5.1.2 Right Bilinear Morphisms of Pointed Sets

Let (X,x0), (Y,y0), and (Z,z0) be pointed sets.

A right bilinear morphism of pointed sets from (X×Y,(x0,y0)) to (Z,z0) is a map of sets

f:X×YZ

satisfying the following condition:1,2

  • Right Unital Bilinearity. The diagram

    commutes, i.e. for each xX, we have

    f(x,y0)=z0.


1Slogan: The map f is right bilinear if it preserves basepoints in its second argument.
2Succinctly, f is bilinear if we have

f(x,y0)=z0

for each xX.

The set of right bilinear morphisms of pointed sets from (X×Y,(x0,y0)) to (Z,z0) is the set HomSets,R(X×Y,Z) defined by

HomSets,R(X×Y,Z)=def{fHomSets(X×Y,Z) | f is right bilinear}.


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