Smash product

In topology, a branch of mathematics, the smash product of two pointed spaces (i.e. topological spaces with distinguished basepoints) ("X," "x"0) and ("Y", "y"0) is the quotient of the product space "X" × "Y" under the identifications ("x", "y"0) ∼ ("x"0, "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 ("x"0, "y"0). The smash product is usually denoted "X" ∧ "Y" or "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" × {"y"0} and {"x"0} × "Y". These subspaces intersect at a single point: ("x"0, "y"0), 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 shows up 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.
As a symmetric monoidal product.
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 formula_5 and formula_6 found by Dieter Puppe. 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.
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. One can therefore think of the smash product as a kind of tensor product in an appropriate category of pointed spaces.
Adjoint relationship.
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 formula_7 is left adjoint to the internal Hom functor formula_8, so that
In the category of pointed spaces, the smash product plays the role of the tensor product in this formula: if formula_10 are compact Hausdorff then we have an adjunction
where formula_12 denotes continuous maps that send basepoint to basepoint, and formula_13 carries the compact-open topology.
In particular, taking formula_14 to be the unit circle formula_15, we see that the reduced suspension functor formula_16 is left adjoint to the loop space functor formula_17: