Coherence theorems for Monoidal Categories

Abstract

In this thesis we look at two statements relating to coherence of monoidal categories and give the categorical background needed to formulate it. The first one is Mac Lane’s coherence theorem, which states that all diagrams containing only α, λ, ρ, 1, – ⊗ – commute. The second one is strictification, which says that all monoidal categories are monoidally equivalent to a strict monoidal category. We prove this second theorem using an adapted version of the Yoneda lemma for 2-categories to monoidal categories. We also briefly discuss the possibility of proving coherence from strictification.