[This question][1] reminded me of a possibly stupid idea that I had a while back.

On page 2 of [this paper][2], 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.

  [1]: http://mathoverflow.net/questions/640/what-is-cohomology-and-how-does-a-beginner-gain-intuition-about-it
  [2]: http://arXiv.org/abs/math/0502016v1