The smash product is the canonical tensor product of pointed objects in an ambient monoidal category. It is essentially given by taking the tensor product of the underlying objects and then identifying all pieces that contain the base point of either with a new basepoint.
An archetypical special case is the smash product of pointed topological spaces and hence of pointed homotopy types. Under stabilization this induces the important smash product of spectra in stable homotopy theory.
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:
This is a special case of the general discussion below.
regarded as a pointed object via the induced co-projection from .
In this generality this appears as (Elmendorf-Mandell 07, construction 4.19).
A proof appears as (Elmendorf-Mandell 07, lemma 4.20). For more of these details see at Pointed object – Closed and monoidal structure. For base change functoriality of these structures see at Wirthmüller context – Examples – On pointed objects.
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.
There is a general abstract way to obtain this smash product monoidal structure:
The category of pointed objects is the Eilenberg-Moore category of algebras over a monad for the “maybe monad”, . This being a suitably monoidal monad it canonically induces a monoidal structure on its EM-category, and that is the smash product.
For more on this see at maybe monad – EM-Category and Relation to pointed objects.
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.
On commutativity of smashing with homotopy limits: