From Coherent Structures to Universal Properties

TL;DR

This paper develops a theoretical framework transforming coherent structures into universal properties within 2-category theory, enabling classification and coherence results for pseudo-algebras and morphisms.

Contribution

It constructs a new 2-category with an adjoint-pseudo-algebra property, establishing an equivalence between pseudo-algebras of different 2-monads and providing intrinsic characterizations.

Findings

Classification of lax and strong morphisms

Coherence results for pseudo-algebras

Application to internal categories and monoidal structures

Abstract

Given a 2-category $\twocat{K}$ admitting a calculus of bimodules, and a 2-monad T on it compatible with such calculus, we construct a 2-category $\twocat{L}$ with a 2-monad S on it such that: (1)S has the adjoint-pseudo-algebra property. (2)The 2-categories of pseudo-algebras of S and T are equivalent. Thus, coherent structures (pseudo-T-algebras) are transformed into universally characterised ones (adjoint-pseudo-S-algebras). The 2-category $\twocat{L}$ consists of lax algebras for the pseudo-monad induced by T on the bicategory of bimodules of $\twocat{K}$. We give an intrinsic characterisation of pseudo-S-algebras in terms of representability. Two major consequences of the above transformation are the classifications of lax and strong morphisms, with the attendant coherence result for pseudo-algebras. We apply the theory in the context of internal categories and examine monoidal and monoidal globular categories (including their monoid classifiers) as well as pseudo-functors into $\Cat$.