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.

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 $\mathcal{C}$

$$\bigvee_i X_i \coloneqq (\coprod_i X_i) \coprod_{\coprod_{i} *} *$$

in

$$\array{ \coprod_{i} * &\stackrel{(x_i)}{\to}& \coprod_i X_i \\ \downarrow && \downarrow \\ * &\to& \bigvee_i X_i } \,,$$

if this exists.

Equivalently (see at overcategory – limits and colimits) this is just the coproduct in the undercategory $\mathcal{C}\backslash\ast$ of pointed objects.

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.

Related concepts

• wedge sum type

• loop space of a wedge of circles

References

Texbook accounts:

• James Munkres, §71 of: Topology, Prentice Hall (1975, 2000) $[$pdf$]$

• Tammo tom Dieck, p. 31 of: Algebraic topology, European Mathematical Society, Zürich (2008) $[$doi:10.4171/048, pdf$]$

See also:

• Wikipedia Wedge sum