In algebraic topology, a branch of mathematics, a spectrum is an object representing a generalized cohomology theory.
is equivalent to computing the homotopy classes of maps to the space
This is one of the key points for introducing spectra because they form a natural home for stable homotopy theory.
The following is due to Frank Adams (1974): a spectrum (or CW-spectrum) is a sequence
showing the category of spectra keeps track of the derived information of commutative rings, where the smash product acts as the derived tensor product.
Moreover, Eilenberg–Maclane spectra can be used to define theories such as topological Hochschild homology for commutative rings, a more refined theory than classical Hochschild homology.
is defined to be the Grothendieck group of the monoid of complex vector bundles on 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.
Moreover, if considering the category of symmetric spectra, this forms the initial object, analogous to
The construction of the suspension spectrum implies every space can be considered as a cohomology theory.
Unfortunately, these two structures, with the addition of the smash product, lead to significant complexity in the theory of spectra because there cannot exist a single category of spectra which satisfies a list of five axioms relating these structures.
There are three natural categories whose objects are spectra, whose morphisms are the functions, or maps, or homotopy classes defined below.
, where two such functions represent the same map if they coincide on some cofinal subspectrum.
This gives the category of spectra (and maps), which is a major tool.
(associativity of the smash product yields immediately that this is indeed a spectrum).
Many other definitions of spectrum, some appearing very different, lead to equivalent stable homotopy categories.
Thus homotopy classes from one spectrum to another form an abelian group.
Furthermore the stable homotopy category is triangulated (Vogt (1970)), the shift being given by suspension and the distinguished triangles by the mapping cone sequences of spectra The smash product of spectra extends the smash product of CW complexes.
It makes the stable homotopy category into a monoidal category; in other words it behaves like the (derived) tensor product of abelian groups.
A major problem with the smash product is that obvious ways of defining it make it associative and commutative only up to homotopy.
Some more recent definitions of spectra, such as symmetric spectra, eliminate this problem, and give a symmetric monoidal structure at the level of maps, before passing to homotopy classes.
The smash product is compatible with the triangulated category structure.
One of the canonical complexities while working with spectra and defining a category of spectra comes from the fact each of these categories cannot satisfy five seemingly obvious axioms concerning the infinite loop space of a spectrum
denote the category of based, compactly generated, weak Hausdorff spaces, and
A version of the concept of a spectrum was introduced in the 1958 doctoral dissertation of Elon Lages Lima.
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.
(This is essentially the category described above, and it is still used for many purposes: for other accounts, see Adams (1974) or Rainer Vogt (1970).)
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 Michael Mandell et al. (2001) for a unified treatment of these new approaches.