Category Theory: Functor Algebra

Given a category $\mathcal{C}$ and an endo-functor $F : \mathcal{C} \to \mathcal{C}$, an $F$-algebra is a pair $(A, \alpha)$ where:

• $A$ is an object of $\mathcal{C}$, and is called the carrier of the algebra.
• $\alpha : F(A) \to A$ is a morphism in $\mathcal{C}$, called the structure map of the algebra.

Example: Boolean Conjunction Algebra

• The category $\mathcal{C}$ is the category of sets, $\mathbf{Set}$ where objects are sets and morphisms are functions between sets.
• The endo-functor $F : \mathbf{Set} \to \mathbf{Set}$ is the cartesian product $F(X) = X \times X$.
• The carrier $A$ of the algebra is the boolean set $A = \mathbf{Bool} = \{ \text{True}, \text{False} \}$.
• The structure map $\alpha : F(A) \to A$ is defined as the boolean conjunction operation: $\alpha (x, y) = x \land y$.
• The algebra