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I've made this a new question, rather than expanding the first one.

Torsten gives a good answer, and it partially illustrates in practice the 'second approach' I outlined in my other question. (You don't need to know the fiddly details of my other question, but you are welcome to look at it).

I'm well aware of the machinery of stacks of categories, a la Giraud, but what I want to know is: why can't these stacks be given some geometry? And not just 'because we don't'. Here's a sketch of how it might work:

Consider the 2-category of stacks of categories over some site $S$, considered as functors $X \to S$. Call a stack representable if it is representable in the usual way (equivalent to $S/a$ for some $a\in S$). Call a map of stacks $p:X\to Y$ lax-representable if for every representable stack $a$ and map $f:a \to Y$ the comma object $(f/p)$ is representable (comma objects coincide the the usual 2-pullback one uses with stacks in groupoids when all the categories involved are groupoids).

A comma object fits into a 2-commuting square, but the 2-arrow filling the square isn't necessarily invertible. We then say as per usual that a lax-representable map has property P if all the projections $(f/p) \to a$ have property P for representable $a$. Then one says a stack in categories $X$ is geometric if given a notion of cover (like smooth maps of schemes) if it admits a lax-representable cover $j:u\to X$ by a representable stack $u$ (and the diagonal $X\to X\times X$ should be lax-representable as well). Then the comma object $(j/j)$ together with $u$ should form an internal category $(j/j) \rightrightarrows u$ in $S$.

This is all well and good, but what stops this happening in practice? Perhaps lax-representable maps just do not exist? Or not enough of them? One potential barrier, in the algebraic case is that the collection of not-necessarily-invertible maps between two objects might be too big to the algebraic. Consider the automorphisms of your favourite algebro-geometric object, such as a vector bundle, or algebraic curve. Such things can again be formed into geometric objects in nice cases. But the categories of such things may not be cartesian closed. I haven't thought too much about this, but it seems to be a possible barrier.

To be concrete, my question is this:

What goes wrong/can go wrong with the above recipe? If you like, consider a concrete example, like the stack of quasicoherent sheaves, or the stack of vector bundles (with all morphisms of vector bundles).

One more thing... to make this theory analogous to that of ordinary geometric stacks (in groupoids), one thing that is lacking is the notion of stackification. The stackification $st(G)$ of a algebraic groupoid $G$ is essentially the category of torsors for that groupoid. Then $G_0$ (the objects of $G$) comes with a (fairly) canonical map $G_0 \to st(G)$ which is the presentation described above.

The problem is how one wants to stackify. What is a torsor for an algebraic category $C$? If one is thinking in terms of descent, it might be a torsor for the maximal algebraic groupoid contained in $C$. The hint I am getting is that this might be the right choice, but it really depends on coming up with working examples, as in my question.

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What does "the diagonal should be lax-representable" mean in this context? Do we use the lax fiber product of $U\to X \leftarrow U$? –  Charles Rezk Mar 2 '11 at 0:50
@Charles: yes, I believe that is what David means: the Hom's in X should be varieties of some kind. @David: Maybe a good starting point would be to answer that question: What is a function on a ``categorical algebraic stack''? If you want these to be considered as geometric objects, I believe you will need to know what are (analogues of) functions, differential forms and so on. Otherwise, I agree with you that a significant part of the machinery, translated in the lax language, seems to work well enough. –  Denis-Charles Cisinski Mar 2 '11 at 1:32
@Charles - cleared up the bit about the diagonal. @Denis-Charles - good point. One could argue (analogously to differentiable stacks) that differential forms on one of these beasts should be given by the total complex of the usual double complex arising from applying $\Omega^\ast$ to the nerve of the category object $(j/j) \rightrightarrows u$ in $Sch$ (or even in some other base site for which differential forms are defined), but I'm not sure this is right. In particular, consider the case of an algebraic monoid acting smoothly on a scheme. Is a function supposed to be equivariant wrt the... –  David Roberts Mar 2 '11 at 5:48
(cont.)... action of the invertible elements, or 'equivariant' wrt the full action? I think examples in the wild should determine this, but I am not an algebraic geometer, hence asking here. –  David Roberts Mar 2 '11 at 5:50
Why should "the diagonal $X\to X\times X$ is lax-representable" imply that "the comma object $(j/j)$ is representable whenever $u$ is representable"? That works for groupoid-stacks, but it's not so clear for category-stacks. –  Charles Rezk Mar 2 '11 at 12:57
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2 Answers

up vote 5 down vote accepted

I think the right thing to do with category-valued stacks is to keep the notion of "representability" the same (that is, use (pseudo) 2-pullbacks rather than comma objects), but to replace representability of the diagonal by representability of $X^2 \to X\times X$, where $X^2$ is the power (cotensor) of X by the free-living arrow (which is equivalent to X in case X is groupoid-valued). This does imply that $(j/j)$ is representable whenever the domain of $j$ is so, since $(j/j)$ is the pullback of $X^2 \to X\times X$ along $j\times j$.

One reason I think this is the right thing is that in the well-developed theory of "indexed categories" over a topos S regarded as "large categories relative to S as a universe of sets", representability of $X^2 \to X\times X$ is equivalent to the standard notion of "local smallness". (More generally, various kinds of "comprehensibility" for indexed categories can be rephrased as the representability of certain functors in the above sense.) Therefore, the resulting notion of "geometricity" would coincide with "essential smallness", as one would expect.

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Nice. I knew what local smallness was meant to mean, but the definition (iirc) was confusing to me at the time. –  David Roberts Oct 6 '11 at 4:03
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I don't have an answer to this question. Instead, I'll propose a different kind of recipe.

It's well-known that if you have a category $C$, you can extract from it a simplicial groupoid $G_\bullet=(G_0\leftleftarrows G_1\cdots)$, where $G_n=\mathrm{gpd}(\mathrm{Func}([n],C))$, where $[n]=(0\to1\to\cdots\to n)$, and "$\mathrm{gpd}(D)$" is the maximal subgroupoid of a category $D$. You can recover $C$ (up to equivalence) from $G_\bullet$.

Likewise, given a category-stack $\mathcal{N}$ on some site, you can similarly extract a simplical groupoid-stack $\mathcal{M}_\bullet$, which contains all the information about $\mathcal{N}$.

The proposal is that "geometric" structure about $\mathcal{N}$ should be encoded in terms of geometric structure of the groupoid-stacks $\mathcal{M}_n$, and/or the maps $\mathcal{M}_m\to \mathcal{M}_n$ in the simplicial diagram. Of course, there is still a lot of freedom as to what "geometric" can mean.

Here's an example:

Let $\mathcal{N}$ be the category-stack over schemes, which represents elliptic curves and isogenies. There is an associated simplicial groupoid-stack $\mathcal{M}_\bullet$. The stack $\mathcal{M}_0$ is just the moduli stack of smooth elliptic curves. The stack $\mathcal{M}_1$ classifies isogenies, i.e., objects are maps $f:E\to E'$ of elliptic curves over a base $S$, such that $f$ is finite and flat, and morphisms are isomorphisms of the data. $\mathcal{M}_2$ is the moduli stack of composable sequences $E\to E'\to E''$ of isogenies, etc.

I am not an algebraic geometer, so I'm not entirely confident I have the following right. But I believe that

  1. each stack $\mathcal{M}_n$ is a Deligne-Mumford stack,
  2. each of the simplicial maps $\mathcal{M}_m\to \mathcal{M}_n$ is representable.

So 1. is a kind of geometric condition on $\mathcal{N}$. I'm not sure what geometric conditions (if any) should be required for the simplicial maps. In this example, the map $\mathcal{M}_0\to \mathcal{M}_1$ is an open/closed immersion; the two maps $\mathcal{M}_1\to \mathcal{M}_0$ are flat (but not etale, unless we restict to schemes in characteristic $0$). The "composition" map $\mathcal{M}_2\to \mathcal{M}_1$ is not even flat.

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