10 votes

When did the Joyal model structure on simplicial sets originate?

My suspicion is now that it was some time between 2004 and 2006. I have a lot more citations in this blog post, but I note three points, in reverse chronological order: Multiple experts are referring ...
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8 votes

Complexity of coherence diagrams in an $n$-category

This is not a full answer to the question but it was too long for a comment. Depending on your conventions, "weak $n$-categories" might mean "$(n,n)$-categories" which are a ...
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  • 7,007
5 votes
Accepted

$\infty$-groupoid iff Kan condition

A good place to start is the following characterization of nerves of ordinary groupoids: A category is a groupoid if and only if its nerve is a Kan complex. Filling the outer horns is precisely what ...
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4 votes

Complexity of coherence diagrams in an $n$-category

Here are some off-the cuff ideas. First, as I think Saal's answer starts to hint at, there's no particular reason to restrict attention to $(n,n)$-categories. It seems more natural to think about $(n+...
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  • 49.5k
3 votes

Is there an analog of Kan's $Ex^\infty$ functor for quasicategories?

$\def\Cnec{{\frak C}^{\rm nec}} \def\Exi{{\rm Ex}^∞} \def\N{{\rm N}} \def\sCat{{\sf sCat}} \def\sSet{{\sf sSet}_{\sf Joyal}}$ As indicated in the answer A combinatorial approximation functor sSet->...
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1 vote
Accepted

Unit of a Quillen equivalence and fibration

If we write down the lifting square for an arbitrary cofibration $f\colon A→B$ and the unit map $η\colon X→RLX$ (with the bottom map being $b\colon B→RLX$), and then use the adjunction to pass to the ...
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