This entry is about the notion of spectrum in stable homotopy theory. For other uses of the term ‘’spectrum’‘ see spectrum - disambiguation.

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Contents

Idea

A topological spectrum is an object in the universal stable (∞,1)-category $Sp(Top) \simeq Sp(\infty Grpd)$ that stabilizes the (∞,1)-category Top or $\simeq$ ∞-Grpd of topological spaces or ∞-groupoids under the operations of forming loop space objects and reduced suspensions: the stable (∞,1)-category of spectra.

More generally, one may consider spectrum objects in any presentable (∞,1)-category.

Connective and non-connective spectra; infinite loop spaces

As opposed to the homotopy groups of a topological space, the homotopy groups of a spectrum are defined and may be non-trivial in negative integer degree. This follows from the fact that the loop space operation is by construction invertible on spectra, which implies that for every spectrum $E$ these and all $n$, the $n$-fold looping $\Omega^n$ has stable homotopy groups given by $\pi_{k-n}(\Omega^n E) \simeq \pi_k(E)$.

Those spectra for which the homotopy groups of spectra in negative degree happen to vanish are called connective spectra. They are equivalent to infinite loop spaces, i.e. grouplike E-∞ spaces.

Connective spectra in the image of the nerve operation of the classical Dold-Kan correspondence: this identifies ∞-groupoids that are not only connective spectra but even have a strict symmetric monoidal group structure with non-negatively graded chain complexes of abelian groups.

The homology groups of the chain complex correspond precisely to the homotopy groups of the corresponding topological space or ∞-groupoid.

The free stabilization of the (∞,1)-category of non-negatively graded chain complexes is simply the stable (∞,1)-category of arbitrary chain complexes. There is a stable Dold-Kan correspondence (see at module spectrum the section stable Dold-Kan correspondence ) that identifies these with special objects in $Sp(Top)$.

So it is the homologically nontrivial parts of the chain complexes in negative degree that corresponds to the non-connectiveness of a spectrum.

Definition

There are many “models” for spectra, all of which present the same (stable) homotopy theory (and in fact, nearly all of them are Quillen equivalent model categories). For more details see at Introduction to Stable Homotopy Theory.

Sequential pre-spectra

A simple first definition is to define a pre-spectrum $\mathbf{E}$ to be a sequence of pointed spaces $(E_n)_{n\in\mathbb{N}}$ together with structure maps $\Sigma{}E_n\to{}E_{n+1}$ (where $\Sigma$ denotes the reduced suspension). See at model structure on sequential spectra.

There are various conditions that can be put on the spaces $E_n$ and the structure maps, for example if the spaces are CW-complexes and the structure maps are inclusions of subcomplexes, the spectrum is called a CW-spectrum.

Without any condition, this is just called a spectrum, or sometimes a pre-spectrum. In order to distinguish from various other richer definitions (such as coordinate-free spectra, one also speaks of sequential spectra).

For details see Introduction to stable homotopy theory – 1.1 Sequential Spectra.

$\Omega$-spectra

If $\Omega$ denotes the loop space functor on the category of pointed spaces, we know that $\Sigma$ is left adjoint to $\Omega$. In particular, given a spectrum $\mathbf{E}$, the structure maps can be transformed into maps $E_n\to\Omega{}E_{n+1}$. If these maps are isomorphisms (depending on the situation it can be weak equivalences or homeomorphisms), then $\mathbf{E}$ is called an $\Omega$-spectrum.

The idea is that $E_0$ contains the information of $\mathbf{E}$ in dimensions $k\ge 0$, $E_1$ contains the information of $\mathbf{E}$ in $k\ge -1$ (but shifted up by one, so that it is modeled by the $\ge 0$ information in the space $E_1$), and so on.

$\Omega$-spectra are special cases of sequential pre-spectra as above, and are in fact the fibrant objects for some model structure on spectra.

Given any sequential pre-spectrum $\mathbf{E}$, it induces an equivalent $\Omega$-spectrum $\mathbf{F}$ (a fibrant replacement of $\mathbf{E}$, its spectrification) given by (Lewis-May-Steinberger 86, p. 3)

(using that $\Omega$ commutes with the filtered colimits).

Coordinate-free spectra

A definition of spectrum consisting of spaces indexed by index sets less “coordinatized” than the integers is a

• coordinate-free spectrum.

See there for details.

Combinatorial spectra

There might be a type of categorical structure related to a spectrum in the same way that $\infty$-categories are related to $\infty$-groupoids. In other words, it would contain $k$-cells for all integers $k$, not necessarily invertible. Some people have called this conjectural object a $Z$-category. “Connective” $Z$-categories could perhaps then be identified with stably monoidal $\infty$-categories.

One realization of this kind of idea is the notion of combinatorial spectrum.

General context

See spectrum object.

Examples

• suspension spectrum

• sphere spectrum

• Eilenberg-MacLane spectrum

• K-theory spectrum

• elliptic spectrum

• tmf

• Thom spectrum, complex cobordism spectrum

Properties

Stabilization

In direct analogy to how topological spaces form the archetypical example, Top, of an (∞,1)-category, spectra form the archetypical example $Sp(Top)$ of a stable (∞,1)-category. In fact, there is a general procedure for turning any pointed (∞,1)-category $C$ into a stable $(\infty,1)$-category $Sp(C)$, and doing this to the category $Top_*$ of pointed spaces yields $Sp(Top)$.

Symmetric monoidal structure

• smash product of spectra

• symmetric monoidal smash product of spectra

Closed structure

• mapping spectrum

Model category structure

• model structure on spectra

Relation to symmetric monoidal ∞-groupoids

• stable homotopy hypothesis

Related concepts

• homotopy group of a spectrum

• sheaf of spectra, parametrized spectrum

  • smooth spectrum

• spectrum with G-action, G-spectrum

• ring spectrum, algebra spectrum, module spectrum

• zero spectrum

• finite spectrum

• p-local spectrum

• Bousfield localization of spectra

• motivic spectrum

• Brown representability theorem

• stable homotopy hypothesis

References

According to (Adams 74, p. 131) the notion of spectrum is due to

• E. L. Lima, Duality and Postnikov invariants, Thesis, University of Chicago, Chicago 1958

It is generally supposed that G. W. Whitehead also had something to do with it, but the latter takes a modest attitude about that. (Adams 74, p. 131)

Early notes include

• Michael Boardman, Stable homotopy theory, mimeographed notes, University of Warwick, 1965 onward

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

See the references at stable homotopy theory.

Lecture notes include

• Introduction to stable homotopy theory – 1.1 Sequential Spectra

More modern developments are due to

• L. Gaunce Lewis, Peter May, M. Steinberger, Equivariant stable homotopy theory, Springer Lecture Notes in Mathematics, 1986 (pdf)

The quick idea is surveyed for instance in

• Cary Malkiewich, The stable homotopy category, 2014 (pdf)

• Aaron Mazel-Gee, An introduction to spectra (pdf)

The first review of stable homotopy theory with symmetric monoidal smash product of spectra is (in terms of S-modules) in

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

A comprehensive account of the symmetric model in terms of symmetric spectra is in

• Stefan Schwede, Symmetric spectra (pdf)

and in terms of orthogonal spectra in

• Stefan Schwede, Global homotopy theory (take $\mathcal{F} = \{1\}$, on p. 4, to be the trivial collection of groups, in order to specialize from global equivariant stable homotopy theory to plain stable homotopy theory).

See also

• Robert Thomason, Symmetric Monoidal Categories Model All Connective Spectra (web)

• Frank Adams, Infinite loop spaces, Princeton University Press, 1978

• Joseph Ayoub, Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique, I. Astérisque, Vol. 314 (2008). Société Mathématique de France. (pdf)