Categories with homotopy determine a closed model structure

Abstract

In a category with homotopy $\mathfrak{K}$ (Definition 1.1), one can define a natural concept of (co)fibrations and weak equivalences (Sec. 2) such that some properties of a closed model category hold. If $\mathfrak{K}$ is a complete and cocomplete category (with respect to finite limits) with simplicial homotopy (Definition 1.3), one achieves a full closed model structure in $\mathfrak{K}$ (Theorem 4.6). For each category with homotopy $\mathfrak{K}$, there exists a category with simplicial homotopy $^s \mathfrak{K} \supseteq \mathfrak{K}$ (see Sec. 5) (the simplicial envelope of $\mathfrak{K}$). If $\mathfrak{K}$ is already a complete category with simplicial homotopy, then $\mathfrak{K}$ is Quillen equivalent to $^s \mathfrak{K}$ (Theorem 5.9).