Subsection 8.4.4 (00L7): Internal Monads

We have a natural identification¹

\[ \left\{ \begin{gathered} \text{Monads in}\\ \text{$\boldsymbol {\mathsf{Rel}}$ on $A$} \end{gathered} \right\} \cong \left\{ \text{Preorders on $A$}\right\} . \]

¹See also for an extension of this correspondence to “relative monads in $\boldsymbol {\mathsf{Rel}}$”.

A monad in $\boldsymbol {\mathsf{Rel}}$ on $A$ consists of a relation $R\colon A\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}A$ together with maps

\begin{align*} \mu _{R} & \colon R\mathbin {\diamond }R \subset R,\\ \eta _{R} & \colon \chi _{A} \subset R \end{align*}

making the diagrams

commute. However, since all morphisms involved are inclusions, the commutativity of the above diagrams is automatic (Chapter 11: Categories, Item 4 of Proposition 11.2.7.1.2), and hence all that is left is the data of the two maps $\mu _{R}$ and $\eta _{R}$, which correspond respectively to the following conditions:

1.
For each $a,b,c\in A$, if $a\sim _{R}b$ and $b\sim _{R}c$, then $a\sim _{R}c$.
2.
For each $a\in A$, we have $a\sim _{R}a$.

These are exactly the requirements for $R$ to be a preorder (, ). Conversely, any preorder $\preceq $ gives rise to a pair of maps $\mu _{\preceq }$ and $\eta _{\preceq }$, forming a monad on $A$.

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8.4.4 Internal Monads