[This question](https://mathoverflow.net/questions/640/what-is-cohomology-and-how-does-a-beginner-gain-intuition-about-it) reminded me of a possibly stupid idea that I had a while back.


On page 2 of [this paper](http://arXiv.org/abs/math/0502016v1), 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.



Homological algebra and calculus (as in Newton)
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.


