===== F-algebra =====
==== Collection ====
| @#55CCEE: context     | @#55CCEE: $F$ in ${\bf C}\longrightarrow{\bf C}$ |
| @#FFBB00: definiendum | @#FFBB00: $\langle A,\alpha\rangle$ in $\text{it}$ |
| @#55EE55: postulate   | @#55EE55: $\alpha:{\bf C}[FA,A]$ |

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

<code/Haskell>
type Algebra f a = f a -> a
</code>

=== 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: [[http://en.wikipedia.org/wiki/F-algebra|F-algbera]]
==== Parents ====
=== Context ===
[[Endofunctor]]