories rather than `semistrict’ ones (whatever that might mean).

Here’s another way of looking at it. In an $n$ -category, you can compose things along a $k$ -cell for any $0\le k < n$ . The composition along $(n-1)$ -cells is always strict (since there’s no room to weaken it), and the composition along $(n-2)$ -cells can be fully strictified (this is coherence for bicategories). The reason weak 3-categories can be strictified to Gray-categories is that after $n-1=2$ and $n-2=1$ , there’s only composition along 0-cells left, and the Gray tensor product builds enough weakness into that. But in a category enriched over strict $\infty$ -categories with any sort of tensor product, only the composition along 0-cells will be at all weak; all the others will inherit strictness from the hom strict $\infty$ -categories.

Posted by: Mike Shulman on February 28, 2008 3:46 PM | Permalink | Reply to this

Re: New Hire at UCR

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If you try to do that for weak 4-categories, the most you can expect is to get a category enriched over ‘semistrict’ 3-categories (that is, Gray-categories)

Yes, I suppose that is the reason why Sjoerd Crans, after having dealt with the monoidal structure on $\omega Cat$ , went on to think about tensor products for Gray categories.

It’s conceivable that one could keep going this way, iteratively, to ever higher $n$ – but intimidating.

So I thought: what if $\omega Cat$ had already a model category structure? Then enriching over it should be much like enriching simplicially or topologically. So maybe there is no need to follow that intimidating route.

But as you remark, a necessary prerequisite for that to have any chance at all is that the tensor product on $\omega Cat$ respect the model category structure on it.

If anyone runs into any information on that, please let me know.

Posted by: Urs Schreiber on February 28, 2008 4:02 PM | Permalink | Reply to this

Re: New Hire at UCR

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I realize that my insistence on trying to understand how far $(\omega Cat,\otimes)-Cat$ goes along the way of general $\infty$ -Cat is based on an assumption which might be wrong:

I think I assumed that $(\omega Cat,\otimes)-Cat$ is itself an $(\omega Cat)$ -category, or maybe CATegory. Is that actually right?

I went back to section 2 of Kelly’s book to find out, but had to realize that I am not up to it.

In particular, if the following isn’t true, then I’ll give up this idea right away:

is it true for $\mathbf{B} G$ an $\omega Cat$ -enriched one-object groupoid, that

$[\mathbf{B}G, \mathbf{B} G]$

is an $\omega Cat$ -category?

(Notation here supposed to be the standard one. I am looking at p. 29 of Kelly’s book.)

That’s my motivation for considering $\omega Cat$ -categories, since I need a setup where for every $\infty$ -group $G$ also $AUT(G)$ is an $\infty$ -group.

Posted by: Urs Schreiber on March 3, 2008 3:32 PM | Permalink | Reply to this

Re: New Hire at UCR

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Wait, what am I saying. It is just about smallness of the objects of $\mathbf{B} G$ , but there is just a single object.

Posted by: Urs Schreiber on March 3, 2008 3:40 PM | Permalink | Reply to this

Re: New Hire at UCR

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Another good question is whether that model structure on strict $\infty$ -categories is compatible with that tensor product. I don’t know enough about either to answer it.

Posted by: Mike Shulman on February 28, 2008 3:47 PM | Permalink | Reply to this

Re: New Hire at UCR

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Following up on my comment on $(\omega Cat, \otimes)$ -enriched categories:

in our discussions Bruce kept pointing out that he finds it remarkable (and I agree that there is a point there) that underlying a Lie $\infty$ -algebra is a strict infinity-category $g$ internal to vector spaces. The entire weakening is in the bracket

$[\cdot,\cdot] : g \times g \to g \,.$

Recalling that we have to think of $g$ here as the hom-space on a single object (the Lie $\infty$ -algebra being a one-object Lie $\infty$ -algebroid) we find a picture quite alike that of $\omega Cat$ -enriched categories:

the Hom-spaces are strict infinity-categories and all the weakening is in the horizontal composition.

That kind of observation makes me think that there are chances that we can obtain an $\infty$ -Lie theorem closely relating smooth $\omega Cat$ -enriched groupoids with Lie $\infty$ -algebras.

Posted by: Urs Schreiber on February 28, 2008 3:51 PM | Permalink | Reply to this

Re: New Hire at UCR

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I’m glad to hear this news!

In case your eyes don’t glaze when you hear the words “complete Segal space”, let me take a stab at a brief explanation/motivation. (I’m fond of these gadgets, and think they should be better known (I’m biased).)

The slogan is that it is a very bad idea to think about the “set of objects” of a category. If $C$ and $D$ are two categories, it is possible for them to be equivalent, yet have non-isomorphic sets of objects. “Set of objects” is not an interesting invariant of categories! Instead of using sets of objects, you should think about the “moduli space” of a category, which does turn out to be an invariant of the category.

Let my try to explain how you get a complete Segal space from an $(\infty,1)$ category, by means of a moduli space construction.

You have an $(\infty,1)$ category, call it $C$ . (Use any model for $(\infty,1)$ categories you like.) Inside $C$ is a maximal infinity groupoid, which I will call $G(C)$ . The idea is that $G(C)$ has all the objects of $C$ , and all the “invertible” 1-morphisms, and all the 2-morphisms between invertible 1-morphisms, etc.

Now take the “functor” $B$ which realizes the equivalence between infinity groupoids and spaces. $B(G(C))$ is thus a space, which deserves to be called the moduli space of $C$ . I’ll call it $M(C)$ for short; it is a space which “parameterizes the objects of $C$ ”. In the case when $C$ is an ordinary $(1,1)$ -category, then $M(C)$ is not the set of objects of $C$ , but rather a space whose set of path components correspond to isomorphism classes of objects in $C$ ; each path component has a possibly non-trivial homotopy type (Exercise: what is this homotopy type?)

If $D$ is another $(\infty,1)$ category, then you can consider $F(D,C)$ , by which I mean the $(\infty, 1)$ category of functors from $D$ to $C$ . Looking at the moduli spaces of the functor categories, you get a contravariant functor $P: [(\infty,1)-Cat]^{op} \to Spaces$ given by sending $D$ to $M(F(D,C))$ .

If you can believe all this, you can probably believe that it should be possible to recover $C$ from the functor $P$ . In fact, it turns out that to recover $C$ , you only need the following information: the spaces $P([m])$ and the maps $P([n]) \to P([m])$ , where we range over the sequential arrow categories and functors between them (i.e., $[m]$ is the category associated to the poset 0<1<…<m).

That is, restricting $P$ to these produces a simplicial space, which I’ll still call $P$ . This simplicial space contains all the information needed to reconstruct $C$ , and furthermore, it is an invariant of $C$ ; equivalent $(\infty,1)$ categories have homotopy equivalent moduli spaces, and in fact have homotopy equivalent simplicial spaces.

As you think about this, you realize that not every simplicial space can arise in this way. The ones which do are the complete Segal spaces (whoever came up with that crappy name should be shot.) Julie proved in her thesis that what I just said actually works as advertised; i.e, that complete Segal spaces model $(\infty,1)$ -categories.

Clark Barwick has (not published) a model for $(\infty,n)$ -categories along the same lines.

Posted by: Charles on February 27, 2008 11:31 PM | Permalink | Reply to this

Re: New Hire at UCR

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Charles wrote:

n case your eyes don’t glaze when you hear the words “complete Segal space”, let me take a stab at a brief explanation/motivation.

Great! I felt guilty fizzling out right when I got to these, but you did a much better job of explaining them than I ever could, so now I’m glad. I learned a lot from what you just wrote, and I’ll learn even more if I read it a few more times.

As you think about this, you realize that not every simplicial space can arise in this way. The ones which do are the complete Segal spaces (whoever came up with that crappy name should be shot.)

Okay — I’ll get a hit man to track you down. With all those pictures of you on the web, you should be easy to find.

Seriously, if this blog entry hadn’t been all about Julie, I would have surely referred to your paper on complete Segal spaces:

Charles Rezk, A model for the homotopy theory of homotopy theory.

Posted by: John Baez on February 28, 2008 12:20 AM | Permalink | Reply to this

Re: New Hire at UCR

Morally the n-cats are there to generate quantum spaces, and QG higher hbar spaces. The T duality of 1 and oo is a nice piece if holography.

Posted by: Kea on February 28, 2008 7:36 AM | Permalink | Reply to this

Re: New Hire at UCR

I tried to update the wiki, but got flummoxed by the wiki-html so will leave the job to some expert.

Posted by: Allen Knutson on February 29, 2008 7:27 AM | Permalink | Reply to this

Read the post Infinity-Groups with Specified Composition
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Re: New Hire at UCR

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Bruce Bartlett and I were talking to each other yesterday about something related to this thread. There are lots of examples of sequences of higher categories with duals that we can construct in principle: for example, $n\!\mathrm{Cob}$ or $n\!\mathrm{Hilb}$ for each natural number $n$ . Assuming we can take the limit $n \to \infty$ in a well-defined way, we then obtain $\infty$ -categories with duals. But if every $\infty$ -category with duals is an $\infty$ -groupoid, then $\infty \mathrm{Cob}$ and $\infty \mathrm{Hilb}$ are $\infty$ -groupoids, and therefore topological spaces!!

Does this make sense? What topological spaces do these $\infty$ -categories give rise to?

Posted by: Jamie Vicary on May 15, 2008 4:05 PM | Permalink | Reply to this

Re: New Hire at UCR

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Jamie wrote:

Does this make sense?

Yes.

What topological spaces do these $\infty$ -categories give rise to?

For $n\Cob$ I’ve thought about this a lot; for $n\Hilb$ , not at all.

If you look at the end of section 7 in the paper Higher-dimensional algebra and topological quantum field theory, you’ll see that Jim and I compared $\infty\Cob$ and the infinite loop space $\Omega^\infty S^\infty$ . We argued that the former was the free stable $\infty$ -category with duals on one object, while the latter was the free stable $\infty$ -groupoid on one object. So, we expected a map

$\infty\Cob \to \Omega^\infty S^\infty$

which promoted the duals to weak inverses.

Only later did we start to suspect that $\infty$ -categories with duals are the same as $\infty$ -groupoids. If this is true, we should have an equivalence

$\infty\Cob \simeq \Omega^\infty S^\infty$

This makes tons of sense, since $\infty\Cob$ means the $\infty$ -category of framed cobordisms, and $\Omega^\infty S^\infty$ is the classifying space for the generalized cohomology theory called ‘framed cobordism theory’ (also known as ‘stable homotopy theory’).

(I say this ‘makes tons of sense’ — you may not agree. The point is, I’m using a lot of jargon, but once you penetrate it you’ll see there’s a simple idea here.)

If this is how the story goes, we can instantly relate many other kinds of cobordism $\infty$ -categories to other spaces! You see, there are lots of ‘cobordism theories’: generalized cohomology theories associated to cobordisms with various sorts of extra structure. Oriented cobordisms, complex cobordisms, etc.

You can see an enormous list of examples here. All this could be ours!

The classifying spaces of these cobordism theories can all be constructed in a systematic way — they’re Thom spaces. In particular, they are all infinite loop spaces — in other words, stable $\infty$ -groupoids. But if stable $\infty$ -groupoids are stable $\infty$ -categories with duals, these spaces secretly are those $\infty$ -categories of cobordisms equipped with extra structure!!!

All this stuff is now a special case of a bigger hypothesis relating $n$ -categories with duals to stratified spaces. We’ve talked about that hypothesis here before.

Now you’ve convinced me that there should be a generalized cohomology theory associated to $\infty Hilb$ . It’s probably something familiar. Maybe complex $K$ -theory.

Posted by: John Baez on May 15, 2008 6:25 PM | Permalink | Reply to this

Re: New Hire at UCR

All this could be ours!

That’s a great turn of phrase John! My uncle Screwtape would have been proud.

Posted by: Wormwood on May 15, 2008 7:54 PM | Permalink | Reply to this

Re: New Hire at UCR

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John said:

$\Omega ^\infty S ^\infty$ is the classifying space for the generalized cohomology theory called ‘framed cobordism theory’ (also known as ‘stable homotopy theory’).

I know what the classifying space of a category is — but what’s the classifying space of a theory?

It will take me a while to appreciate everything you’ve written in your post, but I’ll try my best!

Posted by: Jamie Vicary on May 18, 2008 8:51 PM | Permalink | Reply to this

Re: New Hire at UCR

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Jamie wrote:

I know what the classifying space of a category is — but what’s the classifying space of a theory?

You really want to know what’s the classifying space of a generalized cohomology theory. Very briefly: there are lots of things called generalized cohomology theories. Each one gives you a list of abelian groups $h^n(X)$ for any topological space $X$ . The most famous is plain old-fashioned cohomology $H^n(X,\mathbb{Z})$ , but there are millions more.

For any one of these, you can find a list of spaces $E(n)$ such that

$h^n(X) = [X, E(n)]$

These spaces $E(n)$ are what I was rather fuzzily calling ‘the classifying space’ of the generalized cohomology theory. Each one deserves to be called a classifying space, because of the equation above — but taken together, they form a ‘spectrum’.

Here’s why we call them a ‘spectrum’: the space of based loops in $E(n)$ is $E(n-1)$ . So, they’re all part of a tightly linked structure.

When I said $\Omega^\infty S^\infty$ was the classifying space for framed cobordism theory, I really meant that it was the $E(0)$ for this generalized cohomology theory.

The reason spaces like these $E(n)$ ’s are so important for us is that they’re infinite loop spaces, which are secretly the same as ‘stable $\infty$ -groupoids’. $\Omega^\infty S^\infty$ is especially important because it’s the free stable $\infty$ -groupoid on one generator — in other words, the ‘true integers’, of which the usual set of integers is a shadow.

I’m trying to be very quick and sketchy here. I gave a much more detailed introduction to generalized cohomology theories and spectra in week149. For the relation to $\infty$ -groupoids, then try this.

Posted by: John Baez on May 19, 2008 4:20 AM | Permalink | Reply to this

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February 27, 2008

New Hire at UCR

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