Proposition 8.3.3.7.2 (00K8): Properties of the Internal Hom of $\mathsf{Rel}$

Table of Contents
Part 3: Relations
Chapter 8: Relations
Section 8.3: Categories of Relations
Subsection 8.3.3: The Closed Symmetric Monoidal Category of Relations
Subsubsection 8.3.3.7: The Internal Hom
Proposition 8.3.3.7.2: Properties of the Internal Hom of $\mathsf{Rel}$ (cite)

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8.3.3.7 The Internal Hom

Definition 8.3.3.7.1 ▶ The Internal Hom of $\mathsf{Rel}$

The internal Hom of $\mathsf{Rel}$ is the functor

\[ \mathrm{Rel}\colon \mathsf{Rel}^{\mathsf{op}}\times \mathsf{Rel}\to \mathsf{Rel} \]

defined

•
On objects by sending $A,B\in \operatorname {\mathrm{Obj}}(\mathsf{Rel})$ to the set $\mathrm{Rel}(A,B)$ of of .
•
On morphisms by pre/post-composition defined as in Definition 8.1.3.1.1.

Proposition 8.3.3.7.2 ▶ Properties of the Internal Hom of $\mathsf{Rel}$

Let $A,B,C\in \operatorname {\mathrm{Obj}}(\mathsf{Rel})$.

1.
Adjointness. We have adjunctions
witnessed by bijections

\begin{align*} \mathrm{Rel}(A\times B,C) & \cong \mathrm{Rel}(A,\mathrm{Rel}(B,C)),\\ \mathrm{Rel}(A\times B,C) & \cong \mathrm{Rel}(B,\mathrm{Rel}(A,C)), \end{align*}

natural in $A,B,C\in \operatorname {\mathrm{Obj}}(\mathsf{Rel})$.

Proof of Proposition 8.3.3.7.2.

Item 1: Adjointness

Indeed, we have

\begin{align*} \mathrm{Rel}(A\times B,C) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\mathsf{Sets}(A\times B\times C,\{ \mathsf{true},\mathsf{false}\} )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\mathrm{Rel}(A,B\times C)\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\mathrm{Rel}(A,\mathrm{Rel}(B,C)), \end{align*}

and similarly for the bijection $\mathrm{Rel}(A\times B,C)\cong \mathrm{Rel}(B,\mathrm{Rel}(A,C))$.

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