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1
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0answers
149 views

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 ...
7
votes
0answers
425 views

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 ...
7
votes
2answers
852 views

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 ...
8
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4answers
706 views

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 ...
2
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1answer
498 views

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 ...
6
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3answers
541 views

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 ...
6
votes
0answers
359 views

(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 ...