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### Reference request: Grothendieck construction for $\mathbb V$-distributors?

I'm currently working with an analogue of the Grothendieck construction for enriched categories:
Given a distributor a.k.a. $\mathbb V$-functor $D:X^\mathrm{op}\otimes Y\to \mathbb V$ there is a ...

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### Where else has Proposition B1.3.17 in the Elephant been proved?

(I asked the same question here and got some helpful comments, but thought I'd re-ask in case I get a more direct response.)
This is a sort of reference request. Proposition B1.3.17 in Johnstone's ...

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### Are non-algebraic stacks useful in algebraic geometry?

The title is a bit vague. What I want to know is if there is any geometric application of non-algebraic stacks. I know e.g. the category of coherent sheaves is an example. But I want to ask if people ...

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### Adjunctions form a stack

Let $C$ be a base category, $F,G$ be two categories fibered over $C$ and $F \to G$ be a morphism. The following criterion is used very often: If all the fiber functors $F_U \to G_U$ ($U \in C$) are ...

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### Strong colimits of categories.

Let $\mathcal C$ be a category and let $\mathcal F:\mathcal C\to\mathcal C\textrm{at}$ be a strong bifunctor. Given another category $\mathcal D$, let $\triangle_{\mathcal D}$ denote the constant ...

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### Fibered category with an adjoint inclusion

Suppose $X:D \to C$ is a fibered category (I do not assume the fibers to be groupoids). Suppose that $X$ is actually left adjoint to a fully faithful embedding $C \hookrightarrow D$. Is there a ...

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### (Co-)Limits and fibrations of DG-Categories?

First of all, let me see if I got the 1-categorical version right:
Let $\mathcal F:C\to Cat $ be a
(pseudo-) functor. The 2-colimit
$\mathrm{colim}_C\mathcal F$ is then
given by the Grothendieck
...

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### Explicit description of a fibered category

I found the following exercise in Vistoli's notes. He proves a theorem stating that any category $\mathcal{F}$ fibered over $\mathcal{C}$ is equivalent, as a fibered category, to a split one. Namely ...