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For questions involving one or more categorical dimensions, or involving homotopy coherent categorical structures.

5 votes

When is the category of sheaves on a site compactly assembled/a continuous category?

As far as I'm aware, no such conditions is known - The paper of Anel and Lejay is the closest to an answer available in the litterature. So, this is not an answer to the question, but more of an expan …
Simon Henry's user avatar
  • 42.4k
47 votes

What is the mistake in the proof of the Homotopy hypothesis by Kapranov and Voevodsky?

It's been more than a year and a half since I asked this question and I had a lot of thought about it so I decided I will post my own answer. First I entirely agree with Yonatan that the main problem …
LSpice's user avatar
  • 12.9k
106 votes
4 answers
13k views

What is the mistake in the proof of the Homotopy hypothesis by Kapranov and Voevodsky?

In 1991, Kapranov and Voevodsky published a proof of a now famously false result, roughly saying that the homotopy category of spaces is equivalent to the homotopy category of strict infinity categori …
9 votes

Conservative cocompletion of categories of geometric shapes for homotopy theory

I don't really see a way to give a single answer here, these are 7 different questions (well actually a lot more than 7 if we count all the different flavours of cubes, and of the other ones, probably …
Simon Henry's user avatar
  • 42.4k
9 votes

Is there a good general definition of "sheaves with values in a category"?

$\newcommand\Set{\mathrm{Set}}$The naïve definition of sheaves is very well behaved if you look at functoriality in the $f_*$ direction: Of course, you are going to need to assume that $\mathcal{A}$ h …
LSpice's user avatar
  • 12.9k
9 votes

A possible alternative model for $\infty$-groupoids

It is known that the category of finite non-empty set is a test category, in particular there exists a model structure on the category of presheaf on $Fin_+$ whose cofibrations are the monomorphisms a …
Simon Henry's user avatar
  • 42.4k
7 votes

The category of groupoids vs the category of sets

An important distinction in the "set case" is that there are two things you can refer to when talking about "the category of groupoids": the 1-category of groupoids. the 2-category of groupoids. It …
Alec Rhea's user avatar
  • 10.1k
4 votes

Does $\infty$-categorical localization commute with taking directed fibered products?

Here is a counter example in the general case: Take $E = [1] = 0 \to 1 $. $D = \{0\} \coprod \{1\} $ and $C = \{1\}$. where all maps are weak equivalence. The lax-pullback is $\{id:1 \to 1\}$, and the …
Simon Henry's user avatar
  • 42.4k
22 votes
Accepted

Useful ideas in category theory which violate the principle of equivalence

I would think about this question in this way: If you have a construction that violates the equivalence principle then either (A) it is a strictified or simplified version of something that is compati …
LSpice's user avatar
  • 12.9k
4 votes

Are $\infty$-categories functorially colimits of their simplices?

This is just an expended version of the comment. The answer to the question as asked is no. The problem is that for any ($\infty$-)category $J$ the category $D_J$ of functors $J \to Cat_\infty$ that a …
Simon Henry's user avatar
  • 42.4k
15 votes
1 answer
485 views

Well pointed endofunctors on $\infty$-categories

In $1$-category theory, a well pointed endofunctor of a category $C$, is an endofunctor $F:C \rightarrow C$ endowed with a natural transformation $\sigma : Id \rightarrow F$ such that the two natural …
5 votes
Accepted

Fibrations of sites for $\infty$-topoi

Here is an argument for the 1-categorical version that essentially bypass the use of internal site and should be much easier to generalize to the $\infty$-categorical case. ( I mean you can still see …
Simon Henry's user avatar
  • 42.4k
22 votes
Accepted

Is there a higher analog of "category with all same side inverses is a groupoid"?

Yes, this is possible. The following is a classical result of the theory of quasi-categories (You'll find it in the early part of Lurie's Higher topos theory or in Joyal notes on quasi-categories - w …
Martin Brandenburg's user avatar
5 votes
Accepted

Is there a "geometric definition" of globular $\infty$-groupoids/categories?

In short there isn't: the problem is that if you just have globular sets - and if you want $k$-cells to model $k$-arrows following the globular structure - then globular sets have no way of expressing …
Simon Henry's user avatar
  • 42.4k
7 votes

Generalizing $n$ for $n$-categories

Here are some things I know of that looks like this: I* have argued here that it is possible to extend the notion of strict $\infty$-category to a notion of $P$-category for any poset $P$. At this po …
Simon Henry's user avatar
  • 42.4k

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