f-algebra

F-algebra

context	$F$ in ${\bf C}\longrightarrow{\bf C}$
definiendum	$\langle A,\alpha\rangle$ in $\text{it}$
postulate	$\alpha:{\bf C}[FA,A]$

Think types $\mathrm{a}$ and $\alpha$'s of type

type Algebra f a = f a -> a

The following examples assume that ${\bf C}$ contains all the relevant ingredients (e.g. products).

Addition of natural numbers is a binary relation: $+:\mathbb{N}\times\mathbb{N}\to\mathbb{N}$. Hence $\langle \mathbb{N},+\rangle$ is an $F$-algebra for the endofunctor with object map $FX:=X\times X$.

Fix a monoid $M$. A monoid action on $X$ is a map $\alpha:M\times X\to X$, so consider $FX:=M\times X$. Incidentally, $\langle \mathbb{N},+\rangle$ can also be viewed as an $F$-algebra for $M=\mathbb{N}$.