Highly structured ring spectra have better formal properties than multiplicative cohomology theories – a point utilized, for example, in the construction of topological modular forms, and which has allowed also new constructions of more classical objects such as Morava K-theory.
As not every cohomology theory allows such operations, not every multiplicative structure may be refined to an
The rough idea of highly structured ring spectra is the following: If multiplication in a cohomology theory (analogous to the multiplication in singular cohomology, inducing the cup product) fulfills associativity (and commutativity) only up to homotopy, this is too lax for many constructions (e.g. for limits and colimits in the sense of category theory).
On the other hand, requiring strict associativity (or commutativity) in a naive way is too restrictive for many of the wanted examples.
The most widely used approaches today employ the language of model categories.
[citation needed] All these approaches depend on building carefully an underlying category of spectra.
The theory of operads is motivated by the study of loop spaces.
This situation can be made precise by saying that ΩX is an algebra over the little interval operad.
-operad, i.e. an operad of contractible topological spaces with analogous "freeness" conditions.
An example of such an operad can be again motivated by the study of loop spaces.
To get full coherence of higher homotopies one must assume that the space is (equivalent to) an n-fold loopspace for all n. This leads to the in
Since (commutative) monoids are easier to deal with than algebras over complicated operads, this new approach is for many purposes more convenient.
It should, however, be noted that the actual construction of the category of S-modules is technically quite complicated.
Probably the most famous one of these is the category of symmetric spectra, pioneered by Jeff Smith.
Its basic idea is the following: In the most naive sense, a spectrum is a sequence of (pointed) spaces
Another viewpoint is the following: one considers the category of sequences of spaces together with the monoidal structure given by a smash product.
But the monoid structure of the sphere sequence is not commutative due to different orderings of the coordinates.
The category of symmetric spectra has a monoidal product denoted by
This boils down to giving maps which satisfy suitable equivariance, unitality and associativity (and commutativity) conditions (see Schwede 2007).
On the other hand, symmetric spectra have the advantage that they can also be defined for simplicial sets.
Infinity-categories are a variant of classical categories where composition of morphisms is not uniquely defined, but only up to contractible choice.
The categories of S-modules, symmetric and orthogonal spectra and their categories of (commutative) monoids admit comparisons via Quillen equivalences due to work of several mathematicians (including Schwede).
In spite of this the model category of S-modules and the model category of symmetric spectra have quite different behaviour: in S-modules every object is fibrant (which is not true in symmetric spectra), while in symmetric spectra the sphere spectrum is cofibrant (which is not true in S-modules).
By a theorem of Lewis, it is not possible to construct one category of spectra, which has all desired properties.
A comparison of the infinity category approach to spectra with the more classical model category approach of symmetric spectra can be found in Lurie's Higher Algebra 4.4.4.9.
[dubious – discuss] It is easiest to write down concrete examples of
The most fundamental example is the sphere spectrum with the (canonical) multiplication map
Topological (real or complex) K-theory is also an example, but harder to obtain: in symmetric spectra one uses a C*-algebra interpretation of K-theory, in the operad approach one uses a machine of multiplicative infinite loop space theory.
By a similar (but older) method, it could also be shown that Morava K-theory and also other variants of Brown-Peterson cohomology possess an
Highly structured ring spectra allow many constructions.