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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.

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

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