nLab
algebra for an endomorphism

Definition

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

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

Examples

If $K$ is $\mathrm{Cat}$, an algebra for an endofunctor $F:C\to C$ is the same thing as an $F$-algebra $A:*\to C$ 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.) $H:C⇸C$ on $X:D\to C$ is essentially the same as a $H$-coalgebra $C(1,X)\Rightarrow H\circ C(1,X)$ in $\mathrm{Prof}$, the bicategory of categories and profunctors.