nLab algebra for an endomorphism

Definition

If $K$ is a bicategory and $f \colon a \to a$ is an endomorphism in $K$, then a (left) $f$-algebra or $f$-module is given by a 1-cell $x \colon b \to a$ together with a 2-cell $\lambda \colon f x \Rightarrow x$.

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

Examples

If $K$ is $Cat$, an algebra for an endofunctor $F \colon C \to C$ is the same thing as an $F$-algebra $A \colon \ast \to C$ in the sense above.

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

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