Skew-enriched categories

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.