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,,1,l,r,a)(C,\otimes,1,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).

  • 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.

Revised on December 16, 2009 19:47:02 by Toby Bartels (