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



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,,I,l,r,a)(C,\otimes,I,l,r,a) an actegory is another category DD with a notion of “tensor by object of CC”, i.e., a functor:

:C×DD\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.


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-actegory is a category AA together with a (left or right) coherent action of CC on AA. Depending on an 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-actegories, 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-actegory) is easily extendable to bicategories (see Baković‘s thesis).


A (left) 𝒞\mathcal{C}-(pseudo)actegory 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 an actegory on the ordinary category underlying DD.

Conversely, if an actegory 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.