Model \(\infty\)-categories. III: The fundamental theorem (Q2662943)

Suppose that a relative \(\infty\)-category \((\mathcal{M},\mathbf{W})\) is equipped with the structure of a model \(\infty\)-category. Then the author proves that if \(x\in \mathcal{M}\) is cofibrant and \(y\in \mathcal{M}\) is fibrant, then the hom-space \(\mathcal{M}[[\mathbf{W}^{-1}]](x,y)\) is a ``quotient`'' of the hom-space \(\mathcal{M}(x,y)\) by either a left homotopy relation or a right homotopy relation. For part II, see [the author, ibid. 27, 508--550 (2021; Zbl 1461.18002)].