9 votes
Accepted

Does the Grothendieck construction produce a 2-category or a category?

The usual Grothendieck construction has for $\mathcal C$ an ordinary category, so it doesn't have any 2-cells (or at least, doesn't have any non-identity 2-cells). Moreover, we actually get not only a ...
  • 464
9 votes
Accepted

Does the Grothendieck construction satisfy Fubini's thorem

Yes. See section 4.2 of this paper by Harpaz and Prasma, which also extends the result to the setting of model categories. Another good source on these topics is Emily Riehl's book. The Fubini result ...
  • 22.6k
7 votes

Relationship between two universal properties of the category of elements?

The conceptual explanation is that a comma square exhibits the bottom morphhism as a left Kan extension if the right morphism is dense. There is, of course, a problem what we mean by a "dense ...
3 votes

discrete Grothendieck construction

The name of that article changed (a lot, it seems): the information you seek is in the paper Doctrinal Adjunction by Kelly. It lies on page 257 of the collection Category Seminar, Number 420 of ...
  • 14.8k
3 votes
Accepted

Is there a Grothendieck correspondence for sheaves/stacks?

The essential image of the Grothendieck construction from (weak) functors $C\to\operatorname{Cat}$ which are sheaves with respect to a Grothendieck topology $\mathcal T$ is described in Section 8.4 of ...
2 votes
Accepted

What is the name for the construction of this poset related to coherence of degeneracies of the simplex category?

Here’s one way to see it, if I’m not misunderstanding your definition. For a small category $\newcommand{\C}{\mathbf{C}}\C$, take its categorical nerve $\newcommand{\N}{\mathbf{N}}\N\C$ to be the ...
2 votes
Accepted

Categorical Significance of Fibrations

Yes. In roughly your language, the forgetful $(\infty,1)$-functor $\rm Fib\to Top$ is an $\infty$-fibration, where the fiber over a space $X$ is the category of fibrations over $X$, and descent in ...
  • 60.2k
2 votes
Accepted

Checking that (hyper) sheafification is fibrant in local projective model structure on simplicial presheaves

There are two ways to make this construction work. The first way is to iterate the step $F↦F^†$ transfinitely many times. The reason that a single iteration of $F↦F^†$ is not sufficient is that while $...
2 votes

Explaining the "free left fibration" functor for infinity categories

It's actually something you already know: It is the fibrewise groupoidification of the free cartesian fibration. The free cartssian fibration functor sends a functor $$p:A\to B\mapsto p': A\downarrow ...
  • 18.9k
2 votes

Explaining the "free left fibration" functor for infinity categories

As suggested by David White, I emailed A. Mazel-Gee. Let me paraphrase his answer : We claim that given a cocartesian fibration $F:\mathcal{D}\to\mathcal{C}$, the free left fibration $LF:\mathcal{E}\...
1 vote
Accepted

Stack descent to sheaf descent via Grothendieck construction?

Here's a variation which is true, when interpreted in a suitably non-strict / higher categorical sense (for example, "functor" means "pseudofunctor" below). I'm not sure on which side of the ...
  • 51k
1 vote

Can tangent ($\infty$,1)-categories be described in terms of the higher Grothendieck construction?

See the proof of Proposition 1.1.9 here https://arxiv.org/pdf/0709.3091v2.pdf.
  • 5,248

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