Categories enriched over oplax monoidal categories

Abstract

We define a notion of category enriched over an oplax monoidal category $V$, extending the usual definition of category enriched over a monoidal category. Even though oplax monoidal structures involve infinitely many functors $V^n\to V$, defining categories enriched over $V$ only requires the lower arity maps $(n \leq 3)$, similarly to the monoidal case. The focal point of the enrichment theory shifts, in the oplax case, from the notion of $V$-category (given by collections of objects and hom-objects together with composition and unit maps) to the one of categories enriched over $V$ (genuine categories equipped with additional structures). One of the merits of the notion of categories enriched over $V$ is that it becomes straightforward to define enriched functors and natural transformations. We show moreover that the resulting 2-category $\mathsf{Cat}_V$ can be put in correspondence (via the theory of distributors) with the 2-category of modules over $V$. We give an example of such an enriched category in the framework of operads: every cocomplete symmetric monoidal category $C$ is enriched over the category of sequences in $C$ endowed with an oplax monoidal structure stemming from the usual operadic composition product, whose monoids are still the operads. As an application of the study of the 2-functor $V\mapsto\mathsf{Cat}_V$, we show that when $V$ is also endowed with a compatible lax monoidal structure - thus forming a lax-oplax duoidal category - the 2-category $\mathsf{Cat}_V$ inherits a lax 2-monoidal structure, thereby generalising the corresponding result when the enrichment base is a braided monoidal category. We illustrate this result by discussing the lax-oplax structure on the category of $(R^\mathrm{e}, R^\mathrm{e})$-bimodules, whose bimonoids are the bialgebroids. We also comment on the relations with other enrichment theories (monoidal, multicategories, skew and lax).