Bi-representable bi-multicategories

Abstract

We prove a two-dimensional version of Hermida's equivalence result on representable multicategories and classify its stricter versions. The obtained three-dimensional equivalence of the respective totality categories is of categories enriched over the multicategory $\mathsf{Bicat}$ of Verity. There is a basic enriched equivalence with respect to inherently lax monoidal pseudofunctors, and a one with respect to pseudo-monoidal pseudofunctors, related to the latter via the forgetful enriched functors. We extend Slattery's result on a bi-multicategory structure of the Kleisli bicategory of a 2-multicategory encompassing bi-representability.