Coherence for braided and symmetric pseudomonoids

Abstract

Computads for unbraided, braided, and symmetric pseudomonoids in semistrict monoidal bicategories are defined. Biequivalences characterising the monoidal bicategories generated by these computads are proven. It is shown that these biequivalences categorify results in the theory of monoids and commutative monoids, and generalise the standard coherence theorems for braided and symmetric monoidal categories to braided and symmetric pseudomonoids in any weak monoidal bicategory.

Promonads and String Diagrams for Effectful Categories

“…We will now use the pure tensor of promonads to justify effectful categories as the promonadic counterpart of monoidal categories: effectful categories are pseudomonoids in the monoidal bicategory of promonads with the pure tensor. Pseudomonoids [9,43] are the categorification of monoids. They are still formed by a 0-cell representing the carrier of the monoid and a pair of 1-cells representing multiplication and units.…”

Promonads and String Diagrams for Effectful Categories

“…We will now use the pure tensor of promonads to justify effectful categories as the promonadic counterpart of monoidal categories: effectful categories are pseudomonoids in the monoidal bicategory of promonads with the pure tensor. Pseudomonoids [41,44] are the categorification of monoids. They are still formed by a 0-cell representing the carrier of the monoid and a pair of 1-cells representing multiplication and units.…”

Structured Cospans

“…Finally, from this weak category in SymMonCat, we wish to get a symmetric monoidal double category. Here we need the concept of a 'symmetric pseudomonoid' [34]. To understand the following definitions the reader should keep in mind the example where B is Cat made into a symmetric monoidal bicategory using cartesian products.…”