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Stacks qua moduli spaces were introduced to keep track of nontrivial automorphisms of the objects they parameterize. In essence they are groupoids of objects with some form geometric cohesion. The classic example is principal bundles/torsors, the whole category of which is actually a groupoid. But what about objects one might want to parameterise which have non-invertible maps between them, such as vector bundles, or coherent sheaves? One could imagine a stack of such objects, because they 'glue' as principal bundles do, but if one keeps track of all maps this thing should be a category, not a groupoid. It is certainly deserving of being promoted to something geometric, and so one could present it by a category in algebraic spaces or schemes, much as the stacks we are more familiar with are presented by algebraic groupoids.

Some might argue that we have classifying topoi or similar for these situations, and this is well and good, but what about some geometry on these topoi? I know of two different takes on classifying topoi of a small (internal) category, one approach involving flat functors and the other torsors for the groupoid of all invertible arrows of the category in question. Between these two competing definitions, there are arguments (at least in my own head, too fuzzy to unveil here) both ways, and concrete examples of where one uses stacks of categories in a geometric context would certainly push the balance in one direction or another. This isn't the only reason I would like to know an answer to this question, but it has some bearing - and I've rabbited on long enough.

Question: Do stacks $\operatorname{Sch}^{op} \to \operatorname{Cat}$ of categories come up anywhere in algebraic geometry, such that one considers presentations by internal categories in $\operatorname{Sch}$? If yes, how is the presentation specified? If no, why not? And is there anything stopping us from doing so? (Technical reasons, terminological, or we have other techniques that are better and so on)

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  • $\begingroup$ Interesting question, I also wonder if it is reasonable to generalize the theory of algebraic stacks to some theory where groupoids are replaced by categories. The examples you mentioned already show that this is sometimes more natural. $\endgroup$
    – Martin Brandenburg
    Commented Jun 19, 2011 at 9:00

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The category theoretical definition of stacks (as given for instance in Giraud: Cohomologie non-abélienne) allow for arbitrary categories as targets (the stack condition only involves isomorphisms however). A natural example is the category of quasi-coherent sheaves (which has the category of vector bundles as a subcategory). However, when you are talking about algebraic stacks (which are category theoretic stacks fulfilling extra conditions) they only involve isomorphisms. Note that given any stack restricting to isomorphisms gives a stack in groupoids. This is what one does when one considers the algebraic stack of vector bundles: Start with the stack of vector bundles with arbitrary morphisms. This is not an algebraic stack but restricting to isomorphisms gives one.

General stacks (with non-isomorphisms) are used extensively as they encode the idea of descent. This is somewhat orthogonal to algebraic stacks which try to encode the idea of a moduli problem.

Addendum: All morphisms in a descent datum are isomorphisms (this actually follows and does not have to be assumed). However full descent means that you can descend objects (a descent datum of objects comes from an object downstairs) but also arbitrary morphisms (a descent datum of morphisms comes from a morphisms downstairs). These two properties together can be formulated as an equivalence of categories between the category of objects downstairs and the category of descent data.

Addendum 1: Charles poses an interesting question. One answer can be based on the fact there seems to also be a philosophical difference between general stacks and algebraic stacks. General stacks are based on the idea that we have some objects and relations between them, the morphisms, that can be glued together over some kind of covering. Hence, usually the objects themselves are the things of main interest and the gluing condition is just an extra (though very important) condition on such objects.

Algebraic stacks on the other hand are things that themselves are glued. The relevant idea is that groupoids are a natural generalisation of equivalence relations. (One can more or less arrive at the idea of a groupoid by thinking of bereasoned equivalence relations, elements do not just happen to be equivalent but there are specific, in general several, reasons for them to be equivalent.)

Having said that, one could start with the fact that an algebraic is the stack associated to a smooth algebraic groupoid (i.e., source and target maps are smooth). This gives a candidate generalisation by just looking at smooth algebraic categories instead. However, no natural examples that are not groupoids comes (at least) to my mind. I think the reason might be the above philosophical distinction.

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    $\begingroup$ Dear Torsten: Could you explain a little more about the last paragraph? I thought when doing descent, one still needs "transition isomorphisms" rather than non-isomorphisms, so I got confused by what you mean. Thanks! $\endgroup$
    – shenghao
    Commented Mar 1, 2011 at 14:10
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    $\begingroup$ This leaves the question: is there a place for a theory of "category valued algebraic stacks"? $\endgroup$
    – Charles Rezk
    Commented Mar 1, 2011 at 16:24
  • $\begingroup$ @CharlesRezk as it turns out, Drinfeld has a recent paper talking about "algebraic c-stacks", which is basically the same thing I was thinking of in this and subsequent questions. $\endgroup$
    – David Roberts ♦
    Commented Jul 13, 2021 at 1:37
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Drinfeld's paper Prismatization (Sel. Math. New Ser. 30, no. 49 (2024) https://doi.org/10.1007/s00029-024-00937-3, arXiv:2005.04746) uses objects that he calls algebraic c-stacks, which are exactly the concept I had in mind in 2011.

I think prismatic cohomology is something that people in algebraic geometry find interesting and useful ;-)

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  • $\begingroup$ An additional remark to this answer (since learned this from Drinfeld's paper). Category valued stacks (what Drinfeld calls $c$-stacks) often keep track of "effectivity". For example, sheaves on $B\mathbf{G}_{\mathrm{m}}$, the moduli stack of line bundles (with isomorphisms) are the same as $\mathbf{Z}$-graded abelian groups, while sheaves on $B\mathbf{A}_{\mathrm{m}}^1$, the moduli stack of line bundles (with all maps) are the same as $\mathbf{Z}_{\geq 0}$-graded abelian groups. $\endgroup$
    – lrg
    Commented Jun 17 at 5:19

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