Coherence of Involutive Monoidal Categories

The purpose of this chapter is to study coherence of involutive monoidal categories. The next chapter will deal with the symmetric case. In Section 5.1 and Section 5.2, we give explicit constructions of the free involutive monoidal category and of the free involutive strict monoidal category generated by a category. We observe that they are equivalent via a strict involutive strict monoidal functor. In Section 5.3 we show that in a small involutive monoidal category, every formal diagram is commutative. Here a formal diagram is defined as in the case of monoidal categories, but with the involutive structure also taken into account. In Section 5.4 we show that every involutive monoidal category can be strictified to an involutive strict monoidal category via an involutive adjoint equivalence involving involutive strong monoidal functors. The remaining two sections contain explicit constructions of the free involutive (strict) monoidal category generated by an involutive category.