# nLab spectrum

Contents

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

### Context

#### Stable homotopy theory

stable homotopy theory

Introduction

# 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.

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

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 stable Dold-Kan correspondence (see at module spectrum the section stable Dold-Kan correspondence ) that identifies these with special objects in $Sp(Top)$.

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

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 (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 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)

$F_n \coloneqq \lim_{m\to\infty}\Omega^m E_{n+m}$

(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

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.

## 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 $Sp(Top)$.

### Relation to symmetric monoidal ∞-groupoids

A-∞ operadA-∞ algebra∞-groupA-∞ space, e.g. loop spaceloop space object
E-k operad?E-k algebrak-monoidal ∞-groupiterated loop spaceiterated loop space object
E-∞ operadE-∞ algebraabelian ∞-groupE-∞ space, if grouplike: infinite loop space $\simeq$ ∞-spaceinfinite loop space object
$\simeq$ connective spectrum$\simeq$ connective spectrum object
stabilizationspectrumspectrum object

## 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

See the references at stable homotopy theory.

Lecture notes include

More modern developments are due to

The quick idea is surveyed for instance in

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

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

and in terms of orthogonal spectra in