nLab
smash product

This article is about smash products in topology/homotopy theory. For the notion of Hopf smash product see at crossed product algebra.

Contents

Definition

For pointed sets

Definition

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

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

AB=A×BAB A \wedge B = \frac{A \times B}{A \vee B}
Proposition

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 *(AB,C)Fun *(A,Fun *(B,C)) Fun_*(A \wedge B, C) \cong Fun_*(A, Fun_*(B, C))

where Fun *(A,B)Fun_*(A,B) is the set of basepoint-preserving functions from AA to BB, itself made into a pointed set by taking as basepoint the constant function from all of AA to the basepoint in BB.

This is a special case of the general discussion below.

For general pointed objects

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

Definition

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

XY*(X*)(Y*)(XY) 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).

Proposition

The smash product of def. 2 makes 𝒞 */\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.

Remark

The formula for the smash product in def. 2 can be considered in any category 𝒞\mathcal{C} with finite limits and colimits, but unless 𝒞\mathcal{C} is closed symmetric monoidal, it will not have all these properties.

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

Example

Examples of closed symmetric monoidal categories (𝒞,1 𝒞)(\mathcal{C}, \otimes 1_{\mathcal{C}}) include in particular toposes with their cartesian monoidal structure. For the topos 𝒞=\mathcal{C} = Set the general discussion here reduces to that above.

There is a general abstract way to obtain this smash product monoidal structure:

Proposition

The category of pointed objects is the Eilenberg-Moore category of algebras over a monad for the “maybe monad”, XX*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.

Examples

Of pointed topological spaces

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

Of spectra

See at symmetric smash product of spectra.

References

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

On commutativity of smashing with homotopy limits:

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