Wedge and Smash Products

Let $X$ and $Y$ be pointed simplicial sets. The wedge of $X$ and $Y$, denoted by $X\vee Y$, is the simplicial set obtained by identifying the basepoint of $X$ with the basepoint of $Y$. Hence, we can define $X\vee Y$ by the push-out diagram

$X\vee Y$ can be described as the simplicial subset of $X\times Y$ consisting of $(x,*)$ for $x\in X$ and $(*,y)$ for $y\in Y$.

The smash product $X\wedge Y$ is defined to be the simplicial quotient $X\times Y/(X\vee Y)$.