nLab
spectrum

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


Contents

Idea

A topological spectrum is an object in the universal stable (∞,1)-category Sp(Top)Sp(Grpd)Sp(Top) \simeq Sp(\infty Grpd) that stabilizes“ the (∞,1)-category Top or \simeq ∞-Grpd of topological spaces or ∞-groupoids: the stable (∞,1)-category of spectra.

Recall that the central characterization of a stable (∞,1)-category is that all objects AA have a delooping object BA\mathbf{B}A that is written ΣA\Sigma A in this context and called the suspension of AA. Thus a spectrum is like a topological space or ∞-groupoid that may be delooped indefinitely.

More generally, one can consider spectrum objects in an arbitrary category or (infinity,1)-category.

Connective spectra

In fact all ordinary topological spaces and ∞-groupoids that have the property that all their deloopings exist give rise to special examples of spectra. These are called the

Connective spectra form a sub-(∞,1)-category of spectra

TopConnectSp(Top)Sp(Top). Top \stackrel{\supset}{\leftarrow} ConnectSp(Top) \hookrightarrow Sp(Top) \,.

There are objects in Sp(Top)Sp(Top), though, that do not come from “naively” delooping a space infinitely many times. These are the non-connective spectra. For general spectra there is a notion of homotopy groups of negative degree. The connective ones are precisely those for which all negative degree homotopy groups vanish.

Connective spectra are well familiar in as far as they are in the image of the nerve operation of the 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.

Ch + DoldKannerve ConnectSp(Grp)Grpd (A 2A 1A 000) N(A ) \array{ Ch_+ &\stackrel{Dold-Kan \; nerve}{\to}& ConnectSp(\infty Grp) \subset \infty Grpd \\ (\cdots A_2 \stackrel{\partial}{\to} A_1 \stackrel{\partial}{\to} A_0 \to 0 \to 0 \to \cdots) &\stackrel{}{\mapsto}& N(A_\bullet) }

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 simpy the stable (∞,1)-category of arbitrary chain complexes. There is a stabilized Dold-Kan correspondence (see at module spectrum the section stable Dold-Kan correspondence ) that identifies these with special objects in Sp(Top)Sp(Top).

Ch DoldKannerve Sp(Grp) (A 2A 1A 0A 1A 2) N(A ) \array{ Ch &\stackrel{Dold-Kan \; nerve}{\to}& Sp(\infty Grp) \\ (\cdots A_2 \stackrel{\partial}{\to} A_1 \stackrel{\partial}{\to} A_0 \stackrel{\partial}{\to} A_{-1} \stackrel{\partial}{\to} A_{-2} \to \cdots) &\stackrel{}{\mapsto}& N(A_\bullet) }

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 homotopy theory (and in fact, nearly all of them are Quillen equivalent model categories).

Spectra, CW-spectra

A simple first definition is to define a spectrum E\mathbf{E} to be a sequence of pointed spaces (E n) n(E_n)_{n\in\mathbb{N}} together with structure maps ΣE nE n+1\Sigma{}E_n\to{}E_{n+1} (where Σ\Sigma denotes the reduced suspension).

There are various conditions that can be put on the spaces E nE_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.

Ω\Omega-spectra

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

The idea is that E 0E_0 contains the information of E\mathbf{E} in dimensions k0k\ge 0, E 1E_1 contains the information of E\mathbf{E} in k1k\ge -1 (but shifted up by one, so that it is modeled by the 0\ge 0 information in the space E 1E_1), and so on.

Ω\Omega-spectra are special cases of spectra, and are in fact the fibrant objects for some model structure on the category of spectra. Given any spectrum E\mathbf{E}, it is easy to transform it into an equivalent Ω\Omega-spectrum F\mathbf{F} (a fibrant replacement of E\mathbf{E}) : just take F n=lim mΩ mE n+mF_n=\lim_{m\to\infty}\Omega^m E_{n+m} and use the fact that Ω\Omega commutes with the filtered colimits.

Coordinate-free spectrum

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

See there for details.

Combinatorial definition

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 kk-cells for all integers kk, not necessarily invertible. Some people have called this conjectural object a ZZ-category. “Connective” ZZ-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

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)Sp(Top) of a stable (∞,1)-category. In fact, there is a general procedure for turning any pointed (∞,1)-category CC into a stable (,1)(\infty,1)-category Sp(C)Sp(C), and doing this to the category Top *Top_* of pointed spaces yields Sp(Top)Sp(Top).

Symmetric monoidal structure

Closed structure

Model category structure

(∞,1)-operad∞-algebragrouplike versionin Topgenerally
A-∞ operadA-∞ algebra∞-groupA-∞ space, e.g. loop spaceloop space object
E-k operadE-k algebrak-monoidal ∞-groupiterated loop spaceiterated loop space object
E-∞ operadE-∞ algebraabelian ∞-groupE-∞ space, if grouplike: infinite loop space \simeq Γ-spaceinfinite loop space object
\simeq connective spectrum\simeq connective spectrum object
stabilizationspectrumspectrum object

References

A general treatment is in the last chapter of

  • 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)

Surveys include

Revised on April 13, 2014 05:58:54 by Urs Schreiber (185.37.147.12)