In mathematics, the smash product of two pointed spaces (i.e. topological spaces with distinguished basepoints) X and Y is the quotient of the product space X × Y under the identifications (x, y0) ∼ (x0, y) for all x ∈ X and y ∈ Y. The smash product is usually denoted X ∧ Y. The smash product depends on the choice of basepoints (unless both X and Y are homogeneous).
One can think of X and Y as sitting inside X × Y as the subspaces X × {y0} and {x0} × Y. These subspaces intersect at a single point: (x0, y0), the basepoint of X × Y. So the union of these subspaces can be identified with the wedge sum X ∨ Y. The smash product is then the quotient
The smash product has important applications in homotopy theory, a branch of algebraic topology. In homotopy theory, one often works with a different category of spaces than the category of all topological spaces. In some of these categories the definition of the smash product must be modified slightly. For example, the smash product of two CW complexes is a CW complex if one uses the product of CW complexes in the definition rather than the product topology. Similar modifications are necessary in other categories.
Read more about Smash Product: Examples, As A Symmetric Monoidal Product, Adjoint Relationship
Famous quotes containing the words smash and/or product:
“Whoever sincerely believes that elevated and distant goals are as little use to man as a cow, that “all of our problems” come from such goals, is left to eat, drink, sleep, or, when he gets sick of that, to run up to a chest and smash his forehead on its corner.”
—Anton Pavlovich Chekhov (1860–1904)
“Out of the thousand writers huffing and puffing through movieland there are scarcely fifty men and women of wit or talent. The rest of the fraternity is deadwood. Yet, in a curious way, there is not much difference between the product of a good writer and a bad one. They both have to toe the same mark.”
—Ben Hecht (1893–1964)