# 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 ...

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....

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 ...

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 $...

5
votes

0
answers

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 ...

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 ...

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) ...

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 \...

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, ...

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}]...

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\...

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 ...

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}(...

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
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 ...

4
votes

1
answer

246
views

### 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 ...

6
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: ...

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 @>!>> \...

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 $\Omega(G)$,...

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 ...

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) ...

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
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 ...

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. ...