# Stackification of finite categories

Assume I have a base category $$\mathscr S$$ with finite limits and a geometric morphism $$\gamma :\mathscr S\to Fin$$ into the category of finite sets (for example because $$\mathscr S$$ is positive coherent). Say I have also a topology $$J$$ on $$\mathscr S$$. Given a finite category $$C$$, let $$\underline C$$ denote the fibration above $$\mathscr S$$ each of which fibers is $$C$$. I can also use $$\gamma^\ast$$ to get an internal category $$\gamma^\ast C$$ in $$\mathscr S$$ and use the externalisation operation to get an associate fibration $$[\gamma^\ast C]$$.

Question: Under which conditions on the base and the topology is $$[\gamma^\ast C]$$ the stackification of $$\underline C$$?

Edit. I can deal with discrete finite categories. Say $$C$$ is a discrete finite category, then $$\gamma^\ast C = \mathbf 1 + \mathbf 1 +...+\mathbf 1$$ as a discrete internal category. I know that a fibered functor $$F\in Fib_\mathscr S([\gamma^\ast C],D)$$ is approximately the same thing as a $$D$$-valued diagram above $$\gamma^\ast C$$. In the case that $$C$$ is discrete, this is just an object $$X$$ in $$D$$ lying above $$\mathbf 1 + ...+\mathbf 1$$. On the other hand, an element of $$Fib_\mathscr S(\underline C,D)$$ is the same thing as $$C$$-many objects of $$D$$ lying above $$\mathbf 1$$. So it seems in this case the condition I need is that $$D$$ satisfies effective descent for the coproduct inclusions $$\mathbf 1 \to \mathbf 1 + ...+\mathbf 1$$. Hence, when $$J$$ contains the coproduct inclusions of finite coproducts and $$C$$ is finite discrete, then $$[\gamma^\ast C]$$ will be $$\underline C^{st}$$ (if I did not make a mistake, I did not check the 2-dimensional part of the universal property carefully). I have trouble dealing with the case that $$C$$ has non-trivial morphisms though, hence the question.

• Even for the discrete case you mention, I think you need to assume $J$ is subcanonical (which then forces $\mathcal{S}$ to be extensive). May 22, 2023 at 23:03