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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?

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This is the biggest slam dunk I have ever experienced on MathOverflow. I logged on intending to ask this very question, found it was already asked, and found that Urs's answer was exactly the sort of mind-blowing answer I was hoping for! –  Tim Campion Sep 23 '10 at 21:26
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5 Answers

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 .

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Is it also true for fancy algebraic things like motivic cohomology? –  Kevin H. Lin Nov 19 '09 at 19:49
I'd think so: ncatlab.org/nlab/show/motivic+cohomology –  Urs Schreiber Nov 19 '09 at 20:26
explicitly, notice prop 14.16 on page 126 here: math.rutgers.edu/~weibel/MVWnotes/prova-hyperlink.pdf –  Urs Schreiber Nov 19 '09 at 21:39
In fact, motivic cohomology is precisely an example of the general statement that cohomology is connected components of hom-spaces in an (oo,1)-topos: for motivic cohomology it is the (oo,1)-topos of oo-stacks on the Nisnevich site. This is now described here: ncatlab.org/nlab/show/motivic+cohomology#InfStackDef –  Urs Schreiber Jan 20 '10 at 23:17
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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.)

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Amazing, although I don't really know what to do with this (\infty,1)-stuff...I remember reading your notes/thoughts/questions (andreasholmstrom.org/research/Cohomology1.pdf) some time ago and I was happy that indeed people are thinking about these questions (people always tell me that I'm crazy when I ask such questions!). So, I'm looking forward to whatever you will add to your comment :) –  user717 Nov 25 '09 at 13:44
I wrote a blog post about the difference between generalized and ordinary cohomology in algebraic geometry - it doesn't answer your question but maybe you find it interesting. homotopical.wordpress.com/2009/12/10/… –  Andreas Holmstrom Dec 15 '09 at 2:26
+1 for a cool story. –  Tim Campion Sep 23 '10 at 21:38
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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.

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+1 for 2 n-lab links! –  Andrew Stacey Nov 20 '09 at 12:13
Well, one of them is a mathoverflow link, but Urs has at least half a dozen n-lab links once you get there, so hereditarily speaking, you may be right... +1 for the mathoverflow link. It's good to have these related posts linked together. –  Tim Campion Sep 23 '10 at 21:37
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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)

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I agree. "Cohomology functor" would be much better. However, we seem to be stuck with "theory"... –  Andreas Holmstrom Nov 25 '09 at 23:30
I disagree. Most impprtant examples of cohomology theories like K-theory for example were not defined and established in 2-line slick modern functorial definitions but created in large works and they all did and do have big applications in their fields. I do not see a difference between infinitesimal calculus and say de Rham cohomology theory as being frameworks with purpose, grandior structure and infinite possibilities for futher development and application... –  Zoran Skoda Mar 2 '10 at 1:38
When you mention K-theory, you're talking about a specific example. The zero functor is also a cohomology theory (with trivial coefficient group). The zero functor doesn't tell us anything. So why call it a theory? IMO "trivial theory" should be an oxymoron. –  Ryan Budney Mar 2 '10 at 1:55
Yes, there is something silly about the "theory" in "K-theory". On the other hand, there is something silly about the "K", too. –  Tom Goodwillie Jun 13 '10 at 4:25
For me, the point is that the word "theory", whether referring to a trivial entity or not, should NOT be attached to a definite mathematical object (such as an abelian group or a functor), but to a collection of methods, heuristics and theorems that may help to understand that object (or other objects). The word "K-theory" should not refer to a functor, but to the study of the functor that associates to a space its "K-group". [An exception would occur in Mathematical Logic, in which you want to talk of "theories" as definite mathematical objects] –  Qfwfq Apr 14 '11 at 12:32
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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".

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This is a (the?) universal Weil cohomology theory, right? So all this is just about this particular kind of cohomology theories!? –  user717 Nov 19 '09 at 17:26
-1 for using wikipedia instead of the n-lab (NOT serious) –  Andrew Stacey Nov 20 '09 at 12:14
Well, sometimes Wikipedia is better if you don't know what an (\infty,1)-category is, because the number of entries in nlab you can read without knowing this is close to 0 :) (serious) –  user717 Nov 20 '09 at 16:09
Arminius is right. I've never had any real need for category theory beyond understanding the words "morphism" and "functor", and the nlab is completely opaque to me. –  Steve Huntsman Jun 4 '10 at 17:33
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