===== Category of F-algebras =====
==== Collection ====
| @#55CCEE: context     | @#55CCEE: $F$ in ${\bf C}\longrightarrow{\bf C}$ |
| @#FFBB00: definiendum | @#FFBB00: $\mathcal{A}:\mathrm{Ob}_\mathrm{it}$ |
| @#55EE55: postulate   | @#55EE55: $\mathcal{A}$ ... $F$-algebra |
| @#FFBB00: definiendum | @#FFBB00: $\langle f\rangle:\mathrm{it}[\langle A,\alpha\rangle, \langle B,\beta\rangle]$ |
| @#55EE55: postulate   | @#55EE55: $f\circ\alpha=\beta\circ F(f)$ |

==== Discussion ====
The category of F-algebras and F-algebra homomorphisms. The postulate says that it can't matter if you perform the operation ($\alpha$ resp. $\beta$) before or after the transformation $f$.

Note that $\alpha,\beta,f$ are arrows in ${\bf C}$, while $\langle f\rangle$ denotes the arrow between $F$-algebras $\langle A,\alpha\rangle$ and $\langle B,\beta\rangle$ corresponding to the //homomorphism// $f$. Clearly, $\langle f\rangle$ and $f$ are in bijection and one often just writes $f$ for both.

=== Reference ===
Wikipedia: [[http://en.wikipedia.org/wiki/F-algebra|F-algbera]]
==== Parents ====
=== Context* ===
[[Endofunctor]]
=== Requirements ===
[[F-algebra]]
=== Element of ===
[[Categories]]