7.1.1 Left Bilinear Morphisms of Pointed Sets

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

Definition 7.1.1.1.1 ▶ Left Bilinear Morphisms of pointed sets

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

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

satisfying the following condition:1,2

  • (★)
    • Left Unital Bilinearity. The diagram
    commutes, i.e. for each $y\in Y$, we have
    \[ f(x_{0},y) = z_{0}. \]

  1. 1Slogan: The map $f$ is left bilinear if it preserves basepoints in its first argument.
  2. 2Succinctly, $f$ is bilinear if we have
    \[ f(x_{0},y) = z_{0} \]
    for each $y\in Y$.

Definition 7.1.1.1.2 ▶ The Set of Left Bilinear Morphisms of Pointed Sets

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

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

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