This entry is about the notion of spectrum in stable homotopy theory. For other uses of the term ‘’spectrum’‘ see spectrum - disambiguation.
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.
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 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 $Sp(Top)$.
So it is the homologically nontrivial parts of the chain complexes in negative degree that corresponds to the non-connectiveness of a spectrum.
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).
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).
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).
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).
A definition of spectrum consisting of spaces indexed by index sets less “coordinatized” than the integers is a
See there for details.
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.
See spectrum object.
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 $Sp(Top)$.
According to (Adams 74, p. 131) the notion of spectrum is due to
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
More modern developments are due to
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
A comprehensive account of the symmetric model in terms of symmetric spectra is in
and in terms of orthogonal spectra in
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)