nLab
wedge sum

Idea

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 $A\wedge B$.

The wedge sum can be generalised to pointed objects in any category $C$ with pushouts, and is the coproduct in the category of pointed objects in $C$ (which is the coslice category $*/C$). A very commonly used case is when $C=$Top is a category of topological spaces.

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 $\{x_i:*\to X_i{\}}_i$ a set of pointed objects in a category with colimits, their wedge sum ${\bigvee }_i{X}_i$ is the pushout

$$\underset{i}{\bigvee }{X}_i≔(\coprod _i{X}_i)\coprod _{\coprod _i*}*$$\bigvee_i X_i \coloneqq (\coprod_i X_i) \coprod_{\coprod_{i} *} *

in

$$\begin{array}{ccc}\coprod _i*& \stackrel{(x_i)}{\to }& \coprod _i{X}_i\\ \downarrow & & \downarrow \\ *& \to & \underset{i}{\bigvee }{X}_i\end{array}$$\array{ \coprod_{i} * &\stackrel{(x_i)}{\to}& \coprod_i X_i \\ \downarrow && \downarrow \\ * &\to& \bigvee_i X_i }

Examples

• A wedge sum of pointed circles is also called a bouquet of circles. See for instance at Nielsen-Schreier theorem.

• 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.