nLab
wedge sum

Context

Category theory

Limits and colimits

Contents

Idea

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.

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.

Definition

Definition

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

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

in

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 }

Examples

  • 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.

Revised on October 26, 2012 01:10:32 by Toby Bartels (64.89.53.81)