Lax orthogonal factorisations in monad-quantale-enriched categories

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 $\mathsf{Set}$.

13 pages. Minor changes from previous version

References in corpus (4)

• Algebraic weak factorisation systems I: accessible AWFS
• Lax orthogonal factorisation systems
• Relative injectivity as cocompleteness for a class of distributors
• Cofibrantly generated lax orthogonal factorisation systems