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 ‘tensor product’ functors Vn→V , the definition of categories enriched over V only requires the lower arity maps (n ⩽ 3), similarly to the monoidal case. The focal point of the enrichment theory shifts, in the oplax case, from the notion of V -category (classically 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 both enriched functors and enriched natural transformations. We show moreover that the resulting 2-category CatV 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 (planar) operads. As an application of the study of the 2-functor V 7→ CatV , we show that when V is also endowed with a compatible lax monoidal structure— thus forming a lax-oplax duoidal category—the 2-category CatV 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 in details the lax-oplax structure on the category of (Re,Re)-bimodules, whose bimonoids are the bialgebroids. We conclude by commenting on the relations between the enrichment theory over oplax monoidal categories and other enrichment theories (monoidal, multicategories, skew and lax).