Monad (Category Theory)

Definition

A monad $(T,\eta ,\mu )$ in a category $\mathcal{C}$ consists of:

• A functor $T:\mathcal{C}\to \mathcal{C}$.
• Natural transformations $\eta :1_{\mathcal{C}}\Rightarrow T$ (unit) and $\mu :T^2\Rightarrow T$ (multiplication).
• Such that the following diagrams commute in the category of endofunctors $[\mathcal{C},\mathcal{C}]$:

Associativity Law

We require that $\mu \circ T\mu =\mu \circ \mu T$ as natural transformations from $T^3$ to $T$.

\usepackage{tikz-cd}
\begin{document}
\begin{tikzcd}
T^3 \arrow[r, "T\mu"] \arrow[d, "\mu T"'] & T^2 \arrow[d, "\mu"] \\
T^2 \arrow[r, "\mu"'] & T
\end{tikzcd}
\end{document}

Unit Laws

We require that $\mu \circ T\eta =1_T=\mu \circ \eta T$ as natural transformations from $T$ to $T$.

\usepackage{tikz-cd}
\begin{document}
\begin{tikzcd}
T \arrow[r, "T\eta"] \arrow[d, "\eta T"'] \arrow[rd, "1_T"] & T^2 \arrow[d, "\mu"] \\
T^2 \arrow[r, "\mu"'] & T
\end{tikzcd}
\end{document}

Comonad

A comonad is a monad in the opposite category ${\mathcal{C}}^{op}$.

Relation to Adjunctions

Given any adjunction $F⊣G$ for $F:\mathcal{C}\Rightarrow \mathcal{D},G:\mathcal{D}\Rightarrow \mathcal{C}$ , $(GF,\eta ,G\epsilon F)$ defines a monad in $\mathcal{C}$ .

See also

• Eilenberg-Moore Category
• Adjunction
• Kliesli Category

References

• riehl2016-category-theory
• maclane1978-categories

A monad or a Kleisli triple over a category $\mathcal{C}$ is a triple $(T,\eta ,{\_}^{\ast })$ where $T:\mathcal{C}\to \mathcal{C}$… See capretta2005-partiality.