## Category of F-algebras

### Collection

context | $F$ in ${\bf C}\longrightarrow{\bf C}$ |

definiendum | $\mathcal{A}:\mathrm{Ob}_\mathrm{it}$ |

postulate | $\mathcal{A}$ … $F$-algebra |

definiendum | $\langle f\rangle:\mathrm{it}[\langle A,\alpha\rangle, \langle B,\beta\rangle]$ |

postulate | $f\circ\alpha=\beta\circ F(f)$ |

### 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: F-algbera