Background: I'm a logic student with very little background in cohomology etc., so this question is fairly naive.
Although mathematical logic is generally perceived as sitting off on its own, there are some striking applications of algebraic/geometric/combinatorial ideas to logic. In general, I'm very interested in the following broad question:
"How should I go about looking for pieces of mathematics far from mathematical logic, which have bearing on some piece of mathematical logic?"
Right now, I'm specifically interested in the following:
"When should I think 'cohomology!'?"
The specific example I'm motivated by is a pair of papers by Dan Talayco (http://arxiv.org/pdf/math/9311205.pdf, http://www.sciencedirect.com/science/article/pii/0168007295000240) in which he develops cohomology theories for two purely set-theoretic objects: Hausdorff gaps, and particularly weird infinite trees ("Todorcevic trees").
At the beginning of his paper on Hausdorff gaps, Talayco mentions that
"the original observation that gaps are cohomological in nature is due to Blass."
This is something I want to be able to do! I can tell that e.g. Hausdorff gaps are all about "not being able to fill something in," but it's a long way between that vague statement and the intuition that there should be a cohomology theory around it, let alone coming up with the specifics. So my question is:
Question. When should I suspect that some piece of mathematics (ideally far from algebra/geometry) has a cohomological interpretation, and how should I go about figuring out what the specifics should be?
To clarify: although 'useful' is always good, I'm just asking how I can tell that cohomology can be attached to some piece of mathematics (especially logic), regardless of whether it yields new results.
Best Answer
I guess I will take a crack at this.
First of all, it is probably worthwhile for you to learn some cohomology in its original home so that you have some intuition for it, and some knowledge of what theorems there are, how to compute it, etc. You do not give much indication in your answer as to how much knowledge of cohomology you have currently.
Generally the intuition behind cohomology groups is that they measure the failure of "locally consistent" things to be "globally consistent".
Examples:
The first de Rham cohomology group of the punctured plane is 1 dimensional since there is (up to the gradient of a global function) only 1 vector field on this space which is locally a gradient of a function but not globally the gradient of a function.
The Penrose triangle represents a nontrivial cohomology class over the multiplicative group of positive reals, since it "locally" looks like a perspective drawing, but there is no "global" object realizing that.
If local exchange rates between countries allow arbitrage, then there is no globally consistent exchange rate, so current exchange rates give a nontrivial cohomology class.
The axiom of choice says every surjection splits. In fact, even without the axiom of choice, every surjection splits locally (for every point in the codomain, I can find an inverse image), and so the axiom of choice is a a local to global statement: these local inverses can be assembled into a global section. Blass has written a bit about this in Blass - Cohomology detects failure of the axiom of choice, but there is still a lot more work to be done with this concept.
The moral is just to be on the lookout for situations where things seems to fit together in small bits, but somehow the whole does not work out. There is, more than likely, cohomology playing into this somehow.
I will mention that (from my perspective) sheaf cohomology probably formalizes this intuitive perspective the best, since you do not have to start with a sequence of maps with differentials (where do those come from?) just a notion of local objects and how to patch them. So I would recommend learning some sheaf cohomology if you are planning on looking for cohomology far from algebraic topology.