nLab
smash product

This article is about smash products in topology. For Hopf smash product see crossed product algebra.

Definition

For pointed sets

The smash product $A\wedge B$ of two pointed sets $A$ and $B$ is the quotient set of the cartesian product $A\times B$ where all points with the basepoint as a coordinate (the one from $A$ or the one from $B$) are identified.

The subset that is ‘smashed’ here can be identified with the wedge sum $A\vee B$, so the definition of the smash product can be summarised as follows:

$$A\wedge B=\frac{A\times B}{A\vee B}$$A \wedge B = \frac{A \times B}{A \vee B}

This easily generalizes to the smash products of many spaces, but they do not necessarily agree with iterated version: it is not necessary that $A\wedge (B\wedge C)\cong (A\wedge B)\wedge C$.

The smash product is the tensor product in the closed monoidal category of pointed sets. That is,

$${\mathrm{Fun}}_{*}(A\wedge B,C)\cong {\mathrm{Fun}}_{*}(A,{\mathrm{Fun}}_{*}(B,C))$$Fun_*(A \wedge B, C) \cong Fun_*(A, Fun_*(B, C))

Here, ${\mathrm{Fun}}_{*}(A,B)$ is the set of basepoint-preserving functions from $A$ to $B$, itself made into a pointed set by taking as basepoint the constant function from all of $A$ to the basepoint in $B$.

For general pointed objects

Smash products can be defined for pointed objects in any category $C$ with finite limits and colimits.

Properties

If finite products in $C$ preserve finite colimits, then the smash product is associative, and if $C$ is also cartesian closed, then it makes the category of pointed objects in $C$ closed monoidal. However, if finite products in $C$ do not preserve finite colimits, the smash product can fail to be associative.

Examples

Of pointed topological spaces

The most common case when $C$ is a category of topological spaces. In that case, the natural map $A\wedge (B\wedge C)\to (A\wedge B)\wedge C$ is a homeomorphism provided $C$ is a locally compact Hausdorff space. Thus if both $A$ and $C$ are locally compact Hausdorff, then we have the associativity $A\wedge (B\wedge C)\cong (A\wedge B)\wedge C$.

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.