Medial Commutativity

Abstract

It is shown that all the assumptions for symmetric monoidal categories flow out of a unifying principle involving natural isomorphisms of the type ${(A\otimes B)\otimes(C\otimes D)\to(A\otimes C)\otimes(B\otimes D)}$, called medial commutativity. Medial commutativity in the presence of the unit object enables us to define associativity and commutativity natural isomorphisms. In particular, Mac Lane's pentagonal and hexagonal coherence conditions for associativity and commutativity are derived from the preservation up to a natural isomorphism of medial commutativity by the biendofunctor $\otimes$. This preservation boils down to an isomorphic representation of the Yang-Baxter equation of symmetric and braid groups. The assumptions of monoidal categories, and in particular Mac Lane's pentagonal coherence condition, are explained in the absence of commutativity, and also of the unit object, by a similar preservation of associativity by the biendofunctor $\otimes$. In the final section one finds coherence conditions for medial commutativity in the absence of the unit object. These conditions are obtained by taking the direct product of the symmetric groups $S_{n \choose i}$ for $0\leq i\leq n$.