# Homological algebra and calculus (as in Newton)

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.

-
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
When I think of ∂ I'm necessarily thinking about the fundamental theorem of calculus: take I=[0,1] then $\int_I D[f] = f \vert_{\partial I}$. So wouldn't ∂² then be evaluating $f$ on ∂(∂(I))=∅, which is a strange enough notion that I'm not sure what one would want to do to make sense of it. My way of thinking about it doesn't yield anything nice like $f''$ resulting from ∂∂I, which may make it a bad idea. – isomorphismes Jun 7 '14 at 4:59

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.

-
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

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).

-

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.)

-

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.

-

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

-
It sounds like an isomorphism theorem for functors: replacing map(•) with surj(bij(inj(•))). – isomorphismes Jun 7 '14 at 5:27