Questions tagged [grothendieck-construction]

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Regarding the definition of $f$-morphisms/cartesian arrows in a fibred category $\mathcal{F} \rightarrow \mathcal{C}$

Let $p: \mathcal{F} \rightarrow \mathcal {C}$ be the data of a fibred category. Then, for arrows $f: U \rightarrow V$ in $\mathcal{C}$, a morphism $\phi: \xi \rightarrow \eta$ in $\mathcal{F}$ is said ...
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1answer
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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}]...
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1answer
599 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\...
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1answer
180 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 ...
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174 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}(...
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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
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1answer
237 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 ...
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1answer
201 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 ...
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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: ...
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1answer
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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 @>!>> \...
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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 $\Omega(G)$,...
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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 ...
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2answers
470 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) ...
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161 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^{...
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1answer
773 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 ...
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393 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. ...