Questions tagged [grothendieck-construction]
The grothendieck-construction tag has no usage guidance.
24
questions
4
votes
1
answer
200
views
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 ...
- 1,185
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....
- 1,185
5
votes
1
answer
145
views
What is the name for the construction of this poset related to coherence of degeneracies of the simplex category?
I present you a family of posets here. I don't say the posets themselves have a conventional name. However I'm sure the general construction of this kind has received some terminology, related to ...
- 233
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 $...
- 63.6k
5
votes
0
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199
views
When is a Grothendieck construction a presheaf category?
Let $C$ be a small category, $\mathcal{P}C$ its presheaf category, and $F:\mathcal{P}C^{\rm op} \to \rm Cat$ a pseudofunctor. Are there general conditions we can impose ensuring that the Grothendieck ...
- 63.6k
6
votes
1
answer
367
views
Is there a Grothendieck correspondence for sheaves/stacks?
Given a category $\mathcal{C}$, the category of elements functor sets up an equivalence of categories
$$
\mathsf{DFib}(\mathcal{C})
\cong
\mathsf{PSh}(\mathcal{C}),
$$
whereas the Grothendieck ...
- 9,575
4
votes
1
answer
172
views
Checking that (hyper) sheafification is fibrant in local projective model structure on simplicial presheaves
Context and Notation
Let $X$ be a manifold and $\mathcal{C} = Op(X)$ be the category of open subsets of $X$ with inclusions. I then consider the projective model structure on the (simplicial model) ...
- 1,331
5
votes
2
answers
251
views
Explaining the "free left fibration" functor for infinity categories
This is a cross-post from here
I am reading A. Mazel-Gee's paper "All about the Grothendieck construction". In that paper he explains that the left adjoint ${\mathrm{Cat}_{\infty}}_{/\mathcal{C}}\to \...
- 607
8
votes
2
answers
791
views
Does the Grothendieck construction produce a 2-category or a category?
Let $F : \mathcal{C} \to \mathbf{Cat}$ be a lax 2-functor. Then we can form a category $\int F $ which is the Grothendieck construction on F.
There's a number of resources detailing this construction, ...
- 303
7
votes
1
answer
463
views
Categorical Significance of Fibrations
It is well known that the category $\text{Set}$ classifies covering spaces among $1$-categories. That is, for each topological space $X$, there is an equivalence of categories $[ \Pi (X) , \text{Set}]...
- 4,576
13
votes
1
answer
674
views
Does the Grothendieck construction satisfy Fubini's thorem
Suppose we are given a functor
$F:(A\times B)^{\operatorname{op}}\to \operatorname{Set}$.
It's well-known that the Grothendieck construction in this case evaluates as
$\int_{A\times B}F = (A\...
- 19.2k
3
votes
1
answer
298
views
Stack descent to sheaf descent via Grothendieck construction?
Let S be a Grothendieck site, the (either left or right adjoint to the) Grothendieck construction assigns to each groupoid fibration over S a presheaf valued in groupoids. The following feels it might ...
- 858
9
votes
0
answers
322
views
The category of elements corresponding to a coend as a higher colimit
Let $D: \mathcal{C} \to \mathbf{Set}$ be a diagram of sets, then we can obtain the colimit of $D$ as the set of connected components of the category of elements of $F$, which we denote by $\mathrm{el}(...
- 631
11
votes
0
answers
743
views
Grothendieck construction and coends
In category theory, both the Grothendieck construction and coends are represented by a sort of "integral sign", respectively:
$$
\int F
$$
for a functor $F:C\to\mathbf{Cat}$,
and:
$$
\int^x G(x,x)
$$
...
- 2,079
2
votes
1
answer
280
views
Can tangent ($\infty$,1)-categories be described in terms of the higher Grothendieck construction?
Given a locally presentable ($\infty$,1)-category $C$, can the fibrewise stabilization of it's codomain fibration, also called its tangent category $TC$, be given in terms of the Grothendieck ...
- 703
4
votes
1
answer
246
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discrete Grothendieck construction
In "BASIC CONCEPTS OF ENRICHED CATEGORY THEORY", (version Reprints in Theory and Applications of Categories, No. 10, 2005), chapter 4.7 p.75-76, Kelly introduces the "discrete Grothendieck ...
- 61
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votes
0
answers
757
views
Grothendieck problem
Could you suggest me a book or a link where I can find some information about the Grothendieck problem about differential equations?
The Grothendieck problem that I am reffering to is the following: ...
- 299
10
votes
1
answer
317
views
Relationship between two universal properties of the category of elements?
Let $G: \mathcal{A} \to \mathsf{Set}$ be a functor. Recall that the category of elements $\mathsf{el}G$ is defined by the comma square
$\require{AMScd}$
\begin{CD}
\mathsf{el}G @>!>> \...
- 56.6k
1
vote
0
answers
152
views
Homology and Burnside ring
If $G$ is a finite groupe, we denote $\mathcal{S}(G)$ the category of finite $G$-sets and $\mbox{I}(G)$ the set of isomorphism classes of it's objects. The Burnside ring of $G$, denoted by $\Omega(G)$,...
- 31
1
vote
0
answers
217
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 ...
- 3,113
5
votes
2
answers
652
views
grothendieck construction for profunctors
Given categories $X$ and $Y$ and a strong functor
$$D:X^{op}\times Y\to Cat$$
we can of course build the oplax colimit
$$\mathrm{colim}^{oplax}_{X^{op}\times Y}D$$
via the usual (covariant) ...
- 3,113
1
vote
0
answers
187
views
Composition of Cat-valued distributors - compatible with grothendieck construction?
Let $C$ be a category and $F\in[C^{op}, Cat]$ be a strong functor.
(1) There are functors
$$hom_C(c',c)\times F(c)\to F(c').$$
(2) The grothendieck construction gives a 2-equvalence
$$\int_C: [C^{...
- 3,113
3
votes
1
answer
868
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 ...
- 3,113
7
votes
1
answer
472
views
Reference request: 2-Grothendieck Construction
Hi Folks,
i'm looking for a reference on the 2-grothendieck construction for a functor $F:\mathcal{I}\to \mathcal{B}\mathrm{icat}$ from a bicategory $\mathcal{I}$ to the tricategory of bicategories. ...
- 3,113