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

TL;DR

This paper demonstrates that a model structure on a relative infinity-category provides an effective method to compute hom-spaces in the localized category, especially when the source is cofibrant and the target is fibrant.

Contribution

It establishes that hom-spaces in the localization can be obtained as quotients of original hom-spaces via homotopy relations under certain conditions.

Findings

Hom-spaces in the localization are quotients of original hom-spaces.

Cofibrant sources and fibrant targets enable explicit hom-space calculations.

Provides a practical approach to access hom-spaces in relative infinity-categories.

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 $hom_{M[[W^{-1}]]}(x,y)$ 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 $hom_M(x,y)$ by either of a *left homotopy relation* or a *right homotopy relation*.