nLab biactegory

Let $C$ be a monoidal category (in fact one can easily modify all statements to generalize all statements here to bicategories). One can consider several variants of the 2-category of all categories with monoidal action of $C$, (co)lax monoidal functors and their transformations. A category with an action of $C$ is sometimes called a $C$-actegory. The word ‘module category’ over $C$ is also used, especially when the category acted upon is in addition also additive, like the examples in representation theory.

If a $C$-actegory is a categorification of a module, then for two monoidal categories $C$ and $D$, we should categorify a bimodule, which we call $C$-$D$-biactegory. The two actions on a usual bimodule commute; for biactegories the commuting is up to certain coherence laws, which are in fact the expression of an invertible distributive law between the two monoidal actions. The tensor product of biactegories can be defined (here the invertibility of the distributive law is needed) as a bicoequalizer of a certain diagram.

For very basic outline see section 2 in

• Zoran Škoda, Some equivariant constructions in noncommutative algebraic geometry, Georgian Mathematical Journal 16 (2009), No. 1, 183–-202, arXiv:0811.4770.

and partial writeup

• Zoran Škoda, Biactegories, 2006, pdf)

In a language of “module categories”, a different treatment is now available in

• Justin Greenough, Monoidal 2-structure of bimodule categories, arxiv:0911.4979.

It is one of the notions discussed in the review

• Matteo Capucci, Bruno Gavranović, Actegories for the working amthematician, arXiv:2203.16351