Questions tagged [grothendieck-construction]

Filter by
Sorted by
Tagged with
5 votes
1 answer
104 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 ...
user avatar
  • 173
6 votes
0 answers
135 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 $...
user avatar
  • 59.4k
5 votes
0 answers
181 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 ...
user avatar
  • 59.4k
6 votes
1 answer
304 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 ...
user avatar
  • 6,386
4 votes
1 answer
123 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) ...
user avatar
  • 1,171
5 votes
2 answers
202 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 \...
user avatar
8 votes
2 answers
655 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, ...
user avatar
7 votes
1 answer
445 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}]...
user avatar
  • 4,293
12 votes
1 answer
655 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\...
user avatar
  • 18.8k
3 votes
1 answer
253 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 ...
user avatar
  • 828
7 votes
0 answers
262 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}(...
user avatar
10 votes
0 answers
623 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: $$ \int^x G(x,x) $$ ...
user avatar
  • 2,029
2 votes
1 answer
269 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 ...
user avatar
3 votes
1 answer
225 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 ...
user avatar
  • 51
6 votes
0 answers
723 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: ...
user avatar
  • 299
10 votes
1 answer
299 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 @>!>> \...
user avatar
  • 49.5k
1 vote
0 answers
144 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 $\Omega(G)$,...
user avatar
1 vote
0 answers
212 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 ...
user avatar
5 votes
2 answers
601 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) ...
user avatar
1 vote
0 answers
177 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^{...
user avatar
3 votes
1 answer
849 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 ...
user avatar
7 votes
1 answer
452 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. ...
user avatar