Algebra (Category Theory)

Call the category of algebras $\mathrm{Alg}(\mathcal{C},F)$.
Given a Functor $F:\mathcal{C}\to \mathcal{C}$
A $F$-algebra is $(X:\mathcal{C},\alpha :FX\to X)$
$X$ is called the carrier.

Morphisms

Given two algebras $(X:\mathcal{C},\alpha :FX\to X)$ , $(Y:\mathcal{C},\beta :FY\to Y)$, an algebra homomorphism is a $\mathcal{C}$ arrow $\theta :X\to Y$ s.t. the following commutes:

Initial $F$ algebra

Write $\mu F:\mathcal{C}$ for the Initial Algebra of $F$.

See also

Algebra of Magmas
Coalgebra (Category Theory)
Initial Algebra