nLab action of a monoidal category

Idea

In vertical categorification of how monoids/monoid objects $A$ may act on other objects $N$ (action objects, module objects) inside an ambient monoidal category by maps

$$A \otimes N \longrightarrow N \,.$$

satisfying the action property, so a monoidal category $\mathcal{A}$ may act on other categories $\mathcal{N}$ by functors

$$\oslash \;\colon\; \mathcal{A} \times \mathcal{N} \longrightarrow \mathcal{N}$$

subject to associators, unitors and coherence conditions for action objects coherently internalized into the 2-category Cat, analogous to the laws in a monoidal category.

At least if some linear structure is present and respected (such as when $\mathcal{A}$ qualifies as a 2-ring) it is natural to speak of module categories over $\mathcal{A}$ (see also at $n$-module).

Similarly, compatible actions from both sides, such as for a bimodule, give a notion of bimodule category.

Note that the term actegory, introduced by McCrudden (2000)¹, is often used in the literature for this concept, and consequently biactegory for the two-sided case. However, since “actegory” is a single transposition away from “category”, we prefer to use the explicit terminology on this page and elsewhere.

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) $C$-module category is a category $A$ together with a (left or right) coherent action of $C$ on $A$. Depending on 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.

$C$-module categories, 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 $C$-module category) is easily extendable to bicategories (see Baković‘s thesis).

Definition

A (left) $\mathcal{C}$-module category 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 a module category on the ordinary category underlying $D$.

Conversely, if module category 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.

More generally, supposing $C$ is small, the following data are equivalent: (1) a $C$-module category $D$; (2) an enrichment of $D$ in the category $\hat C$ of presheaves on $C$ (i.e. a locally graded category), with copowers by representables. Here we regard $\hat C$ as a monoidal category with the Day convolution.

Starting from a $C$-module category $D$, consider the enrichment given by $D(d,d')=hom(-\oslash d,d'):C^{op}\to Set$. If these presheaves are representable, this is what it means to be enriched in $C$ for which the action is a copower. If they are not representable, it is still an enrichment, and the copowers by representables are the action.

For this reason, many concepts from enriched category theory make sense for module categories too.

Related Concepts

• action object, module object

• 2-module, n-module

• monoidal actegory

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. McCrudden, Categories of representations of coalgebroids, Advances in Mathematics 154 2 (2000) 299–332 [doi:10.1006/aima.2000.1926]

• Peter 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

• J. R. B. Cockett, Craig Pastro, The logic of message-passing arXiv:math/0703713.

A recent survey of many basic definitions and operations on actegories is

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

A generalisation from monoidal categories to bicategories is studied in, where actions are called representations:

• Robert Gordon, and John Power. Enrichment through variation, Journal of Pure and Applied Algebra 120.2 (1997): 167-185.

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1. From Cockett & Pastro (2007): “The term actegory is used to describe the situation of a monoidal category “acting” on a category. They first appeared (under a different name) in the work of Bénabou as a simple example of a bicategory. B. Pareigis developed the theory of actegories (again under a different name) and showed there usefulness in the representation theory of monoids and comonoids. The word “actegory” was first suggested at the Australian Category Seminar and first appeared in print in the thesis of P. McCrudden where they were used to study categories of representations of coalgebroids.” ↩