In a bicategory this equivalence is an identity. In even higher (and non-semi-strict) category theory, the interchange law becomes a higher morphism itself: the exchanger.
Combinatorics of exchange laws
One way to capture all exchange laws combinatorially is encoded by the cosimplicial -category that induces the homotopy coherent nerve. See there for more details on how this encodes the exchange laws.
Revised on January 5, 2017 07:07:47