Coherence of Involutive Symmetric Monoidal Categories

Abstract

The purpose of this chapter is to study coherence of involutive symmetric monoidal categories. Most of the constructions are adapted from Chapter 5, where we studied coherence of involutive monoidal categories. In Section 6.1 and Section 6.2, we give explicit constructions of the free involutive symmetric monoidal category and of the free involutive strict symmetric monoidal category generated by a category. We observe that they are equivalent via a strict involutive strict symmetric monoidal functor. In Section 6.3 we show that in a small involutive symmetric monoidal category, every suitably defined formal diagram is commutative. In Section 6.4 we show that every involutive symmetric monoidal category can be strictified to an involutive strict symmetric monoidal category via an involutive adjoint equivalence involving involutive strong symmetric monoidal functors. The remaining two sections contain explicit constructions of the free involutive (strict) symmetric monoidal category generated by an involutive category.