Generalization of formal monad theory to lax functors

TL;DR

This paper extends formal monad theory to lax functors between bicategories, introducing new notions and theorems that generalize classical results using 2-monads and distributive laws.

Contribution

It generalizes formal monad theory to lax functors, defining lax doctrinal adjunctions and establishing coreflective embeddings and generalized distributive laws.

Findings

Lax doctrinal adjunctions are introduced for 2-monads.

Coreflective embedding of lax algebras generalizes monad-adjunction relations.

A generalized Beck's theorem for distributive laws is established.

Abstract

We study lax functors between bicategories as a generalized concept of monads and describe generalized notions and theorems of formal monad theory for lax functors. Our first approach is to use the 2-monad whose lax algebras are lax functors. We define lax doctrinal adjunctions for a 2-monad $T$ on a 2-category $\mathcal{K}$, and we show that if $\mathcal{K}$ admits and $T$ preserves certain codescent objects, the 2-category $\mathrm{Lax}\text{-}{T}\text{-}\mathrm{Alg}_{c}$ of lax algebras and colax morphisms can coreflectively be embedded in the 2-category of lax doctrinal adjunctions. This coreflective embedding generalizes the relation between monads and adjunctions. Our second approach is to see a distributive law for monads as a 2-functor from a lax Gray tensor product, and we show a generalized form of Beck's characterization of distributive laws.