TITLE: What's a cohomology that's not defined from a cochain complex?
QUESTION [9 upvotes]: According to Wikipedia:

In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups associated to a topological space, often defined from a cochain complex.

(emphasis mine)
When I read "often defined", I assume that to mean "not always". I've seen that Čech cohomology is defined from a cochain complex. de Rham cohomology the same. I checked a few more and all defined from a cochain complex. So what's an example of a cohomology that is NOT defined from a cochain complex?

REPLY [7 votes]: Wikipedia is talking about generalized cohomology theories. They satisfy a bunch of axioms formally similar to what e.g. singular cohomology satisfies. But the so-called extraordinary cohomology theories (K-theory, cohomotopy, cobordism and so on) do not come from chain complexes. Indeed, by a theorem of Burdick--Conner--Floyd, in some sense if they did "come from chain complexes", then they would be given by a product of ordinary cohomology theories.

R. O. Burdick, P. E. Conner, and E. E. Floyd. “Chain theories and their derived homology”. In: Proc. Amer. Math. Soc. 19 (1968), pp. 1115–1118. issn: 0002-9939.

Roughly speaking, the theorem is that if you have a generalized cohomology theory $h$ given by the cohomology of a (functorial on pairs) cochain complex, and such that the long exact sequence comes from the long exact sequence of that cochain complex, then $$h^n(X,A) = \sum_{i+j=n} H^i(X,A; h^j(\text{point}))$$ is really given by singular cohomology with various coefficients.
But the emphasis here is on "cohomology theory". In a different setting (homological algebra), "cohomology" always means "cohomology of a cochain complex". Honestly, the Wikipedia article should probably be renamed to something like "Cohomology theories" or "Cohomology (topology)" because that's what it's really about.

REPLY [5 votes]: There are various generalized cohomology theories which are typically not described in terms of cochain complexes, perhaps the simplest example being topological K-theory. There's a precise technical sense in which topological K-theory cannot be "described in terms of cochain complexes," but it's not easy to spell out.