User:Rswarbrick/Scratchpad

In algebraic topology, a branch of mathematics, a spectrum is an object representing a generalized cohomology theory. There are several different constructions of categories of spectra, but they are designed so that they all give the same stable homotopy theory.

Motivation from generalized cohomology
Suppose one starts with a reduced generalized cohomology theory, $$E$$ defined on pairs of CW complexes. Write $$E^n(X)$$ for the $$n$$'th cohomology group. Then Brown's representability theorem says that there exist connected CW complexes $$E_n$$ with basepoint such that
 * $$E^n(X) \cong [X, E_n]$$.

Of course, a cohomology theory isn't just a collection of groups: there are also the coboundary maps $$\delta: E^n(X)\to E^{n+1}(X)$$. In particular, there is a suspension isomorphism
 * $$E^{n+1}(\Sigma X) \cong E^n(X)$$,

where $$\Sigma X$$ is the reduced suspension of $$X$$. Now there is the following sequence of natural isomorphisms:

[X, E_n] \cong E^n(X) \cong E^{n+1}(\Sigma X) \cong [\Sigma X, E_{n+1}] \cong [X, \Omega E_{n+1}] $$, where $$\Omega E_{n+1}$$ is the space of based loops on $$E_{n+1}$$. This must come from a weak equivalence $$E_n\to\Omega E_{n+1}$$. One can apply the adjunction between reduced suspension and based loops to obtain a map $$\Sigma E_n \to E_{n+1}$$ instead if that is more useful. .

Formal definition
A prespectrum is a sequence of spaces $$X_n$$ for all integers $$n$$, together with maps $$\sigma_n: \Sigma X_n \to X_{n+1}$$. Under the adjunction mentioned above, $$\sigma_n$$ corresponds to a map
 * $$\tilde{\sigma}_n: X_n \to \Omega X_{n+1}$$

A prespectrum is called a spectrum if this map is a homeomorphism. One simple example of such a prespectrum is the suspension prespectrum of a space $$X$$, written $$\Sigma X$$. Its spaces are repeated suspensions, $$X_n = \Sigma^n X$$, and the structure maps are the identity on $$\Sigma X_n$$.
 * Mention connective spectra here?

Of course, we wish to define a category of spectra. There are various constructions, but


 * Hmm, how do you talk about EKMM or whatever here? They use indexing universes etc...

The 'stable' category

 * Explain here how one has just inverted the suspension homomorphism.

Smash products of spectra

 * Here I want to explain how to do smash products in the EKMM version, since it's much simpler than Adams's one

Examples
Consider singular cohomology $$ H^n(X;A) $$ with coefficients in an abelian group A. By Brown representability $$ H^n(X;A) $$ is the set of homotopy classes of maps from X to K(A,n), the Eilenberg-MacLane space with homotopy concentrated in degree n. Then the corresponding spectrum HA has n'th space K(A,n); it is called the Eilenberg-MacLane spectrum.
 * Mention that HA isn't connective?

As a second important example, consider topological K-theory. At least for X compact, $$ K^0(X) $$ is defined to be the Grothendieck group of the monoid of complex vector bundles on X. Also, $$ K^1(X) $$ is the group corresponding to vector bundles on the suspension of X. Topological K-theory is a generalized cohomology theory, so it gives a spectrum. The zero'th space is $$ \mathbb{Z} \times BU $$ while the first space is $$U$$. Here $$U$$ is the infinite unitary group and $$BU$$ is its classifying space. By Bott periodicity we get $$ K^{2n}(X) \cong K^0(X) $$ and $$ K^{2n+1}(X) \cong K^1(X) $$ for all n, so all the spaces in the topological K-theory spectrum are given by either $$ \mathbb{Z} \times BU $$ or $$U$$. There is a corresponding construction using real vector bundles instead of complex vector bundles, which gives an 8-periodic spectrum.

For many more examples, see the list of cohomology theories.

History
A version of the concept of a spectrum was introduced in the 1958 doctoral dissertation of Elon Lages Lima. His advisor Edwin Spanier wrote further on the subject in 1959. Spectra were adopted by Michael Atiyah and George W. Whitehead in their work on generalized homology theories in the early 1960s. The 1964 doctoral thesis of J. Michael Boardman gave a workable definition of a category of spectra and of maps (not just homotopy classes) between them, as useful in stable homotopy theory as the category of CW complexes is in the unstable case.

Important further theoretical advances have however been made since 1990, improving vastly the formal properties of spectra. Consequently, much recent literature uses modified definitions of spectrum: see Mandell et al. (2001) for a unified treatment of these new approaches.