# nLab smash product

category theory

## Applications

#### Stable Homotopy theory

stable homotopy theory

# Contents

## Definition

### For pointed sets

###### Definition

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:

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

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

$Fun_*(A \wedge B, C) \cong Fun_*(A, Fun_*(B, C))$

where $Fun_*(A,B)$ 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 $B$.

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

### For general pointed objects

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

For $\mathbf{H}$ a topos, with $\mathbf{H}^{\ast/}$ its category of pointed objects, the smash product is the tensor product that makes this a closed monoidal category.

## 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.

## References

On commutativity of smashing with homotopy limits:

Revised on October 12, 2013 01:06:54 by Urs Schreiber (82.113.98.193)