nLab action of a monoidal category



Monoidal categories

monoidal categories

With symmetry

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products



Internal monoids



In higher category theory

Category theory



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

ANN. A \otimes N \longrightarrow N \,.

satisfying the action property,

so a monoidal category 𝒜\mathcal{A} may act on other categories 𝒩\mathcal{N} by functors

:𝒜×𝒩𝒩 \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 𝒜\mathcal{A} qualifies as a 2-ring) it is natural to speak of module categories over 𝒜\mathcal{A} (see also at n 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.


For any category AA, the category of endofunctors End(A)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,,I,l,r,a)(C,\otimes,I,l,r,a) a (left or right) CC-module category is a category AA together with a (left or right) coherent action of CC on AA. Depending on author and context, the left coherent action of CC on AA is a morphism of monoidal categories CEnd(A)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)End(A) with the opposite tensor product.

CC-module categories, colax CC-equivariant functors and natural transformations of colax CC-equivariant functors form a strict 2-category CAct c_C Act^c. A monad in CAct c_C Act^c amounts to a pair of a monad in CatCat and a distributive law between the monad and an action of CC.

The notion of CC-action (hence a CC-module category) is easily extendable to bicategories (see Baković‘s thesis).


A (left) 𝒞\mathcal{C}-module category is

  1. a category 𝒜\mathcal{A};
  2. a functor :𝒞×𝒜𝒜\oslash : \mathcal{C} \times \mathcal{A} \to \mathcal{A} called the action;
  3. a natural isomorphism λ a:aIa\lambda_a : a \to I \oslash a called the unitor;
  4. a natural isomorphism α c,d,a:c(da)(cd)a\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 λ\lambda and α\alpha with the unitors and associators of 𝒞\mathcal{C}.

Connection with enrichment

If a category DD is enriched in CC with copowers, then the copower structure forms a module category on the ordinary category underlying DD.

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


  • 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]

  • 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

  • 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

  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.”

Last revised on December 12, 2022 at 12:52:12. See the history of this page for a list of all contributions to it.