Item 2 (00U0) — The Clowder Project

Table of Contents
Part 3: Relations
Chapter 10: Conditions on Relations
Section 10.2: Reflexive Relations
Subsection 10.2.2: The Reflexive Closure of a Relation
Item 2 (cite)

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10.2.2 The Reflexive Closure of a Relation

Let $R$ be a relation on $A$.

Definition 10.2.2.1.1 ▶ The Reflexive Closure of a Relation

The reflexive closure of $\mathord {\sim }_{R}$ is the relation $\smash {\mathord {\sim }^{\mathrm{refl}}_{R}}$¹ satisfying the following universal property:²

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¹Further Notation: Also written $R^{\mathrm{refl}}$.
²Slogan: The reflexive closure of $R$ is the smallest reflexive relation containing $R$.

Construction 10.2.2.1.2 ▶ The Reflexive Closure of a Relation

Concretely, $\smash {\mathord {\sim }^{\mathrm{refl}}_{R}}$ is the free pointed object on $R$ in $\webleft (\mathbf{Rel}\webleft (A,A\webright ),\chi _{A}\webright )$¹, being given by

\begin{align*} R^{\mathrm{refl}} & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}R\mathbin {\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}^{\mathbf{Rel}\webleft (A,A\webright )}}\Delta _{A}\\ & = R\cup \Delta _{A}\\ & = \left\{ \webleft (a,b\webright )\in A\times A\ \middle |\ \text{we have $a\sim _{R}b$ or $a=b$}\right\} .\end{align*}

¹Or, equivalently, the free $\mathbb {E}_{0}$-monoid on $R$ in $\webleft (\mathrm{N}_{\bullet }\webleft (\mathbf{Rel}\webleft (A,A\webright )\webright ),\chi _{A}\webright )$.

Proof of Construction 10.2.2.1.2.

Clear.

Proposition 10.2.2.1.3 ▶ Properties of the Reflexive Closure of a Relation

Let $R$ be a relation on $A$.

1.
Adjointness. We have an adjunction
witnessed by a bijection of sets

\[ \mathbf{Rel}^{\mathsf{refl}}\webleft (R^{\mathrm{refl}},S\webright ) \cong \mathbf{Rel}\webleft (R,S\webright ), \]

natural in $R\in \operatorname {\mathrm{Obj}}\webleft (\mathbf{Rel}^{\mathsf{refl}}\webleft (A,A\webright )\webright )$ and $S\in \operatorname {\mathrm{Obj}}\webleft (\mathbf{Rel}\webleft (A,A\webright )\webright )$.