The wedge sum of two pointed sets and is the quotient set of the disjoint union where both copies of the basepoint (the one in and the one in ) are identified. The wedge sum can be identified with a subset of the cartesian product ; if this subset is collapsed to a point, then the result is the smash product .
The wedge sum can be generalised to pointed objects in any category with pushouts, and is the coproduct in the category of pointed objects in (which is the coslice category ). A very commonly used case is when Top is a category of topological spaces.
In particular, if itself is a pointed category, then every object is uniquely a pointed object, so that the coproduct in itself may be called a wedge sum. A commonly used case is when 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 has pushouts of that size.
For a set of pointed objects in a category , their wedge sum is the pushout in
in
if this exists.
Equivalently (see at overcategory – limits and colimits) this is just the coproduct in the undercategory of pointed objects.
A wedge sum of pointed circles is also called a bouquet of circles. See for instance at Nielsen-Schreier theorem.
For a CW complex with filtered topological space structure the quotient topological spaces are wedge sums of -spheres.
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:
Last revised on June 11, 2022 at 15:52:40. See the history of this page for a list of all contributions to it.