Definition 10.2.2.1.1 (00TW): The Reflexive Closure of a Relation

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
Definition 10.2.2.1.1: The Reflexive Closure of a Relation (cite)

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

(★)

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 $(\mathbf{Rel}(A,A),\chi _{A})$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}(A,A)}}\Delta _{A}\\ & = R\cup \Delta _{A}\\ & = \left\{ (a,b)\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 $(\mathrm{N}_{\bullet }(\mathbf{Rel}(A,A)),\chi _{A})$.

Proof of Construction 10.2.2.1.2.

Omitted.

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

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

natural in $R\in \operatorname {\mathrm{Obj}}(\mathbf{Rel}^{\mathsf{refl}}(A,A))$ and $S\in \operatorname {\mathrm{Obj}}(\mathbf{Rel}(A,A))$.
2.
The Reflexive Closure of a Reflexive Relation. If $R$ is reflexive, then $R^{\mathrm{refl}}=R$.
3.
Idempotency. We have

\[ (R^{\mathrm{refl}})^{\mathrm{refl}} = R^{\mathrm{refl}}. \]
4.
Interaction With Inverses. We have
5.
Interaction With Composition. We have