A topological spectrum is an object in the universal stable (∞,1)-category that stabilizes” the (∞,1)-category Top or ∞-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 have a delooping object that is written in this context and called the suspension of . Thus a spectrum is like a topological space or ∞-groupoid that may be delooped indefinitely.
Connective spectra form a sub-(∞,1)-category of spectra
There are objects in , 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.
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 .
So it is the homologically nontrivial parts of the chain complexes in negative degree that corresponds to the non-connectiveness of a spectrum.
A simple first definition is to define a spectrum to be a sequence of pointed spaces together with structure maps (where denotes the reduced suspension).
There are various conditions that can be put on the spaces 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.
If denotes the loop space functor on the category of pointed spaces, we know that is left adjoint to . In particular, given a spectrum , the structure maps can be transformed into maps . If these maps are isomorphisms (depending on the situation it can be weak equivalences or homeomorphisms), then is called an -spectrum.
The idea is that contains the information of in dimensions , contains the information of in (but shifted up by one, so that it is modeled by the information in the space ), and so on.
-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 , it is easy to transform it into an equivalent -spectrum (a fibrant replacement of ) : just take and use the fact that 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 -categories are related to -groupoids. In other words, it would contain -cells for all integers , not necessarily invertible. Some people have called this conjectural object a -category. “Connective” -categories could perhaps then be identified with stably monoidal -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 of a stable (∞,1)-category. In fact, there is a general procedure for turning any pointed (∞,1)-category into a stable -category , and doing this to the category of pointed spaces yields .
|(∞,1)-operad||∞-algebra||grouplike version||in Top||generally|
|A-∞ operad||A-∞ algebra||∞-group||A-∞ space, e.g. loop space||loop space object|
|E-k operad||E-k algebra||k-monoidal ∞-group||iterated loop space||iterated loop space object|
|E-∞ operad||E-∞ algebra||abelian ∞-group||E-∞ space, if grouplike: infinite loop space Γ-space||infinite loop space object|
|connective spectrum||connective spectrum object|
A general treatment is in the last chapter of
Frank Adams, Infinite loop spaces, Princeton University Press, 1978