nLab
actegory

For any category $A$, the category of endofunctors $\mathrm{End}(A)$ is monoidal with respect to the (horizontal) composition (the composition of functors and the Godement product for natural transformations).

Given a monoidal category $(C,\otimes ,1,l,r,a)$ a (left or right) $C$-actegory is a category $A$ together with a (left or right) coherent action of $C$ on $A$. Depending on an author and context, the left coherent action of $C$ on $A$ is a morphism of monoidal categories $C\to \mathrm{End}(A)$ in the lax, colax, pseudo or strict sense (most often in pseudo-sense) or, in another terminology, a monoidal, comonoidal, strong monoidal or strict monoidal functor. Right coherent actions correspond to the monoidal functors into the category $\mathrm{End}(A)$ with the opposite tensor product.

$C$-actegories, colax $C$-equivariant functors and natural transformations of colax $C$-equivariant functors form a strict 2-category ${}_C{\mathrm{Act}}^c$. A monad in ${}_C{\mathrm{Act}}^c$ amounts to a pair of a monad in $\mathrm{Cat}$ and a distributive law between the monad and an action of $C$.

The notion of $C$-action (hence a $C$-actegory) is easily extendable to bicategories (see Baković’s thesis).

• Bodo Pareigis, Non-additive ring and module theory I. General theory of monoids, Publ. Math. Debrecen 24 (1977), 189–204. MR 56:8656; Non-additive ring and module theory II. C-categories, C-functors, and C-morphisms, Publ. Math. Debrecen 24 (351–361) 1977.

• M. Kelly, G. Janelidze, A note on actions of a monoidal category, Theory and Applications of Categories, Vol. 9, 2001, No. 4, pp 61–91 link

• P. Schauenburg, Actions of monoidal categories and generalized Hopf smash products, J. Algebra 270 (2003) 521–563 (remark: actegories with action in the strong monoidal sense)

• Z. Škoda, Distributive laws for actions of monoidal categories, arxiv:0406310, Equivariant monads and equivariant lifts versus a 2-category of distributive laws, arxiv:0707.1609

If an actegory is like a module, then a biactegory is like a bimodule.