# Lax orthogonal factorisations in monad-quantale-enriched categories

## TL;DR

This paper develops a framework for constructing weak factorisation systems in enriched categorical settings using lax orthogonal factorisation systems derived from presheaf monads on $( ext{T},V)$-categories, with applications to topological categories.

## Contribution

It introduces a method to obtain lax orthogonal factorisation systems in $V$-enriched categories via presheaf monads, extending the theory to topological categories.

## Key findings

- Presheaf monads are simple and induce lax orthogonal factorisation systems.
- The approach applies to well-known topological categories over Set.
- Presheaf submonads define additional LOFSs, enriching the categorical structure.

## Abstract

We show that, for a quantale $V$ and a $\mathsf{Set}$-monad $\mathbb{T}$ laxly extended to $V$-$\mathsf{Rel}$, the presheaf monad on the category of $(\mathbb{T},V)$-categories is simple, giving rise to a lax orthogonal factorisation system (lofs) whose corresponding weak factorisation system has embeddings as left part. In addition, we present presheaf submonads and study the LOFSs they define. This provides a method of constructing weak factorisation systems on some well-known examples of topological categories over $\mathsf{Set}$.