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
wedge smash
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).

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