# Questions tagged [fibered-categories]

The fibered-categories tag has no usage guidance.

27
questions

4
votes

1
answer

200
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### Morphisms of fibered categories which are compatible with the chosen cleavages

Let $\pi \colon F \rightarrow C$ and $\pi' \colon F' \rightarrow C$ be two fibered categories over the category $C$. A morphism from $\pi \colon F \rightarrow C$ to $\pi' \colon F' \rightarrow C$ is a ...

4
votes

1
answer

200
views

### What is a correct notion of an internal pseudofunctor?

Let $C$ be a category internal to a category $K$. It is well known (for example see Proposition 2.4 in the paper Higher Dimensional Algebra VI: Lie 2-Algebra by Baez and Crans https://digitalcommons....

3
votes

1
answer

288
views

### Schemes as categories fibered in thin groupoids

Every time I start to read about schemes from a birds-eye view (like in the introduction to The Geometry of Schemes by Eisenbud and Harris) I get really excited; they sound like a categorical approach ...

6
votes

0
answers

149
views

### Freely adding comprehensions

If $P:T^{\rm op}\to \rm Cat$ is a hyperdoctrine with at least products in the fiber categories, then there is a way of "freely" adding Lawvere-style comprehension to it. The base category $...

6
votes

1
answer

263
views

### Can we show that a functor is a fibration without choosing a cleavage?

Is there a standard method for showing that a functor $F:\mathcal{C}\to\mathcal{D}$ is a fibration, aside from constructing a cleavage?
In the proof of the Grothendieck construction, the fibration we ...

18
votes

2
answers

765
views

### Relationship between enriched, internal, and fibered categories

In this question, let $(\mathcal{V}, \otimes, [-,-], e)$ be a nice enough symmetric monoidal closed bicomplete category.
The usual set-based Category theory has been generalized in many directions, ...

3
votes

0
answers

226
views

### Sheaf $\operatorname{Isom}(x,y)$ isomorphic to fibered product $U \times_{(x,y),X \times X, \Delta} X$

$\DeclareMathOperator\Mor{Mor}\DeclareMathOperator\Isom{Isom}$Let $S$ be a scheme and $C$ be the category $(Sch/S)$. Let moreover $p:X \to C$ be an algebraic stack over $C$. Consider an arbitrary ...

2
votes

1
answer

118
views

### Fiberwise skeleton vs. category of isomorphism types

What is the relationship (if any) between the process of taking a skeleton of a category, taking a fibered skeleton of a fibered category, and taking isomorphism classes of a category as objects of a ...

0
votes

0
answers

84
views

### Left/right adjoints to core/Cartesian inclusions

Let $p:E\to B$ be a fibration and let $Cart(E)$ denote the subcategory of $E$ with all objects but only Cartesian arrows. Since all isomorphisms in $E$ are Cartesian, we naturally have inclusion ...

6
votes

1
answer

291
views

### Fibration on the category of Lie pseudoalgebras implementing comorphisms

I am trying to understand comorphisms of Lie pseudoalgebras from the point of view of fibred categories, but failing miserably so far. My question would be:
Is there a (op)fibration $\mathrm{LiePs} \...

2
votes

0
answers

119
views

### Infinititesimal Automorphisms intuition (algebraic stacks)

Let $F$ be a category cofibered in groupoids over category $C$. Given a morphism $x'\to x$ in $F$ lying over a morphism $A′\to A$ in $C$, there is an induced homomorphism
$\operatorname{Aut} A'(x')\to ...

2
votes

1
answer

124
views

### Is there any relation between two pseudofunctors associated to two different cleavages of the same fibered category?

It is well known that given a Fibered category $P_F: E \rightarrow C$ with a cleavage $K$ we can construct a pseudofunctor $F_K: C^{op} \rightarrow Cat$. Now if one chooses a different cleavage $L$ ...

4
votes

0
answers

96
views

### Partially fibered categories vs T-Multicategories

Short version: This is a reference request question. I would like to know if something has been written on the connection between $T$-multicategory (for $T$ a monad on a category $\mathcal{E}$), and ...

26
votes

3
answers

1k
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### What is the geometric significance of fibered category theory in topos theory?

Often in topos theory, one starts with a geometric morphism $f: \mathcal Y \to \mathcal X$, but quickly passes to the Grothendieck fibration $U_f: \mathcal Y \downarrow f^\ast \to \mathcal X$, which ...

3
votes

1
answer

125
views

### Internal equality for Eq-fibrations' morphisms

I have posted this question here on M.SE but since it received little attention and since it seems difficult to find helpfule references I reposting it here.
In Jacob's Categorical logic and Type ...

1
vote

0
answers

165
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### Choice free definition of category of descent data w.r.t a fibration?

Let $\mathsf C$ be a category and consider a pseudofunctor (non-strict 2-functor) $P:\mathsf C^{\text{op}}\to\mathsf{Cat}$. Given an arrow $f:X\to Y$ in $\mathsf C$, define the category of descent ...

8
votes

4
answers

805
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### English Reference for the Bénabou-Roubaud theorem

The Bénabou-Roubaud theorem links fibrational descent theory with monadicity. Particularly, it says that given a bifibration satisfying the Beck-Chevalley condition w.r.t some arrow $p$ in the base ...

9
votes

1
answer

642
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### Yoneda Lemma for internal presheaves

I'm looking for a reference explaining under what conditions the internal Yoneda lemma holds; in particular, I am wondering if it is known what properties of the ($2$-)category of categories are ...

15
votes

1
answer

1k
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### Difficulties with descent data as homotopy limit of image of Čech nerve

Apologies if this question is inappropriate for MO. It is not a research level question in any of the topics it addresses, I just don't see how a novice can go about answering it alone (I've tried ...

1
vote

0
answers

217
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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 ...

7
votes

0
answers

515
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 ...

13
votes

2
answers

1k
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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 ...

9
votes

4
answers

819
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 ...

3
votes

1
answer

868
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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 ...

6
votes

3
answers

617
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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 ...

8
votes

0
answers

619
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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
...

8
votes

2
answers

670
views

### Explicit description of a fibered category

I found the following exercise in Vistoli's notes. He proves a theorem (Theorem $3.45$, page number $64$) stating that any category $\mathcal{F}$ fibered over $\mathcal{C}$ is equivalent, as a fibered ...