nLab
algebra for an endomorphism

Definition

If KK is a bicategory and f:aaf \colon a \to a is an endomorphism in KK, then a (left) ff-algebra or ff-module is given by a 1-cell x:bax \colon b \to a together with a 2-cell λ:fxx\lambda \colon f x \Rightarrow x.

One can also define right modules/algebras, comodules/coalgebras and bimodules as for monads.

Examples

If KK is CatCat, an algebra for an endofunctor F:CCF \colon C \to C is the same thing as an FF-algebra A:*CA \colon \ast \to C in the sense above.

Every module over a monad (t,η,μ)(t, \eta, \mu) is an algebra over the underlying endomorphism tt.

An algebra for a profunctor (q.v.) H:CCH \colon C ⇸ C on X:DCX \colon D \to C is essentially the same as a HH-coalgebra C(1,X)HC(1,X)C(1,X) \Rightarrow H \circ C(1,X) in ProfProf, the bicategory of categories and profunctors.

Revised on September 23, 2010 21:13:48 by Finn Lawler (86.41.31.112)