algebra of a functor

categories

category theory

An algebra of an endofunctor \( F : C \to C \) is an object \( x \in C_0 \)—the carrier of the algebra—together with an arrow \( \alpha : F(x) \to x \).

Dually, a coalgebra of \( F \) is an object \( x \in C_0 \) together with an arrow \( x \to F(x) \).

An algebra of a functor \( F \) is usually called an \( F \)-algebra. Often the functor \( F \) is left implicit, so that the specific term “\( F \)-algebra” means “algebra of some functor \( F \)”.

Algebras of a (fixed) functor \( F \) form a category. An arrow from \( (x, \alpha) \) to \( (y, \beta) \) is an arrow \( m : x \to y \) such that the following square commutes:

\begin{tikzcd} {F(x)} & {F(y)} \\ x & y \arrow["{F(m)}", from=1-1, to=1-2] \arrow["\alpha"', from=1-1, to=2-1] \arrow["\beta", from=1-2, to=2-2] \arrow["m"', from=2-1, to=2-2] \end{tikzcd}

Likewise, \( F \)-coalgebras form a category; an arrow from \( (x, \alpha) \) to \( (y, \beta) \) is an arrow \( m : x \to y \) such that the following square commutes:

\begin{tikzcd} {F(x)} & {F(y)} \\ x & y \arrow["{F(m)}", from=1-1, to=1-2] \arrow["\alpha", from=2-1, to=1-1] \arrow["\beta"', from=2-2, to=1-2] \arrow["m"', from=2-1, to=2-2] \end{tikzcd}