The Clowder Project

The left tensor product of pointed sets may be described as follows:

• The left tensor product of $\webleft (X,x_{0}\webright )$ and $\webleft (Y,y_{0}\webright )$ is the pair $\webleft (\webleft (X\lhd Y,x_{0}\lhd y_{0}\webright ),\iota \webright )$ consisting of
  • A pointed set $\webleft (X\lhd Y,x_{0}\lhd y_{0}\webright )$;
  • A left bilinear morphism of pointed sets $\iota \colon \webleft (X\times Y,\webleft (x_{0},y_{0}\webright )\webright )\to X\lhd Y$;
  satisfying the following universal property:
  • Given another such pair $\webleft (\webleft (Z,z_{0}\webright ),f\webright )$ consisting of
    • A pointed set $\webleft (Z,z_{0}\webright )$;
    • A left bilinear morphism of pointed sets $f\colon \webleft (X\times Y,\webleft (x_{0},y_{0}\webright )\webright )\to X\lhd Y$;
    there exists a unique morphism of pointed sets $X\lhd Y\overset {\exists !}{\to }Z$ making the diagram
    

    commute.