smash product

This article is about smash products in topology/homotopy theory. For the notion of

Hopf smash productsee at crossed product algebra.

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:

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

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 is a special case of the general discussion below.

Let $(\mathcal{C}, \otimes, 1_{\mathcal{C}})$ be a closed symmetric monoidal category with (finite) limits and colimits. Write $\ast \in \mathcal{C}$ for the terminal object of $\mathcal{C}$. Write $\mathcal{C}^{\ast/}$ for the category of pointed objects in $\mathcal{C}$.

For $X,Y \in \mathcal{C}^{\ast/}$ two pointed objects in $\mathcal{C}$, their *smash product* is given by the following pushout of pushouts and tensor products all formed in $\mathcal{C}$

$X \wedge Y
\coloneqq
\ast \underset{(X \otimes \ast)\coprod (Y \otimes \ast)}{\coprod} (X \otimes Y)$

regarded as a pointed object via the induced co-projection from $\ast$.

In this generality this appears as (Elmendorf-Mandell 07, construction 4.19).

The smash product of def. 2 makes $\mathcal{C}^{\ast/}$ be a closed symmetric monoidal category with (finite) limits and colimits.

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*.

The formula for the smash product in def. 2 can be considered in any category $\mathcal{C}$ with finite limits and colimits, but unless $\mathcal{C}$ is closed symmetric monoidal, it will not have all these 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 closed symmetric monoidal categories $(\mathcal{C}, \otimes 1_{\mathcal{C}})$ include in particular toposes with their cartesian monoidal structure. For the topos $\mathcal{C} =$ Set the general discussion here reduces to that above.

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”, $X \mapsto X \coprod \ast$. 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 $C$ 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.

See at *symmetric smash product of spectra*.

On the general definition of smash products via closed monoidal category structure on pointed objects

- Anthony Elmendorf, Michael Mandell,
*Permutative categories, multicategories, and algebraic K-theory*, Algebraic & Geometric Topology 9 (2009) 2391-2441 (arXiv:0710.0082)

On commutativity of smashing with homotopy limits:

- Wolfgang Lueck, Holger Reich, Marco Varisco,
*Commuting homotopy limits and smash products*, K-Theory, 30 (2): 137–165, 2003 (arXiv:math/0302116)

Revised on February 12, 2014 18:29:48
by Toby Bartels
(75.88.85.132)