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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 $\alpha:{\bf C}[FA,A],\beta{\bf C}[FB,B]$ were arrows in ${\bf C}$, the function $f:A\to B$ can be concatenated with those, and $\langle f\rangle$ denotes the arrow between $F$-algebras $\langle A,\alpha\rangle$ and $\langle B,\beta\rangle$ corresponding to the homomorphism $f:A\to B$. Clearly, $\langle f\rangle$ and $f$ are in bijection and one often just write $f$ for both.
Reference
Wikipedia: F-algbera