Pointed Algebra

A pointed category of some category $\mathcal{C}$ is just the over $1/\mathcal{C}$. That is the category with pairs of objects and elements of those objects, along with morphisms between objects that preserve points.

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Let $\mathcal{C}$ be a category with a terminal object and finite Coproduct (Category Theory). Let $F:\mathcal{C}\Rightarrow \mathcal{C}$, then an Algebra is an object $X:\mathcal{C}$ and morphism $\alpha :FX\to X$. A pointed algebra additionally has $x:1\to X$, making $(X,\alpha :FX\to X,x:1\to X)$ which is isomorphic to,

$$(X,\beta :FX+1\to X)$$