In topology, a branch of mathematics, the smash product of two pointed spaces (i.e. topological spaces with distinguished basepoints) and is the quotient of the product space under the identifications for all in and in . The smash product is itself a pointed space, with basepoint being the equivalence class of The smash product is usually denoted or . The smash product depends on the choice of basepoints (unless both X and Y are homogeneous).
X\veeY=(X\amalgY) /{\sim}
X\veeY
X\veeY
x0\simy0
X\wedgeY=(X x Y)/(X\veeY).
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.
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
\begin{align} X\wedgeY&\congY\wedgeX,\\ (X\wedgeY)\wedgeZ&\congX\wedge(Y\wedgeZ). \end{align}
X=Y=Q
Z=N
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 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
(- ⊗ RA)
Hom(A,-)
Hom(X ⊗ A,Y)\congHom(X,Hom(A,Y)).
A,X
Maps*(X\wedge |
A,Y)\cong
Maps*(X,Maps*(A,Y)) |
where
\operatorname{Maps*}
Maps*(A,Y) |
In particular, taking
A
S1
\Sigma
\Omega
Maps*(\Sigma |
X,Y)\cong
Maps*(X,\Omega |
Y).