Proposition 6.3.4.1.1 (00L8): Monads in $\textbf{Rel}$

Style	Class	Font	Theorem Environments
Style 1	`book`	Alegreya Sans	`tcbthm`
Style 2	`book`	Alegreya Sans	`amsthm`
Style 3	`book`	Arno*	`amsthm`
Style 4	`book`	Computer Modern	`amsthm`

We have a natural identification¹

\[ \webleft\{ \begin{gathered} \text{Monads in}\\ \text{$\textbf{Rel}$ on $A$} \end{gathered} \webright\} \cong \webleft\{ \text{Preorders on $A$}\webright\} . \]

¹See also

for an extension of this correspondence to “relative monads in $\textbf{Rel}$”.

A monad in $\textbf{Rel}$ 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, 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:

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