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
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... The smash product of spectra extends the smash product of CW complexes ... homotopy category into a monoidal category in other words it behaves like the tensor product of abelian groups ... A major problem with the smash product is that obvious ways of defining it make it associative and commutative only up to homotopy ...
... Adjoint functors make the analogy between the tensor product and the smash product more precise ... the internal Hom functor Hom(A,–) so that In the category of pointed spaces, the smash product plays the role of the tensor product ...
... The smash product algebra is the vector space with the product , and we write for in this context ... In this case the smash product algebra is also denoted by ... The cyclic homology of Hopf smash products has been computed ...
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