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.
7.1.3 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.
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
-
1.
Left Unital Bilinearity. The diagram
commutes, i.e. for each $y\in Y$, we have
\[ f\webleft (x_{0},y\webright ) = z_{0}. \] -
2.
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 $y\in Y$.
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 $\smash {\operatorname {\mathrm{Hom}}^{\otimes }_{\mathsf{Sets}_{*}}\webleft (X\times Y,Z\webright )}$ defined by
\[ \operatorname {\mathrm{Hom}}^{\otimes }_{\mathsf{Sets}_{*}}\webleft (X\times Y,Z\webright ) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\left\{ f\in \operatorname {\mathrm{Hom}}_{\mathsf{Sets}}\webleft (X\times Y,Z\webright )\ \middle |\ \text{$f$ is bilinear}\right\} . \]
