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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 …

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 …

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 …

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 …

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 …

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. …

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 …

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 …

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 …

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 …

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 …

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 …

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 …

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 …