The wedge sum$A \vee B$ of two pointed sets$A$ and $B$ is the quotient set of the disjoint union$A \uplus B$ where both copies of the basepoint (the one in $A$ and the one in $B$) are identified. The wedge sum $A \vee B$ can be identified with a subset of the cartesian product$A \times B$; if this subset is collapsed to a point, then the result is the smash product$A \wedge B$.

In particular, if $C$ itself is a pointed category, then every object is uniquely a pointed object, so that the coproduct in $C$ itself may be called a wedge sum. A commonly used case is when $C=$Spectra is a category of spectra.

Also, the wedge sum also makes sense for any family of pointed objects, not just for two of them, as long as $C$ has pushouts of that size.

Definition

Definition

For $\{x_i \colon * \to X_i\}_i$ a set of pointed objects in a category$\mathcal{C}$, their wedge sum$\bigvee_i X_i$ is the pushout in $\mathcal{C}$

For $X$ a CW complex with filtered topological space structure $X_0 \hookrightarrow \cdots \hookrightarrow \X_k \hookrightarrow X_{k+1} \hookrightarrow \cdots \hookrightarrow X$ the quotient topological spaces $X_{k+1}/X_k$ are wedge sums of $(k+1)$-spheres.