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

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 …
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 …
Simon Henry's user avatar
  • 42.4k
36 votes
Accepted

Current status of Grothendieck's homotopy hypothesis and Whitehead's algebraic homotopy prog...

The problem is that the question is highly dependent on the definition of $n$-groupoids. The notion of strict $n$-groupoid is very clear and precise but we know very well (and Grothendieck knew that) …
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 …
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 …
Simon Henry's user avatar
  • 42.4k
21 votes

A sheaf is a presheaf that preserves small limits

This has nothing to do with $\infty$-categories, but with the fact that we look at the full topos and not an arbitrary site of definition: Theorem: If $T$ is a (Grothendieck) ($1$-)topos, then a "she …
Simon Henry's user avatar
  • 42.4k
19 votes
Accepted

"Spatial (geometrical)" realization of Elementary topos?

I would like to explain why I think the answer is no, but of course there is no way to prove this, and probably some way to use some geometric insight when talking about elementary toposes. My main p …
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  • 42.4k
19 votes
2 answers
1k views

A "universally non Hypercomplete" $\infty$-topos via Goodwillie calculus?

My question is : Is there a classifying $\infty$-topos for $\infty$-connected objects ? Does this $\infty$-topos has a nice description (as an $\infty$-category ) ? What I mean by $\infty$-connected …
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18 votes
1 answer
656 views

Equivalences of categories of sheaves vs categories of $\infty$-Stack

Let say I have two different sites $(\mathcal{C},I)$ and $(\mathcal{D},J)$ for an ordinary topos $\mathcal{T}$. I.e. $$Sh(\mathcal{C},I) \simeq \mathcal{T} \simeq Sh(\mathcal{D},J)$$ And we want to …
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  • 42.4k
16 votes
1 answer
432 views

Descent of Higher categorical structures along geometric morphisms

Let $f: \mathcal{E} \rightarrow \mathcal{T}$ be a geometric morphism between two (Grothendieck) toposes (or maybe more generally a bounded geometric morphism between elementary toposes). It is well k …
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 …
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14 votes
Accepted

Higher $\infty$-categories

Using Street's "one type" definition of strict $\infty$-category one can see that the concept of "strict $P$-category" makes sense not justs for any ordinals $P$ but in fact for any posets $P$ (though …
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  • 42.4k
14 votes
Accepted

Truncation of infinity-categories

There is a bit of notation to be careful about here: $\mathcal{X}_{\leqslant 1}$ is often used to denote the full subcategory of $\mathcal{X}$ of set-truncated object. For example if $\mathcal{X}$ is …
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  • 42.4k
14 votes
Accepted

A multicategory is a ... with one object?

This has been called a "fc-multicategory" by Tom Leinster, for example here. I think this as also been called a "Hypervirtual double category" here, but I don't remember if this is exactly the same …
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  • 42.4k
13 votes
Accepted

Why are finite cell complexes also finite as infinity-categories?

Start from a finite simplicial set $K$ which is homotopicaly equivalent to a Kan complex $X$. Then by applying a finite number of pushout of outer horn inclusion to $K$, you can build homotopy equiva …
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