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* 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 **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 April 15, 2016 10:35:21
by Urs Schreiber
(131.220.184.222)