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.

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? stably monoidal $\mathrm{\infty }$-groupoids

• or symmetic 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.

Non-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

• complex cobordism spectrum