A Constructive Proof of Coherence for Symmetric Monoidal Categories Using Rewriting

A symmetric monoidal category is a category equipped with an associative and commutative (binary) product and an object which is the unit for the product. In fact, those properties only hold up to natural isomorphisms which satisfy some coherence conditions. The coherence theorem asserts the commutativity of all linear diagrams involving the left and right unitors, the associator and the braiding. We prove the coherence for symmetric monoidal categories using a constructive homotopical method based on higher dimensional rewriting. For that scope, we detail the convergence proof of Lafont’s string diagram rewriting system which presents the isomorphisms of these theories.