nLab
actegory

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 (173.60.119.197)