## F-algebra

### Collection

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

### Discussion

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

type Algebra f a = f a -> a

#### Example

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 $M=\mathbb{N}$.

#### Reference

Wikipedia: F-algbera