The n-Category Café

do you think that there might be any interesting sense in which richard garner’s example of v = _cocomplete category_ might be the “universal” way to provide adjoints for pro-v-functors if the “adjoints” are allowed to be of a certain “weak” (maybe “lax” in some sense, or maybe just “2-weak”?) kind, while v = _spectrum_ might be the universal way if they’re allowed to be weak in a certain different way (maybe “(infinity,1)-weak” or something?)?

this is just a vague silly guess…

do the adjoints for pro-v-functors where v = _cocomplete poset_ enjoy some stronger formal property than the ones for where v = _cocomplete category_?

i still haven’t really understood these other two examples (v = _cocomplete poset_ and v = _cocomplete category_) in any detail yet so i could be off on the wrong track here.

Those are interesting questions; I don’t know the answers right now. But you are right, I think, that the “adjoints” in the cocomplete-category case are kind of lax. I was thinking last night about how you would construct that adjunction, and it seemed to me that you would need both (a) morphisms of 2-profunctors to be lax natural transformations, and (b) the “tensor product” of 2-profunctors to be a lax colimit. I’m not sure in what formal context one could make sense of that (e.g. is “composition” of 2-profunctors via lax colimits associative, unital, and functorial on lax natural transformations?).

i haven’t yet been able to understand what you wrote well enough for me to answer your question. which isn’t surprising; i often have lots of trouble understanding mathematical writing.

to try to see whether we can figure out what each other is talking about, it might be a good idea to mostly stay away from higher categories at first. in the paragraph of mine that you quoted, i’m pretty much only talking about a single groupoid-enriched category of structured 1-categories.

(i need a convenient short term here for “groupoid-enriched category” so i’ll use “g-category” for this purpose.)

the g-category that i’m talking about is the g-category of symmetric monoidal finitely cocomplete small k-algebroids over some fixed commutative ring k; are you reasonably satisfied for the moment that we both agree what that means?

(i need a convenient short name for this g-category so for the moment i’ll call it “x”.)

one of my main reasons for being interested in x is that it has a substantial overlap with certain g-categories that algebraic geometers study, such as the g-category of “deligne-mumford stacks”. (that is, there are pretty big full sub-g-categories of x that are equivalent to pretty big full sub-g-categories of those other g-categories.)

however, i’m not too interested in the fact that x-objects can be viewed as “stacks” (in the sense of “categorified sheaves”) over some “site”. for one thing, the objects of _any_ g-category can be viewed as stacks over some (higher) site, by turning the yoneda embedding crank. thus, viewing objects of a g-category as stacks of some sort is something that you might resort to if you can’t find any better way of thinking about those objects. but there’s already a very good way of thinking about what x-objects are: they’re symmetric monoidal finitely cocomplete algebroids.

similarly, the objects of any category can be viewed as sheaves over some site by turning the yoneda embedding crank, and thus viewing objects of some category as sheaves of some sort is something that you might resort to if you can’t find any better way of thinking about those objects. thus for example “coherent sheaves” can indeed be thought of as sheaves of a sort but in general there are much better ways of thinking about them.

so even though a lot of the things that i’m studying have traditional names (like “coherent sheaf” or “deligne-mumford stack”) that suggest a connection to the study of sites and sheaves and stacks and so forth, those names are somewhat misleading, mostly just unfortunate historical accidents. i think that this is one of the things that’s making it difficult for me to connect what you wrote (which seems to focus a lot on things like sites and sheaves and stacks and so forth) with what i was talking about. the connection might very well be there; i just didn’t manage to find it yet.

so, one of the ideas that i was talking about is the idea that under good conditions, an “algebraic stack” (such as a “deligne-mumford stack”) can be recovered from its symmetric monoidal finitely cocomplete algebroid of coherent sheaves. this is a very standard idea; for example, in the recent thread here on the “3-fold way”, theo alluded to the fact that the “two point” stack and the “half a point” stack share the same finitely cocomplete algebroid of coherent sheaves, but are distinguished from each other by having different symmetric monoidal structures. being able to recover a stack from an x-object in this way makes the actual stack (and the site where it lives, and so forth) superfluous if we so choose, and i personally often do so choose.

to try to give an idea of how i’m suggesting to think about coherent sheaves instead of thinking of them as sheaves of some sort, it might be helpful to recall a thread here some time ago where tom leinster suggested that the appreciation and exploitation of universal properties of structured categories constitutes a shibboleth that marks a “professional” category-theorist; that although the idea of a universal property has become the property of the wider math community, the special case where the object possessing the universal property is itself a category of some sort has not. in that thread, mike shulman mentioned the relationship of such “categorified universal properties” to the ideas about the use of doctrines in algebraic geometry that i’ve been exploring. but it’s striking that category-theorists aren’t the only mathematical subculture who’ve made an exclusionary shibboleth out of the concept of “categorified universal property”- algebraic geometers speaking their own mathematical dialect have done so too. such a double shibboleth dividing two subcultures constitutes a failure to communicate, an unexploited opportunity to combine the insights of both cultures.

the example that i mentioned of the coherent sheaves over projective n-space as “the free symmetric monoidal finitely cocomplete algebroid on a one-dimensional object equipped with a linear embedding into standard linear [n+1]-space” is an excellent example to work through in detail if you haven’t already done so. algebraic geometers sometimes describe this categorified universal property by saying that projective n-space is “the classifying space for line bundles generated by n+1 sections”.

in trying to understand this categorified universal property, it’s probably more of a hindrance than a help to think of coherent sheaves as sheaves over some site. instead, there’s a way, described in lots of basic algebraic geometry books, to think of coherent sheaves over a projective variety as special graded modules of the “homogeneous coordinate algebra” of the variety (which in the example of projective n-space is just polynomials in n+1 variables). the categorified universal property can be understood very directly from this viewpoint.

category-theorists are perhaps familiar with exploiting the idea of categorified universal properties in the context of “tannaka-krein” theorems, about recovering a group from its representation category; for example, the idea that the finite-dimensional representations of gl(n) form the free symmetric monoidal finitely cocomplete algebroid on one “n-dimensional” object. my impression though is that they’re less familiar with exploiting such categorified universal properties in the context of “gabriel-rosenberg” theorems, a vast generalization of tannaka-krein theorems, about recovering an algebraic stack from its coherent sheaf category. my sense of this estrangement between category-theorists and algebraic geometry is based largely on my own personal experiences, my failure for such a long time to understand what algebraic geometers were saying. i don’t really know for sure whether it’s mainly just me who experienced this estrangement, or whether other category-theorists have had a similar experience.

Thanks for the reply.

I will try to answer, but not in detail right now. (It’s 2:30 am, I have to get up early, teach a seminar, then run to catch a train to berlin where I visit my brother over the weekend. So I might be offline for a bit.)

Just a few quick remarks.

(i need a convenient short term here for “groupoid-enriched category” so i’ll use “g-category” for this purpose.)

There is by now a fairly “established” term for this: “$(2,1)$-category”.

the g-category that i’m talking about is the g-category of symmetric monoidal finitely cocomplete small k-algebroids over some fixed commutative ring k; are you reasonably satisfied for the moment that we both agree what that means?

Yes, I know exactly what you mean.

(i need a convenient short name for this g-category so for the moment i’ll call it “$x$”.)

I like descriptive terms. It’s like putting books into the shelf in the right order: it facilitates retrieving information without losing time on inessential searches.

Therefore I’d tend to suggest that we call this $(2,1)$-category by a different name, eventually. Something like $2Alg_k$ for instance. This is still not entirely descriptive, as there are various different concepts that can all justly be called a “2-algebra over $k$”, but this here is one of them.

At the same time, trying to agree on such a more or less descriptive term may help us get on the same page and find out if we agree on what we are talking about.

however, i’m not too interested in the fact that x-objects can be viewed as “stacks” (in the sense of “categorified sheaves”) over some “site”. for one thing, the objects of any g-category can be viewed as stacks over some (higher) site, by turning the yoneda embedding crank. thus, viewing objects of a g-category as stacks of some sort is something that you might resort to if you can’t find any better way of thinking about those objects. but there’s already a very good way of thinking about what x-objects are: they’re symmetric monoidal finitely cocomplete algebroids.

Sure. The point of identifying something as a stack on something, or better as an object in some $(2,1)$-topos, is not that it can be done at all. The point of it is that is specifies an ambient context in which to reason about the objects under consideration. This becomes relevant as soon as we start considering certain operations on which our objects may not be closed.

There is a differential geometric analog to all that we are discussing here, comparison to which might be useful:

we can talk about orbifolds without thinking of them as stacks on anything. We can think about effective Lie groupoids without thinking of them as stacks on anyting. We can think about general Lie groupoids without thinking of them as stacks on anything…. But at some point, when working with these objects, we will notice that we tend to drop out of the $(2,1)$-categories that they form. At that point, we may feel the need to pass to larger $(2,1)$-categories that contains all them in a useful way. Experience (and Lawvere) indicate that this large ambient $(2,1)$-category should always be taken to be some $(2,1)$-topos. And that will typically have some site of definition, even if we may not care about it.

so, one of the ideas that i was talking about is the idea that under good conditions, an “algebraic stack” (such as a “deligne-mumford stack”) can be recovered from its symmetric monoidal finitely cocomplete algebroid of coherent sheaves. this is a very standard idea;

Yes, as I mentioned in the other thread: this is the idea that notably Kontsevich has been following for more than two decades: describe all things geometric by their collections of quasicoherent sheaves. (Which ultimately always form a stable $(\infty,1)$-category, which Kontsevich models as a linear $A_\infty$- or – once rectified/semi-strictified – as a dg-category).

[…] shibboleth […]

I am not a big fan of the shibboleth story. But I strongly agree that it is most curious that after Grothendieck there was this schism in the community, and I am all in favor of discussion that aims to bring together again what belongs together.

a vast generalization of tannaka-krein theorems, about recovering an algebraic stack from its coherent sheaf category.

By the way, there is a nice account of this in

Jacob Lurie, Tannaka duality for geometric stacks (pdf)

It also announces a followup article where the analogue is dicussed for $\infty$-stacks. But if this followup has already been made available, then I am not aware of it.

my sense of this estrangement between category-theorists and algebraic geometry is based largely on my own personal experiences, my failure for such a long time to understand what algebraic geometers were saying. i don’t really know for sure whether it’s mainly just me who experienced this estrangement, or whether other category-theorists have had a similar experience.

I have the same impression. I was once talking to somebody who made his career in algebraic geometry, and who told me that he just has no chance of understanding synthetic differential geometry. How curious that is: he has been spending his whole academic life inside a topos that models sdg!

What I find most curious also is the widespread habit in algebraic geometry to happily use plenty of general abstract constructions, but at the same time always pick one specific site (or two), without checking much what of the constructions really depend on the choice of that site. I guess that’s also one of the points you have been making here, maybe.

Recently I supervised a master-student who had some fun with showing (here) that a good deal (the higher $(\mathcal{O} \dashv Spec)$-Isbell duality adjunction) of the story of (derived/higher) algebraic geometry in fact works with the theory of ordinary algebras replaced by any abelian Lawvere theory.