Proposition 5.2.2.7.2 (00K8): Properties of the Internal Hom of $\mathsf{Rel}$

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Table of Contents
Part 2: Relations
Chapter 5: Relations
Section 5.2: Categories of Relations
Subsection 5.2.2: The Closed Symmetric Monoidal Category of Relations
Subsubsection 5.2.2.7: The Internal Hom
Proposition 5.2.2.7.2: Properties of the Internal Hom of $\mathsf{Rel}$ (cite)

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Proposition 5.2.2.7.2 ▶ Properties of the Internal Hom of $\mathsf{Rel}$

Let $A,B,C\in \text{Obj}\webleft (\mathsf{Rel}\webright )$.

Adjointness. We have adjunctions
witnessed by bijections
\begin{align*} \mathrm{Rel}\webleft (A\times B,C\webright ) & \cong \mathrm{Rel}\webleft (A,\mathrm{Rel}\webleft (B,C\webright )\webright ),\\ \mathrm{Rel}\webleft (A\times B,C\webright ) & \cong \mathrm{Rel}\webleft (B,\mathrm{Rel}\webleft (A,C\webright )\webright ), \end{align*}

natural in $A,B,C\in \text{Obj}\webleft (\mathsf{Rel}\webright )$.

Proof of Proposition 5.2.2.7.2.

Item 1: Adjointness

Indeed, we have

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

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

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