stacks that are not necessarily fibered in groupoids appearing in algebraic geometry and differential geometry

Question

Question:

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.

Answer 1

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 $U$).

Answer 2

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.