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Stream: theory: category theory

Topic: Kan enriched categories and localisation

Lukas Heidemann (Jul 04 2022 at 16:31):

When $\mathcal{C}$ is a Kan enriched category, then there is a functor of $(\infty, 1)$-categories $\mathcal{C}_0 \to \mathcal{C}$ where $\mathcal{C}_0$ is the underlying 1-category of $\mathcal{C}$. Is it always true that this functor exhibits a localisation of $\mathcal{C}_0$ at the homotopy equivalences? If not, are there some tools to check this which are ideally more lightweight than finding a simplicial model category?

Mike Shulman (Jul 04 2022 at 16:39):

It's not always true. For instance, let $M$ be a simplicial group with a single 0-simplex; then the corresponding 1-object Kan-enriched category has a terminal underlying 1-category, which it cannot be the localization of.

Lukas Heidemann (Jul 11 2022 at 11:53):

In the case someone comes across this question in the future, Proposition 1.3.4.7. in Higher Algebra gives a simple way to check if a Kan enriched category is the localisation of its underlying 1-category.