Axiomatization of the Coherence Property for Categories of Symmetries

Given an equational theory (Σ, E), a relaxed (Σ, E)-system is a category S enriched with a Σ-algebra structure on both objects and arrows such that a natural isomorphism σ S ⇒ t′S), called natural symmetry, exists for each t = E t′. A symmetry is an instance of a natural symmetry. A category of symmetries, which includes only symmetries, is a free object in the category of relaxed (Σ, E)-systems. The coherence property states that the diagrams in a category of symmetries are commutative. In this paper we present a method for expressing the coherence property in an axiomatic way.