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Stream: learning: questions

Topic: simplicial localization of model categories

Daniel Teixeira (Nov 15 2021 at 21:11):

I have recently been reading a lot about relative categories and simplicial localization, and while simplicial localization gives all simplicial categories up to DK equivalence (there's a Quillen equivalence to the Bergner model structure), I am particularly interested in the case when we localize a [[relative category]] that is actually a model category. This is a broad reference-request question, specific tools or general theorems are equally appreciated.

Couple findings:

• The case of presentable simplicial model categories is solved at the appendix of HTT (these are locally presentable quasicategories).

• Simplicial localization sends $C_{cf}\hookrightarrow C$ to a DK equivalence, i.e. $LC_{cf}\cong LC$.

Daniel Teixeira (Nov 15 2021 at 21:12):

I also feel that $LC_{cf}$ is already a Kan-enriched category, but couldn't write this down precisely (yet)

Zhen Lin Low (Nov 15 2021 at 22:10):

What gives you the impression that simplicial localisation gives you a Kan-enriched category? My impression is that this basically never happens.

Dmitri Pavlov (Nov 16 2021 at 05:25):

The relative category of combinatorial model categories is Dwyer–Kan equivalent to the relative category of presentable quasicategories: https://arxiv.org/abs/2110.04679

Daniel Teixeira (Nov 16 2021 at 14:16):

Zhen Lin Low said:

What gives you the impression that simplicial localisation gives you a Kan-enriched category? My impression is that this basically never happens.

That was because at the original DK paper a "homotopy" between zig-zags is a hammock e.g.

image.png

In general there is nothing more to say, but from Whitehead's theorem in $C_{cf}$ the vertical weak equivalences are actually homotopy equivalences, implying there is at least a "partial hammock"

image.png

where the vertical arrows are homotopy equivalent to the identity. However, I think that the bottom square isn't commutative with the dashed arrow. At most there is a huge diagram

image.png

(where again the square with a dashed arrow doesn't seem to commute)

Daniel Teixeira (Nov 16 2021 at 14:18):

this idea probably stemmed from the desire of having a oo-category with the objects and morphisms of the model category, and as 2-morphisms homotopy equivalences; it's likely another case of slogan-pushing, so maybe it shouldn't be taken much seriously