Eilenberg-Moore Category

Given a monad $(T,\eta ,\mu )$ on a category $\mathcal{C}$, the Eilenberg-Moore category ${\mathcal{C}}^T$ is the category of algebras for that monad. It provides a way to study the algebraic structure defined by the monad.

Algebra over a Monad

Definition of a $T$-algebra

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

• An object $A\in \mathcal{C}$ (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, analogous to the associativity and unit laws of a group action:

1. Associativity: $\alpha \circ T\alpha =\alpha \circ {\mu }_A$.
2. 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}

Algebra 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}

The Eilenberg-Moore Category

The Eilenberg-Moore category ${\mathcal{C}}^T$ has:

• Objects: All $T$-algebras $(A,\alpha )$.
• Morphisms: All $T$-algebra homomorphisms between them.
• Composition and Identities: Inherited from the underlying category $\mathcal{C}$.

Forgetful and Free Functors

There is a canonical adjunction between $\mathcal{C}$ and ${\mathcal{C}}^T$:

• The forgetful functor $U^T:{\mathcal{C}}^T\to \mathcal{C}$ sends $(A,\alpha )$ to $A$ and $f$ to $f$.
• The free functor $F^T:\mathcal{C}\to {\mathcal{C}}^T$ sends an object $X$ to the free algebra $(TX,{\mu }_X)$ and a morphism $g$ to $Tg$.

This adjunction $F^T⊣U^T$ recovers the original monad $T$ on $\mathcal{C}$.

References

• riehl2016-category-theory