The Clowder Project

Concretely, the tensor of $\webleft (X,x_{0}\webright )$ by $A$ is the pointed set $A\odot \webleft (X,x_{0}\webright )$ consisting of:

• The Underlying Set. The set $A\odot X$ given by
  \[ A\odot X\cong \bigvee _{a\in A}\webleft (X,x_{0}\webright ), \]

  where $\bigvee _{a\in A}\webleft (X,x_{0}\webright )$ is the wedge product of the $A$-indexed family $\webleft (\webleft (X,x_{0}\webright )\webright )_{a\in A}$ of Chapter 4: Pointed Sets, Definition 4.3.2.1.1.

• The Basepoint. The point $\webleft [\webleft (a,x_{0}\webright )\webright ]=\webleft [\webleft (a',x_{0}\webright )\webright ]$ of $\bigvee _{a\in A}\webleft (X,x_{0}\webright )$.