nLab suspension spectrum

Definition

For $X$ a pointed topological space, its suspension spectrum $\Sigma^\infty X$ is the spectrum given by the pre-spectrum whose degree-$n$ space is the $n$-fold reduced suspension of $X$:

$$(\Sigma^\infty X)_n = \Sigma^n X \,.$$

(e.g. Elmendorf-Kriz-May, example 1.1)

As a symmetric spectrum: (Schwede 12, example I.2.6)

Properties

Completion to an $\Omega$-spectrum

See at Omega spectrum – Completion of a suspension spectrum.

Relation to looping and stabilization

As an infinity-functor $\Sigma^\infty\colon Top_* \to Spec$ the suspension spectrum functor exhibits the stabilization of Top.

$$(\Sigma^\infty \dashv \Omega^\infty)\colon Top_* \stackrel{\overset{\Omega^\infty}{\leftarrow}}{\underset{\Sigma^\infty}{\to}} Spec$$

Strong monoidalness

The suspension spectrum functor is strong monoidal.

On the one hand, this is the case for its incarnation as a 1-functor with values in structured spectra (this Prop.) Via the corresponding symmetric monoidal model structure on structured spectra this exhibits strong monoidalness also as an (infinity,1)-functor.

More abstractly this follows from general properties of stabilization when regarding stable homotopy theory as the result of inverting smash product with the circle, via Robalo 12, last clause of Prop. 4.1 with last clause of Prop. 4.10 (1). For emphasis see also Hoyois 15, section 6.1, specifically Hoyois 15, Def. 6.1.

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

• free structured spectrum

• equivariant suspension spectrum

References

• Frank Adams, part III, section 2 of Stable homotopy and generalised homology, 1974

• Anthony Elmendorf, Igor Kriz, Peter May, example 1.1 of Modern foundations for stable homotopy theory, in Ioan Mackenzie James, Handbook of Algebraic Topology, Amsterdam: North-Holland (1995) pp. 213–253, (pdf)

• Nicholas J. Kuhn, Suspension spectra and homology equivalences, Trans. Amer. Math. Soc. 283, 303–313 (1984) (JSTOR)

• John Klein, Moduli of suspension spectra (arXiv:math/0210258, MO)

• Stefan Schwede, Example I.2.6 in Symmetric spectra, 2012 (pdf)

Suspension spectra of infinite loop spaces are discussed (in a context of Goodwillie calculus and chromatic homotopy theory) in

• Nicholas J. Kuhn, section 6.2 of Goodwillie towers and chromatic homotopy: An overview (pdf)

Formalization of suspension spectra in dependent linear homotopy type theory (see also on the “exponential modality” here):

• Mitchell Riley, Def. 2.1.29 & Rem. 2.4.13 in: A Bunched Homotopy Type Theory for Synthetic Stable Homotopy Theory, PhD Thesis (2022) [doi:10.14418/wes01.3.139]