Skew-enriched categories

TL;DR

This paper develops a new framework for enriched categories over skew-monoidal categories, extending existing equivalences and introducing concepts like locally weak comonads to advance the theory of enriched algebraic weak factorisation systems.

Contribution

It introduces skew-enriched categories, extending classical notions to a skew setting and establishing new equivalences with skew actegories and proactegories.

Findings

Defined skew-enriched categories suitable for skew-monoidal enrichment.

Extended the equivalence between tensored categories and actegories to the skew context.

Provided foundational tools for enriched algebraic weak factorisation systems.

Abstract

This paper introduces a skew variant of the notion of enriched category, suitable for enrichment over a skew-monoidal category, the main novelty of which is that the elements of the enriched hom-objects need not be in bijection with the morphisms of the underlying category. This is the natural setting in which to introduce the notion of locally weak comonad, which is fundamental to the theory of enriched algebraic weak factorisation systems. The equivalence, for a monoidal closed category $\mathcal{V}$, between tensored $\mathcal{V}$-categories and hommed $\mathcal{V}$-actegories is extended to the skew setting and easily proved by recognising both skew $\mathcal{V}$-categories and skew $\mathcal{V}$-actegories as equivalent to special kinds of skew $\mathcal{V}$-proactegory.