A bilinear morphism of pointed sets from $(X\times Y,(x_{0},y_{0}))$ to $(Z,z_{0})$ 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 $(X,x_{0})$, $(Y,y_{0})$, and $(Z,z_{0})$ be pointed sets.
In detail, a bilinear morphism of pointed sets from $(X\times Y,(x_{0},y_{0}))$ to $(Z,z_{0})$ is a map of sets
\[ f \colon (X\times Y,(x_{0},y_{0})) \to (Z,z_{0}) \]
satisfying the following conditions:1,2
-
1.
Left Unital Bilinearity. The diagram
commutes, i.e. for each $y\in Y$, we have
\[ f(x_{0},y) = z_{0}. \] -
2.
Right Unital Bilinearity. The diagram
commutes, i.e. for each $x\in X$, we have
\[ f(x,y_{0}) = z_{0}. \]
- 1Slogan: The map $f$ is bilinear if it preserves basepoints in each argument.
-
2Succinctly, $f$ is bilinear if we have \begin{align*} f(x_{0},y) & = z_{0},\\ f(x,y_{0}) & = z_{0} \end{align*}for each $x\in X$ and each $y\in Y$.
The set of bilinear morphisms of pointed sets from $(X\times Y,(x_{0},y_{0}))$ to $(Z,z_{0})$ is the set $\smash {\operatorname {\mathrm{Hom}}^{\otimes }_{\mathsf{Sets}_{*}}(X\times Y,Z)}$ defined by
\[ \operatorname {\mathrm{Hom}}^{\otimes }_{\mathsf{Sets}_{*}}(X\times Y,Z) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\left\{ f\in \operatorname {\mathrm{Hom}}_{\mathsf{Sets}}(X\times Y,Z)\ \middle |\ \text{$f$ is bilinear}\right\} . \]
