Question

This question has bugged me for a long time. Is there a unifying concept behind everything that is called a "cohomology theory"?

I know that there exist generalized cohomology theories, Weil cohomology theories and perhaps one might include delta-functors, which describe (some of) the properties of explicit cohomology theories. But is there now a concept that underlies all these concepts? Or has the term "cohomology theory" been used so inflationary that the best thing one could say is that a cohomology theory is a sequence of functors into an "algebraic category" (whatever that is)?

Moreover, what is the difference between a cohomology theory and a homology theory? Of course I am aware of the examples like singular (co-)homology and I know the difference in this situation, but in general?

Answer 1

I have now newly written a detailed "Idea"-section at the nLab entry on cohomology, which should give a helpful overview on the observation that and how every flavor of cohomology ever considered is nothing but the study of connected components in the hom-spaces of some (oo,1)-topos:

nLab:Cohomology -- Idea .

Answer 2

I promised to write a longer answer, but I simply don't have time this week - sorry. What I wanted to point our was that although the idea that "every flavor of cohomology ever considered is nothing but the study of connected components in the hom-spaces of some (oo,1)-topos" is one of the most amazing ideas ever (imho), it is still not clear (at least to me) exactly how this works in all cases, even for abelian sheaf cohomology. For example, most people seem to believe that the right (oo,1)-topos for cohomology theories in algebraic geometry should be given by A1 (or "motivic") homotopy theory, but there is nothing in the literature about representability of p-adic cohomology theories such as rigid cohomology. I believe this might be because there is some technical problem, but I am not sure. There are also other issues and examples which are not clear (to me!).

The other thing I wanted to do was to clarify various pieces of terminology related to cohomology in algebraic geometry, for example, "generalized cohomology" means different things in different articles, and there are many different notions of "universal cohomology". Maybe I can expand on this later.

One small remark: Motivic cohomology is usually thought of as the universal Bloch-Ogus cohomology, while the universal Weil cohomology should probably be pure motives with respect to rational equivalence ("probably", because it depends on what exactly you mean by "universal" and "Weil cohomology"). The two notions are closely related though.

(Aside: The reason I am very busy this week is that I suddenly find myself writing job applications, after essentially solving my thesis problem last week, and one of the main reasons I could solve my thesis problem was that I applied Urs' unified point of view on cohomology in a new setting.)

Answer 3

Urs Schreiber has already addressed parts of this question here. There's lots of good stuff to mine through in the nLab entry on cohomology.

Answer 4

I'd say a cohomology theory is a misnomer. A theory really ought to be significant, make predictions, help us think about things, help us prove theorems. At some point mathematicians decided to start giving away the word "theory" for free. Newtonian mechanics, evolution, calculus -- those are theories.

Is it a lack of imagination on our part? It seems like anything that hasn't earned a proper name gets called "X theory" nowadays, for various values of X. I'm glad differential geometry was invented in a previous era. Our contemporaries would have saddled the subject with some glorious name like "geometry theory" or "G-theory".

(not exactly sure if tongue is in cheek)

Answer 5

A complementary answer to Kevin's that is provided by the interpretation of motivic cohomology as a "mother of all cohomology theories in algebraic geometry".