nLab
spectrum

Idea

A topological spectrum is an object in the universal stable (∞,1)-category $\mathrm{Sp}(\mathrm{Top})\simeq \mathrm{Sp}(\mathrm{\infty }\mathrm{Grpd})$ 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 $A$ have a delooping object $BA$ that is written $\Sigma A$ in this context and called the suspension of $A$. Thus a spectrum is like a topological space or ∞-groupoid that may be delooped indefinitely.

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

• or infinite loop spaces

• or grouplike symmetric monoidal ∞-groupoids.

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

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

There are objects in $\mathrm{Sp}(\mathrm{Top})$, 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.

$$\begin{array}{ccc}{\mathrm{Ch}}_{+}& \stackrel{\mathrm{Dold}-\mathrm{Kan}\mathrm{nerve}}{\to }& \mathrm{ConnectSp}(\mathrm{\infty }\mathrm{Grp})\subset \mathrm{\infty }\mathrm{Grpd}\\ (\cdots A_2\stackrel{\partial }{\to }{A}_1\stackrel{\partial }{\to }{A}_0\to 0\to 0\to \cdots )& \stackrel{}{\mapsto }& N(A_{•})\end{array}$$\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 that identifies these with special objects in $\mathrm{Sp}(\mathrm{Top})$.

$$\begin{array}{ccc}\mathrm{Ch}& \stackrel{\mathrm{Dold}-\mathrm{Kan}\mathrm{nerve}}{\to }& \mathrm{Sp}(\mathrm{\infty }\mathrm{Grp})\\ (\cdots A_2\stackrel{\partial }{\to }{A}_1\stackrel{\partial }{\to }{A}_0\stackrel{\partial }{\to }{A}_{-1}\stackrel{\partial }{\to }{A}_{-2}\to \cdots )& \stackrel{}{\mapsto }& N(A_{•})\end{array}$$\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).

$\Omega$-spectra

One fairly simple, and quite useful, approach is to define a spectrum $E$ to be a sequence of based spaces $E_n$, for all natural numbers $n$, together with isomorphisms $E_n\cong \Omega E_{n+1}$, where $\Omega$ denotes the based loop space. The idea is that $E_0$ contains the information of $E$ in dimensions $k\ge 0$, $E_1$ contains the information of $E$ 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.

This is called an $\Omega$-spectrum.

Coordinate-free spectrum

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 definition

There might be a type of categorical structure related to a spectrum in the same way that $\mathrm{\infty }$-categories are related to $\mathrm{\infty }$-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 $\mathrm{\infty }$-categories.

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

Remarks

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

Examples of spectra

• Eilenberg-MacLane spectrum

• K-theory spectrum

• tmf

• Thom spectrum, complex cobordism spectrum