Proposition 6.3.5.1.1 (00LA): Comonads 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 identification

\[ \webleft\{ \begin{gathered} \text{Comonads in}\\ \text{$\textbf{Rel}$ on $A$} \end{gathered} \webright\} \cong \webleft\{ \text{Subsets of $A$}\webright\} . \]

A comonad 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*} \Delta _{R} & \colon R \subset R\mathbin {\diamond }R,\\ \epsilon _{R} & \colon R \subset \chi _{A} \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 $\Delta _{R}$ and $\epsilon _{R}$, which correspond respectively to the following conditions:

For each $a,b\in A$, if $a\sim _{R}b$, then there exists some $k\in A$ such that $a\sim _{R}k$ and $k\sim _{R}b$.
For each $a,b\in A$, if $a\sim _{R}b$, then $a=b$.

Taking $k=b$ in the first condition above shows it to be trivially satisfied, while the second condition implies $R\subset \Delta _{A}$, i.e. $R$ must be a subset of $A$. Conversely, any subset $U$ of $A$ satisfies $U\subset \Delta _{A}$, defining a comonad as above.