Coherent homotopical algebras: ``Special gamma-categories'' (Q1372683)

This paper derives from the author's doctoral dissertation. The author introduces a theory of coherence for symmetric monoidal categories in the spirit of \textit{G. Segal} [Topology 13, 293-312 (1974; Zbl 0284.55016)] and shows that it is equivalent, in an appropriate sense, to \textit{S. MacLane}'s original notion [Rice Univ. Stud. 49, No. 4, 28-46 (1963; Zbl 0244.18008)]. More precisely, the author proves that ``special categories'', the analogue of special spaces, and coherently symmetric monoidal categories are one and the same. This is analogous to the situation in topology where special spaces are precisely homotopical commutative monoids. In the light of the observation that the category of small categories Cat bears a functorial Quillen model structure [\textit{D. G. Quillen}, Homotopical algebra, Lect. Notes Math. 43 (1967; Zbl 0168.20903)] with respect to the class of categorial equivalences: in fact is a homotopy theory in the sense of \textit{A. Heller} [Homotopy theories, Mem. Am. Math. Soc. 71, No. 383 (1988; Zbl 0643.55015); Trans. Am. Math. Soc. 272, 185-202 (1982; Zbl 0508.55022)], the theorem can be reinterpreted as stating that coherently symmetric monoidal categories are precisely the homotopical commutative monoids within this new homotopy theory. If \(G\) is an algebraic 2-sketch a homotopical algebra over \(G\) is a 2-functor \(X:G \to\text{Cat}\) such that for all \(m,n\) \(X_{m+n} @>\langle X_{m+0}, X_{0+n} \rangle>> X_m \times X_n\) is an equivalence of categories. The corresponding full subcategory of \(\text{Ho(Cat}^G)\) is denoted by \(\text{Ho Alg} (G,\text{Cat})\). In particular, homotopical algebras over \(\Gamma_{\text{mon}} (\Gamma_{\text{cm}})\) are homotopical (commutative) monoids, while homotopical algebras over \(G_{\text{mon}} (G_{\text{cm}})\) are homotopical coherent (symmetric) monoidal categories. An algebra over \(G\) is a fortiori a homotopical algebra; these algebras determine the full subcategory \(\text{Ho(Alg} (G,\text{Cat})) \subset (\text{Ho Alg}) (G,\text{Cat})\). If \(\Phi: G\to H\) is a morphism of algebraic 2-sketches then \(\text{Ho} \Phi^*: \text{Ho(Cat}^H) \to\text{Ho(Cat}^G)\) restricts to a functor \(\text{Ho Alg}: \text{Ho Alg} (H,\text{Cat}) \to \text{Ho Alg} (G,\text{Cat})\). If \(\Phi\) is an equivalence of 2-diagram schemes then the following theorem holds: Theorem 5.1. If \(\Phi: G\to H\) is an equivalence of algebraic 2-sketches then Ho Alg is an equivalence of categories. The main result of the paper is the following theorem: Theorem 5.3. The inclusions \(\text{Ho(Alg}(G_{\text{mon}},\text{Cat}))\subset (\text{Ho Alg}) (G_{\text{mon}}, \text{Cat})\) and \(\text{Ho(Alg} (G_{\text{cm}}, \text{Cat})) \subset (\text{Ho Alg}) (G_{\text{cm}}, \text{Cat})\) are equivalences of categories. Corollary 5.4. \(\text{Ho(Alg} (\Gamma_{\text{mon}}, \text{Cat})) \to(\text{Ho Alg}) (\Gamma_{\text{mon}}, \text{Cat})\) is an equivalence of categories.