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f-algebra [2014/09/22 23:21] nikolaj |
f-algebra [2014/09/23 10:36] (current) nikolaj |
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| === Example === | === Example === | ||
| - | Addition of natural numbers is a binary relation: | + | The following examples assume that ${\bf C}$ contains all the relevant ingredients (e.g. products). |
| - | $+:\mathbb{N}\times\mathbb{N}\to\mathbb{N}.$ | + | * 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 $FX:=X\times X$. |
| - | 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 === | === Reference === | ||
