nLab wedge sum



Category theory

Limits and colimits



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

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

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



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

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


i* (x i) iX i * iX i, \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 𝒞\*\mathcal{C}\backslash\ast of pointed objects.


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

  • For XX a CW complex with filtered topological space structure X 0X kX k+1XX_0 \hookrightarrow \cdots \hookrightarrow \X_k \hookrightarrow X_{k+1} \hookrightarrow \cdots \hookrightarrow X the quotient topological spaces X k+1/X kX_{k+1}/X_k are wedge sums of (k+1)(k+1)-spheres.


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Last revised on June 11, 2022 at 15:52:40. See the history of this page for a list of all contributions to it.