Smash product

In topology, a branch of mathematics, the smash product of two pointed spaces (i.e. topological spaces with distinguished basepoints) (X, x0) and (Y, y0) is the quotient of the product space X × Y under the identifications (x, y0) ~ (x0, y) for all x in X and y in Y.

The smash product is itself a pointed space, with basepoint being the equivalence class of (x0, y0).

The smash product depends on the choice of basepoints (unless both X and Y are homogeneous).

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

, subspaces X and Y intersect in the single point

The smash product is then the quotient The smash product shows up in homotopy theory, a branch of algebraic topology.

In some of these categories the definition of the smash product must be modified slightly.

For any pointed spaces X, Y, and Z in an appropriate "convenient" category (e.g., that of compactly generated spaces), there are natural (basepoint preserving) homeomorphisms However, for the naive category of pointed spaces, this fails, as shown by the counterexample

[1] A proof due to Kathleen Lewis that Puppe's counterexample is indeed a counterexample can be found in the book of Johann Sigurdsson and J. Peter May.

[2] These isomorphisms make the appropriate category of pointed spaces into a symmetric monoidal category with the smash product as the monoidal product and the pointed 0-sphere (a two-point discrete space) as the unit object.

Adjoint functors make the analogy between the tensor product and the smash product more precise.

In the category of R-modules over a commutative ring R, the tensor functor

is left adjoint to the internal Hom functor

, so that In the category of pointed spaces, the smash product plays the role of the tensor product in this formula: if

is left adjoint to the loop space functor