This article is about smash products in topology/homotopy theory. For the notion of Hopf smash product see at crossed product algebra.
Stable Homotopy theory
For pointed sets
The smash product of two pointed sets and is the quotient set of the cartesian product where all points with the basepoint as a coordinate (the one from or the one from ) are identified.
The subset that is ‘smashed’ here can be identified with the wedge sum , so the definition of the smash product can be summarised as follows:
The smash product is the tensor product in the closed monoidal category of pointed sets.
That is, it is characterized by the existence of natural isomorphisms
where is the set of basepoint-preserving functions from to , itself made into a pointed set by taking as basepoint the constant function from all of to the basepoint in .
This easily generalizes to the smash products of many spaces, but they do not necessarily agree with iterated version: it is not necessary that .
For general pointed objects
Smash products can be defined for pointed objects in any category with finite limit and colimit.
For a topos, with its category of pointed objects, the smash product is the tensor product that makes this a closed monoidal category.
If finite products in preserve finite colimits, then the smash product is associative, and if is also cartesian closed, then it makes the category of pointed objects in closed monoidal. However, if finite products in do not preserve finite colimits, the smash product can fail to be associative.
Of pointed topological spaces
The most common case when is a category of topological spaces. In that case, the natural map is a homeomorphism provided is a locally compact Hausdorff space. Thus if both and are locally compact Hausdorff, then we have the associativity .
Associativity fails in general for the category Top of all topological spaces; however, it is satisfied for pointed objects in any convenient category of topological spaces, since such a category is cartesian closed. In particular, the smash product is associative for pointed compactly generated spaces.
See at symmetric smash product of spectra.
On commutativity of smashing with homotopy limits: