What are (some of) the stacks (occurring in algebraic/differential geometry) that are fibered in arbitrary categories and not necessarily in groupoids?

In the notes Notes on Grothendieck topologies,fibered categories and descent theory Angelo Vistoli introduce the notion of a stack over a site $(\mathcal{C},\mathcal{J})$ to be a fibered category (not necessarily fibered in groupoids) over $\mathcal{C}$ satisfying some "locally determined" condition.

But, examples of stacks of interest in algebraic geometry and differential geometry (a small set of examples I have seen) are always fibered in groupoids. So, what could be a justification or necessity for introducing the notion of stacks fibered over arbitrary categories, if "almost all" stacks that occur in Algebraic geometry (that I know) are fibered in groupoids.

There might be interesting examples of stacks outside algebraic geometry of differential geometry that are not necessarily fibered in groupoids. I would be happy to see such examples (please add them as answers if you wish) but as for this question, I would like to learn about situations in algebraic geometry or differential geometry.

  • $\begingroup$ The categories of continuous fields of Hilbert spaces, Banach spaces and $C^*$-algebras should all be stacks over the appropriate base category of spaces (for instance, locally compact Hausdorff). I've not seen this mentioned, since the people who deal with these don't usually think about stacks. $\endgroup$ – David Roberts Aug 4 at 2:05
  • $\begingroup$ @DavidRoberts are you counting those as algebraic or differential geometry? $\endgroup$ – Praphulla Koushik Aug 5 at 12:25
  • $\begingroup$ I would say differential topology, maybe. $\endgroup$ – David Roberts Aug 5 at 12:33
  • $\begingroup$ @DavidRoberts fair enough.. $\endgroup$ – Praphulla Koushik Aug 6 at 6:34

What you are referring to are sometimes called stacks or sheaves of categories. A famously important example is the stack $\mathrm{QCoh}$ sending a scheme $U$ to the category of quasi-coherent sheaves over it (if you want to work with fibered categories, an object of this fibered category is a pair $(U,F)$ where $U$ is a scheme and $F$ a quasi-coherent sheaf on $U$).

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  • $\begingroup$ Yes, I have seen this example in Angelo Vistoli's notes (Example 3.2.1 page 53). Do you have any personal choice of appearance of this stack in any construction or some concept.. $\endgroup$ – Praphulla Koushik Aug 3 at 11:59
  • $\begingroup$ @PraphullaKoushik: One obvious occurrence is the definition of quasi-coherent sheaves on an arbitrary stack S: these are defined as (derived) maps S→QCoh. $\endgroup$ – Dmitri Pavlov Aug 7 at 4:45

Virtually any kind of algebraic structure (e.g., group, ring, module, vector space, affine space, etc.) leads to a stack in categories whose objects are bundles of such structures and morphisms are fiberwise homomorphisms of such structures.

For example, the stack Vect of (finite-dimensional, say) vector bundles is a stack in categories over the site of cartesian smooth manifolds.

Likewise, the stack BGrb^n_A of bundle n-gerbes with structure group A is a stack in (n+1)-categories.

As a practical application, one can immediately define the category of vector bundles or bundle n-gerbes on a given stack or ∞-stack S as the category of (derived) sections S→Vect or S→BGrb^n_A. This also captures the symmetric monoidal structure, and in the case of vector bundles, the bimonoidal structure.

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  • $\begingroup$ I understood most of first three paragraphs. I have difficulty in understanding last 2 lines.. I will read it again and ask if I have any specific question. Thanks for the answer.. $\endgroup$ – Praphulla Koushik Aug 5 at 12:23

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