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generalized (Eilenberg-Steenrod) cohomology

What is called generalized Eilenberg-Steenrod cohomology is really the general fully abelian subcase of cohomology.

This means that generalized Eilenberg-Steenrod cohomology is the cohomology in an (∞,1)-category $H$ that happens to be a stable (∞,1)-category.

The archetypical example of this is $H=\mathrm{Sp}(\mathrm{Top})$, the stable (∞,1)-category of spectra and this is the context in which generalized Eilenberg-Steenrod cohomology is usually understood. So

Generalized Eilenberg-Steenrod cohomology is cohomology $H(X,A)$ with coefficient object $A$ a spectrum.

the Eilenberg-Steenrod axioms

remark Originally Eilenberg and Steenrod had written down axioms that characterized the behaviour of ordinary integral cohomology, what is now understood to be cohomology with coefficients in the Eilenberg-MacLane spectrum. Generalized Eilenberg-Steenrod cohomology is originally defined as anything that satisfies this list of axioms except the first one. Later it was proven, by the Brown representability theorem, that all the models for these axioms arise in terms of homotopy classes of maps into a spectrum. In our revisionist perspective above, we take this historically secondary point of view as the conceptually primary one.

The Eilenberg-Steenrod axioms are this:

Definition

A cohomology theory is a collection $\{A^n{\}}_{n\in \mathbb{Z}}$ of functors

$$A^n:{\mathrm{Top}}^{\mathrm{op}}\to \mathrm{Ab}$$A^n : Top^{op} \to Ab

from the category Top of topological spaces to the category Ab of abelian groups, that satisfies the following axioms, for all $n\in \mathbb{Z}$:

1. if $f:X\to Y$ is a weak homotopy equivalence then $A^n(f):A^n(Y)\to A^n(X)$ is an isomorphism;

   i.e. $A^n$ is a homotopical functor with respect to the standard structure of a homtopical category? on Top,

1. …

1. …

1. …

1. …

Examples

• Eilenberg–Mac Lane spectrum
• K-theory spectrum
• tmf

References

A pedagogical introduction to spectra and generalized (Eilenberg-Steenrod) cohomology is in

• John Baez, TWF 149 .

A general discussion of cohomology with an emphasis on nonabelian cohomology and on Postnikov systems is in

• John Baez, Mike Shulman, Lectures on n-Categories and Cohomology (pdf, nLab)

In particular a remark on the relation between Postnikov systems and Eilenberg-MacLane cohomology is in the discussion beginning at the bottom of page 24.