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Part of higher category theory that for instance in Algebraic Topology enables us to capture finer homotopic distinctions. As in say Eilenberg-Maclane spaces.

9 votes

Model categories with uniqueness

As Denis-Charles said in a comment, if you require the unique lifting property on a weak factorization system, it becomes a "unique" or "orthogonal" factorization system. In the paper Bousfield local …
Mike Shulman's user avatar
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5 votes

What is the functoriality of the $\infty$-categorical slice construction?

I can't speak directly to the $\infty$-categorical version, but I can say something about the 2-categorical phenomena, which may suggest an approach to the $\infty$-categorical question. The general 2 …
Mike Shulman's user avatar
  • 66.8k
26 votes
Accepted

Grothendieck derivators vs $\infty$-categories

The short answer is that $(\infty,1)$-categories are the "real" object of interest. Derivators are a tool for working with them that is sometimes (for some people) easier to use, but doesn't remember …
Mike Shulman's user avatar
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12 votes

How should I think about presentable $\infty$-categories?

Just to be a little bit contrary, let me point out one concrete reason that locally presentable categories are not aesthetic: it is unclear whether they have any analogue in constructive mathematics. …
Mike Shulman's user avatar
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17 votes

Internal categories in simplicial sets

It appears that Geoffroy Horel has solved this problem completely: Geoffroy Horel, A model structure on internal categories in simplicial sets, Theory and Applications of Categories 30 No. 20 (201 …
David Roberts's user avatar
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3 votes

Describing fiber products in stable $\infty$-categories

A proof of this in the context of stable derivators can be found in my paper Mayer-Vietoris sequences in stable derivators with Moritz Groth and Kate Ponto. It is rather more verbose than Denis's, bu …
Mike Shulman's user avatar
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5 votes

Lex $\infty$-colimits

Probably not. Claim: Let $C$ be a small $(\infty,1)$-category with finite limits and colimits which admits an embedding $V:C\to E$ into a Grothendieck $(\infty,1)$-topos preserving finite limits and …
Mike Shulman's user avatar
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5 votes
Accepted

When the global section functor is a Cartesian fibration?

In fact I believe something quite general can be said: if $p:C\to D$ is any functor that preserves finite limits, then it is a Street fibration if and only if it has a fully faithful right adjoint. A …
Mike Shulman's user avatar
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7 votes
Accepted

Type Theory to Study $(\infty,n)$-Categories and $(r,n)$-Categories

I don't know of anyone who is specifically working on generalizing our paper to the $(\infty,n)$-case. But there have been other attempts to design a type theory for higher categories, such as Finste …
Mike Shulman's user avatar
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14 votes

$\infty$-categorical interpretation of type theory

More generally, the issue with such interpretation is that substitution in type theory is interpreted by pullback in category theory, and substitution in ordinary type theory preserves all type-theore …
Mike Shulman's user avatar
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16 votes

generalisations of the Seifert-van Kampen Theorem?

I don't actually see how to deduce the version of the classical SvKT on a set of base points $A$ directly from Lurie's version. It seems that to apply Lurie's theorem, we would need the stronger hypo …
Mike Shulman's user avatar
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3 votes

Limitations on model-categorical presentations

Here is another answer that involves adding extra properties. If we have a model category which is locally cartesian closed, as a category (such as if it is a presheaf category) has its cofibratio …
Mike Shulman's user avatar
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10 votes
Accepted

(∞,1) vs Category weakly enriched over spaces

If "space" means the same thing in the two cases, as seems to be implied, then there is no difference, at least not at that level of precision. There are many different models of $(\infty,1)$-categor …
Mike Shulman's user avatar
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16 votes
Accepted

(∞, 1)-categorical description of equivariant homotopy theory

I think a good reference for the first paragraph is "Equivariant Homotopy and Cohomology Theory" by Peter May and a bunch of other people. Chapter 5 includes "Elmendorf's theorem" that this homotopy …
Mike Shulman's user avatar
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