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.

Idea

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 with a new basepoint all pieces that contain the base point of either factor.

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.

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:

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

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

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

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.

For general pointed objects

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 $\mathcal{C}$.

Definition

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).

Proposition

The smash product of def. 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.

Remark

The formula for the smash product in def. 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 $C$ distribute over 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 distribute over finite colimits, the smash product can fail to be associative.

Example

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:

Proposition

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.

Examples

Of pointed topological spaces

The most common case when $\mathcal{C}$ 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.

Proposition

(one-point compactification intertwines Cartesian product with smash product)

On the subcategory $Top_{LCHaus}$ in Top of locally compact Hausdorff spaces with proper maps between them, the functor of one-point compactification (Prop. )

$$(-)^{cpt} \;\colon\; Top_{LCHaus} \longrightarrow Top^{\ast/}$$

1. sends coproducts, hence disjoint union topological spaces, to wedge sums of pointed topological spaces;

2. sends Cartesian products, hence product topological spaces, to smash products of pointed topological spaces;

hence constitutes a strong monoidal functor for both monoidal structures of these distributive monoidal categories in that there are natural homeomorphism

$$\big( X \sqcup Y \big)^{cpt} \;\simeq\; X^{cpt} \vee Y^{cpt} \,,$$

and

$$\big( X \times Y \big)^{cpt} \;\simeq\; X^{cpt} \wedge Y^{cpt} \,.$$

This is briefly mentioned in Bredon 93, p. 199. The argument is spelled out in: MO:a/1645794, Cutler 20, Prop. 1.6.

Of spectra

See at symmetric smash product of spectra.

Properties

Smash-monoidal diagonals

Write

(1)$$\big( PointedTopologicalSpaces, S^0, \wedge \big) \;\;\in\; SymmetricMonoidalCategories$$

• for the category of pointed topological spaces (with respect to some convenient category of topological spaces such as compactly generated topological spaces or D-topological spaces)

• regarded as a symmetric monoidal category with tensor product the smash product and unit the 0-sphere $S^0 \,=\, \ast_+$.

This category also has a Cartesian product, given on pointed spaces $X_i = (\mathcal{X}_i, x_i)$ with underlying $\mathcal{X}_i \in TopologicalSpaces$ by

(2)$$X_1 \times X_2 \;=\; (\mathcal{X}_1, x_1) \times (\mathcal{X}_2, x_2) \;\coloneqq\; \big( \mathcal{X}_1 \times \mathcal{X}_2 , (x_1, x_2) \big) \,.$$

But since this smash product is a non-trivial quotient of the Cartesian product

(3)$$X_1 \wedge X_1 \,\coloneqq\, \frac{X_1 \times X_2}{ X_1 \vee X_2 }$$

it is not itself cartesian, but just symmetric monoidal.

However, via the quotienting (3), it still inherits, from the diagonal morphisms on underlying topological spaces

(4)$$\array{ \mathcal{X} &\overset{ \Delta_{\mathcal{X}} }{\longrightarrow}& \mathcal{X} \times \mathcal{X} \\ x &\mapsto& (x,x) }$$

a suitable notion of monoidal diagonals:

Definition

[Smash monoidal diagonals]

For $X \,\in\, PointedTopologicalSpaces$, let $D_X \;\colon\; X \longrightarrow X \wedge X$ be the composite

of the Cartesian diagonal morphism (2) with the coprojection onto the defining quotient space (3).

It is immediate that:

Proposition

The smash monoidal diagonal $D$ (Def. ) makes the symmetric monoidal category (1) of pointed topological spaces with smash product a monoidal category with diagonals, in that

1. $D$ is a natural transformation;

2. $S^0 \overset{\;\;D_{S^0}\;\;}{\longrightarrow} S^0 \wedge S^0$ is an isomorphism.

While elementary in itself, this has the following profound consequence:

Remark

[Suspension spectra have diagonals]

Since the suspension spectrum-functor

$$\Sigma^\infty \;\colon\; PointedTopologicalSpaces \longrightarrow HighlyStructuredSpectra$$

is a strong monoidal functor from pointed topological spaces (1) to any standard category of highly structured spectra (by this Prop.) it follows that suspension spectra have monoidal diagonals, in the form of natural transformations

(5)$$\Sigma^\infty X \overset{ \;\; \Sigma^\infty(D_X) \;\; }{\longrightarrow} \big( \Sigma^\infty X \big) \wedge \big( \Sigma^\infty X \big)$$

to their respective symmetric smash product of spectra, which hence makes them into comonoid objects, namely coring spectra.

For example, given a Whitehead-generalized cohomology theory $\widetilde E$ represented by a ring spectrum

$$\big(E, 1^E, m^E \big) \;\; \in \; SymmetricMonoids \big( Ho(Spectra), \mathbb{S}, \wedge \big)$$

the smash-monoidal diagonal structure (5) on suspension spectra serves to define the cup product $(-)\cup (-)$ in the corresponding multiplicative cohomology theory structure:

$$\begin{aligned} & \big[ \Sigma^\infty X \overset{c_i}{\longrightarrow} \Sigma^{n_i} E \big] \,\in\, {\widetilde E}{}^{n_i}(X) \\ & \Rightarrow \;\; [c_1] \cup [c_2] \, \coloneqq \, \Big[ \Sigma^\infty X \overset{ \Sigma^\infty(D_X) }{\longrightarrow} \big( \Sigma^\infty X \big) \wedge \big( \Sigma^\infty X \big) \overset{ ( c_1 \wedge c_2 ) }{\longrightarrow} \big( \Sigma^{n_1} E \big) \wedge \big( \Sigma^{n_2} E \big) \overset{ m^E }{\longrightarrow} \Sigma^{n_1 + n_2}E \Big] \;\; \in \, {\widetilde E}{}^{n_1+n_2}(X) \,. \end{aligned}$$

Related concepts

• pointed object

• wedge sum

• pointed mapping space

• While the smash product is not cartesian, it does admit diagonals.

References

Basic accounts:

• Glen Bredon, p. 199 of: Topology and Geometry, Graduate Texts in Mathematics 139, Springer 1993 (doi:10.1007/978-1-4757-6848-0, pdf)

Review:

• Tyrone Cutler, The category of pointed topological spaces, 2020 (pdf, pdf)

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)