Questions tagged [fibered-categories]
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27
questions
4
votes
1
answer
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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 ...
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4
votes
1
answer
200
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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....
- 1,185
3
votes
1
answer
288
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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 ...
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6
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0
answers
149
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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 $...
- 63.6k
6
votes
1
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263
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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 ...
- 8,760
18
votes
2
answers
765
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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, ...
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3
votes
0
answers
226
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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 ...
- 5,272
2
votes
1
answer
118
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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 ...
- 8,760
0
votes
0
answers
84
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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 ...
- 8,760
6
votes
1
answer
291
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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} \...
- 309
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0
answers
119
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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 ...
- 5,272
2
votes
1
answer
124
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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 $L$ ...
- 1,185
4
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0
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96
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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 ...
- 38.5k
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3
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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 ...
- 56.6k
3
votes
1
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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 ...
- 3,251
1
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0
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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 ...
- 10.2k
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 ...
- 10.2k
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 ...
- 570
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votes
1
answer
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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 ...
- 10.2k
1
vote
0
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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 ...
- 3,113
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votes
0
answers
515
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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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votes
2
answers
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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 ...
- 3,688
9
votes
4
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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 ...
- 59.2k
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 ...
- 3,113
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 ...
- 15.2k
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
...
- 3,113
8
votes
2
answers
670
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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 ...
- 14.2k