Proposition 8.5.8.1.1 (02F1): Eilenberg–Moore and Kleisli Objects in $\boldsymbol {\mathsf{Rel}}$

Table of Contents
Part 3: Relations
Chapter 8: Relations
Section 8.5: Properties of the 2-Category of Relations
Subsection 8.5.8: Eilenberg–Moore and Kleisli Objects
Proposition 8.5.8.1.1: Eilenberg–Moore and Kleisli Objects in $\boldsymbol {\mathsf{Rel}}$ (cite)

Let $X$ be a set.

Let $R$ be a preorder on $X$, viewed as an internal monad on $X$ via Proposition 8.5.4.1.1.

1.
Eilenberg–Moore Objects in $\boldsymbol {\mathsf{Rel}}$. The Eilenberg–Moore object for $R$ exists iff it is an equivalence relation, in which case it is the quotient $X/\mathord {\sim }_{R}$ of $X$ by $R$.
2.
Kleisli Objects in $\boldsymbol {\mathsf{Rel}}$. […]

Proof of Proposition 8.5.8.1.1.

Omitted.

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