Model $\infty$-categories III: the fundamental theorem

Abstract

We prove that a model structure on a relative $\infty$-category $(M,W)$ gives an efficient and computable way of accessing the hom-spaces $ho{m}_{M[[W^{-1}]]}(x,y)$ in the localization. More precisely, we show that when the source $x\in M$ is *cofibrant* and the target $y\in M$ is *fibrant*, then this hom-space is a "quotient" of the hom-space $ho{m}_M(x,y)$ by either of a *left homotopy relation* or a *right homotopy relation*.

References in corpus (5)

• Higher Topos Theory
• Quillen adjunctions induce adjunctions of quasicategories
• All about the Grothendieck construction
• Model $\infty$-categories I: some pleasant properties of the $\infty$-category of simplicial spaces
• Model $\infty$-categories II: Quillen adjunctions

Cited by in corpus (2)

• Model $\infty$-categories I: some pleasant properties of the $\infty$-category of simplicial spaces
• Goerss--Hopkins obstruction theory for $\infty$-categories