Algebra over a Monad

Definition

A $T$-algebra (or an algebra for the monad $T$) is a pair $(A,\alpha )$ consisting of:

• An objects $A:\mathcal{C}$ (the underlying object).
• A morphism $\alpha :TA\to A$ in $\mathcal{C}$ (the structure map or algebra action).

This pair must satisfy the following two conditions:

• Associativity: $\alpha \circ T\alpha =\alpha \circ {\mu }_A$.
• Unit Law: $\alpha \circ {\eta }_A=1_A$.

\usepackage{tikz-cd}
\begin{document}
\begin{tikzcd}
T^2 A \arrow[r, "T\alpha"] \arrow[d, "\mu_A"'] & TA \arrow[d, "\alpha"] & & A \arrow[r, "\eta_A"] \arrow[rd, "1_A"'] & TA \arrow[d, "\alpha"] \\
TA \arrow[r, "\alpha"'] & A & & & A
\end{tikzcd}
\end{document}

Homomorphisms

A morphism of $T$-algebras (or $T$-homomorphism) from $(A,\alpha )$ to $(B,\beta )$ is a morphism $f:A\to B$ in $\mathcal{C}$ that commutes with the algebra actions. That is, $f\circ \alpha =\beta \circ Tf$.

\usepackage{tikz-cd}
\begin{document}
\begin{tikzcd}
TA \arrow[r, "Tf"] \arrow[d, "\alpha"'] & TB \arrow[d, "\beta"] \\
A \arrow[r, "f"'] & B
\end{tikzcd}
\end{document}

References

• riehl2016-category-theory