Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

This question reminded me of a possibly stupid idea that I had a while back.

On page 2 of this paper, while discussing Euclid's axioms of plane geometry and spatial geometry, Manin makes an extremely interesting comment:

Euclid misses a great opportunity here: if he stated the principle

“The extremity of an extremity is empty”,

he could be considered as the discoverer of the

BASIC EQUATION OF HOMOLOGICAL ALGEBRA: d^2 = 0.

Ever since I read this, I've had a suspicion that the equation "d^2 = 0" of homological algebra is somehow related to the equation "epsilon^2 = 0" of (first-order) calculus (as in Newton)*, since the latter equation can be interpreted as saying "a very very small quantity is zero" which at least superficially seems similar to "the extremity of an extremity is empty". I once explained my suspicion to Dan Erman over beers, and he responded by asking another question: Can we do some sort of homological algebra using the equation d^n = 0 rather than d^2 = 0? Perhaps if d^2 = 0 can be related to first-order calculus, then d^3 = 0 can be related to second-order calculus, and so on...

I don't really have a specific question to ask -- I just thought I might put this idea out there. Maybe someone can tell me why this idea is stupid, or why it is not stupid.


*or the ring of dual numbers k[epsilon]/(epsilon^2) if you're an algebraist or an algebraic geometer.

share|improve this question
3  
As a more direct explanation of what d^2=0 has to do with \epsilon^2=0, in de Rham cohomology d^2=0 amounts to the fact that partial derivatives commute with each other. If you write this down in terms of infinitesimals it ought to involve some \epsilon^2=0 somewhere. –  Eric Wofsey Oct 15 '09 at 21:01
    
@Eric: Oh yeah, I had had that same thought, I forgot to mention it. –  Kevin H. Lin Oct 15 '09 at 21:07
add comment

5 Answers

up vote 4 down vote accepted

I saw the notion of an "n-complex" once in this preprint by Peter Olver: http://www.math.umn.edu/~olver/a_/hyper.pdf

I only ever studied sections 5 and 6 of this paper, so I don't know what he's actually doing in the other sections (he introduces "hypercomplexes" which contain "n-complexes" as subcomplexes in section 7). The introduction mentions that he is interested in higher-order versions of de Rham complexes.

share|improve this answer
    
Thanks for the reference. I just very quickly skimmed some of it and it seems very interesting, and probably closely related to my idea. –  Kevin H. Lin Oct 15 '09 at 22:59
add comment

A while ago I worked on the question of what we can say if $d^n=0$, but I got distracted by more concrete problems. A few people have certainly thought about this question. One place to start looking is "$d^N=0$: generalized homology" or "Generalized homologies for $d^N=0$ and graded $q$-differential algebras" both by Michel Dubois-Violette.

(Sorry for the lack of links; I'm off-campus so I can't actually get to the MathSciNet entries right now.)

share|improve this answer
add comment

Another reference for the same idea is the paper "On the q-analog of homological algebra" by Misha Kapranov (arXiv:q-alg/9611005).

In the theory of cyclic homology, which is (among other things) an algebraic/homotopical interpretation of calculus, the fact that d^2=0 comes directly from the structure of the homology of the circle, which is indeed the dual numbers (except that epsilon has degree -1). The ring structure here is the Pontryagin product -- ie it comes from the group structure on S^1 via convolution (it's the "group algebra" of S^1).

share|improve this answer
add comment

I'm answering this from a much more amateurish standpoint than Kevin's, and so everything I say here should be taken with liberal application of sodium chloride. If I'm completely wrong, please let me know so I can remove the incriminating evidence. :)

That said, I seem to recall having convinced myself at one point that a "homological algebra" with d^n = 0 wouldn't give much new information. The reason, I think, was essentially that homology groups measure the failure of a chain complex to be exact, and so intuitively the homology of something with d^n = 0 would measure the failure of longer subsequences to be exact. But longer exact sequences can be built up from short ones, so it should be possible to associate a "standard" chain complex to such a weird beast, that contains basically the same information.

Again, this was all thought out way before I had any real concept of how homology behaves in the wild, so it's very likely incorrect, but I'm mentioning it anyway on the off chance that there's something to it.

share|improve this answer
add comment

There is a whole theory of calculus of functors started by Goodwillie. There are Taylor approximations of functors and so on. Here's the wikipedia page which contains over references:

http://en.wikipedia.org/wiki/Calculus_of_functors

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.