Lax orthogonal factorisation systems

TL;DR

This paper develops a framework for lax orthogonal algebraic weak factorisation systems in 2-categories, introducing a construction method based on simple 2-monads and exploring associated lifting operations.

Contribution

It introduces lax orthogonal algebraic weak factorisation systems on 2-categories and a construction method using simple 2-monads, extending previous reflection concepts.

Findings

Each simple 2-monad yields a lax orthogonal algebraic weak factorisation system.

The paper defines and studies KZ lifting, lax natural lifting, and lax orthogonality.

An example is provided via completion under a class of colimits.

Abstract

This paper introduces lax orthogonal algebraic weak factorisation systems on 2-categories and describes a method of constructing them. This method rests in the notion of simple 2-monad, that is a generalisation of the simple reflections studied by Cassidy, H\'ebert and Kelly. Each simple 2-monad on a finitely complete 2-category gives rise to a lax orthogonal algebraic weak factorisation system, and an example of a simple 2-monad is given by completion under a class of colimits. The notions of KZ lifting operation, lax natural lifting operation and lax orthogonality between morphisms are studied.