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:

\usepackage{tikz-cd}
\begin{document}
\begin{tikzcd}
F X \arrow[r, "F\theta"] \arrow[d, "\alpha"']
  & F Y \arrow[d, "\beta"] \\
X \arrow[r, "\theta"']
  & Y
\end{tikzcd} 
\end{document}

Initial $F$ algebra

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

See also

Algebra of Magmas Coalgebra (Category Theory) Initial Algebra

Duplicate Content 1

Call the category of algebras it $\mathrm{Alg}(\mathcal{C},F)$. 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:

$$\text{require}AMScd\begin{array}{ccc}FX& \stackrel{F\theta }{\to }& FY\\ \alpha \downarrow & & \beta \downarrow \\ X& \stackrel{\theta }{\to }& Y\end{array}$$

Initial $F$ algebra

Write $\mu F:\mathcal{C}$ for the initial algebra of $F$, and let ${\mathrm{in}}_F:\mathcal{C}(T\mu T,\mu T)$ s.t. for any $F$ algebra, there is a pair of

See also

Algebra of Magmas Coalgebra (Category Theory) Initial Algebra