Construction 5.5.1.1.6 (01VX): A Second Construction of the Smash Product of Pointed Sets

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Table of Contents
Part 1: Sets
Chapter 5: Tensor Products of Pointed Sets
Section 5.5: The Smash Product of Pointed Sets
Subsection 5.5.1: Foundations
Construction 5.5.1.1.6: A Second Construction of the Smash Product of Pointed Sets (cite)

Construction 5.5.1.1.6 ▶ A Second Construction of the Smash Product of Pointed Sets

Alternatively, the smash product of $\webleft (X,x_{0}\webright )$ and $\webleft (Y,y_{0}\webright )$ may be constructed as the pointed set $X\wedge Y$ given by

\begin{align*} X\wedge Y & \cong \bigvee _{x\in X^{-}}Y\\ & \cong \bigvee _{y\in Y^{-}}X.\end{align*}

Proof of Construction 5.5.1.1.6.

Indeed, since $X\cong \bigvee _{x\in X^{-}}S^{0}$, we have

\begin{align*} X\wedge Y & \cong \webleft (\bigvee _{x\in X^{-}}S^{0}\webright )\wedge Y\\ & \cong \bigvee _{x\in X^{-}}S^{0}\wedge Y\\ & \cong \bigvee _{x\in X^{-}}Y, \end{align*}

where we have used that $\wedge $ preserves colimits in both variables by for the second isomorphism above, since it has right adjoints in both variables by Item 2.

A similar proof applies to the isomorphism $X\wedge Y\cong \bigvee _{y\in Y^{-}}X$.

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