Subsection 5.1.3 (00CB): Bilinear Morphisms of Pointed Sets

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Table of Contents
Part 1: Sets
Chapter 5: Tensor Products of Pointed Sets
Section 5.1: Bilinear Morphisms of Pointed Sets
Subsection 5.1.3: Bilinear Morphisms of Pointed Sets (cite)

5.1.3 Bilinear Morphisms of Pointed Sets

Let $\webleft (X,x_{0}\webright )$, $\webleft (Y,y_{0}\webright )$, and $\webleft (Z,z_{0}\webright )$ be pointed sets.

Definition 5.1.3.1.1 ▶ Bilinear Morphisms of pointed sets

A bilinear morphism of pointed sets from $\webleft (X\times Y,\webleft (x_{0},y_{0}\webright )\webright )$ to $\webleft (Z,z_{0}\webright )$ is a map of sets

\[ f \colon X\times Y \to Z \]

that is both left bilinear and right bilinear.

Remark 5.1.3.1.2 ▶ Unwinding Definition 5.1.3.1.1

In detail, a bilinear morphism of pointed sets from $\webleft (X\times Y,\webleft (x_{0},y_{0}\webright )\webright )$ to $\webleft (Z,z_{0}\webright )$ is a map of sets

\[ f \colon \webleft (X\times Y,\webleft (x_{0},y_{0}\webright )\webright ) \to \webleft (Z,z_{0}\webright ) \]

satisfying the following conditions:¹ ²

Left Unital Bilinearity. The diagram

commutes, i.e. for each $y\in Y$, we have

\[ f\webleft (x_{0},y\webright ) = z_{0}. \]
Right Unital Bilinearity. The diagram

commutes, i.e. for each $x\in X$, we have

\[ f\webleft (x,y_{0}\webright ) = z_{0}. \]

¹Slogan: The map $f$ is bilinear if it preserves basepoints in each argument.

²Succinctly, $f$ is bilinear if we have

\begin{align*} f\webleft (x_{0},y\webright ) & = z_{0},\\ f\webleft (x,y_{0}\webright ) & = z_{0} \end{align*}

for each $x\in X$ and each $y\in Y$.

Definition 5.1.3.1.3 ▶ The Set of Bilinear Morphisms of Pointed Sets

The set of bilinear morphisms of pointed sets from $\webleft (X\times Y,\webleft (x_{0},y_{0}\webright )\webright )$ to $\webleft (Z,z_{0}\webright )$ is the set $\smash {\textup{Hom}^{\otimes }_{\mathsf{Sets}_{*}}\webleft (X\times Y,Z\webright )}$ defined by

\[ \textup{Hom}^{\otimes }_{\mathsf{Sets}_{*}}\webleft (X\times Y,Z\webright ) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft\{ f\in \textup{Hom}_{\mathsf{Sets}}\webleft (X\times Y,Z\webright )\ \middle |\ \text{$f$ is bilinear}\webright\} . \]

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