nLab
actegory

Idea

As monoidal categories are a vertical categorification of monoids, actegories are a vertical categorification of actions of a monoid. So given a monoidal category $(C,\otimes,I,l,r,a)$ an actegory is another category $D$ with a notion of “tensor by object of $C$”, i.e., a functor:

$$\oslash : C \times D \to D$$

that is associative and unital up to natural isomorphism with respect to $\otimes$ in ways that generalize actions of a monoid, and satisfy coherence laws similar to those of a monoidal category.

Definition

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

Given a monoidal category $(C,\otimes,I,l,r,a)$ 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 $C\to End(A)$ 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 $End(A)$ with the opposite tensor product.

$C$-actegories, colax $C$-equivariant functors and natural transformations of colax $C$-equivariant functors form a strict 2-category $_C Act^c$. A monad in $_C Act^c$ amounts to a pair of a monad in $Cat$ 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).

Definition

A (left) $\mathcal{C}$-(pseudo)actegory is

1. a category $\mathcal{A}$;
2. a functor $\oslash : \mathcal{C} \times \mathcal{A} \to \mathcal{A}$ called the action;
3. a natural isomorphism $\lambda_a : a \to I \oslash a$ called the unitor;
4. a natural isomorphism $\alpha_{c,d,a} : c \oslash (d \oslash a) \to (c \otimes d) \oslash a$ called the actor;

satisfying a pentagonal and two triangular laws (see KJ01, diagg. (1.1)-(1.3)) that witness the coherence of $\lambda$ and $\alpha$ with the unitors and associators of $\mathcal{C}$.

Connection with enrichment

If a category $D$ is enriched in $C$ with copowers, then the copower structure forms an actegory on the ordinary category underlying $D$.

Conversely, if an actegory is such that the functor $(-)\oslash d:C\to D$ has a right adjoint for all objects $d$ of $D$, then the right adjoints $D(d,-):D\to C$ provide an enrichment of $D$ in $C$ for which the action is a copower. See KJ01.

Related Concepts

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

References

• 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.

• Max Kelly, George 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)

• Zoran 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