Subsection 6.4.2 (00MR): The Left Skew Monoidal Unit

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Table of Contents
Part 2: Relations
Chapter 6: Relations
Section 6.4: The Left Skew Monoidal Structure on $\mathbf{Rel}\webleft (A,B\webright )$
Subsection 6.4.2: The Left Skew Monoidal Unit (cite)

Let $A$ and $B$ be sets and let $J\colon A\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}B$ be a relation.

The left $J$-skew monoidal unit of $\mathbf{Rel}\webleft (A,B\webright )$ is the functor

\[ \mathbb {1}^{\mathbf{Rel}\webleft (A,B\webright )}_{\lhd _{J}} \colon \mathsf{pt}\to \mathbf{Rel}\webleft (A,B\webright ) \]

picking the object

\[ \mathbb {1}^{\lhd _{J}}_{\mathbf{Rel}\webleft (A,B\webright )}\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}J \]

of $\mathbf{Rel}\webleft (A,B\webright )$.

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