Recently, I'm learning $F$-algebra.
In wikipedia, it says that
If $C$ is a category, and $F: C → C$ is an endofunctor of $C$, then an $F$-algebra is a tuple $(A, α)$, where $A$ is an object of $C$ and $α$ is a $C$-morphism $F(A) → A$. The object $A$ is called carrier of the algebra.
A homomorphism from an $F$-algebra $(A, α)$ to an $F$-algebra $(B, β)$ is a $C$-morphism $f: A→ B$ such that $f \circ α = β \circ F(f)$.
The category of $F$-algebras seems to be very similar to some kind of comma category. (One exception is that objects in comma category are triples, but objects in category of $F$-algebras are pairs).
Can I describe category of $F$-algebras by some kind of comma category?
Thanks.
Let $F : \mathscr C \to \mathscr C$ be a functor. The comma category $F \downarrow \mathrm{Id}_{\mathscr C}$ has as objects triples $(A \in \mathscr C, A' \in \mathscr C', \alpha : FA \to A')$ and as morphisms from $(A \in \mathscr C, A' \in \mathscr C', \alpha : FA \to A')$ to $(B \in \mathscr C, B' \in \mathscr C', \beta : FB \to B')$ pairs $(f : A \to B, f' : A' \to B')$ such that the following diagram commutes.
Here, we see the problem with expressing the category of $F$-algebras as a comma category: to restrict to the subcategory in which $A = A'$ and $B = B'$, we need to be able to choose the two objects from $\mathscr C$ in such a way that one depends on another, but in a comma category the choices are entirely independent. So, despite the seeming similarity between the two constructions, we cannot simply describe the category of $F$-algebras as a comma category.
(We can trivially describe the category of $F$-algebras as a comma category by considering $F\text{-}\mathbf{alg} \to \mathbb 1 \downarrow \mathrm{Id}_{\mathbb 1}$, but not in a way that takes advantage of the seeming similarity.)