Recently, answering a question [here](http://mathoverflow.net/questions/68339/motivating-the-category-of-chain-complexes/), [Dror Bar-Natan](http://mathoverflow.net/users/8899/dror-bar-natan) observed that «way too often two-step complexes have a natural extension to become many-step complexes». By such a thing I mean (and I think Dror means too!) a "complex" in which $d^N=0$ and not $d^2=0$, as usual.

Many-step complexes have appeared in recent work of Mikhail Kapranov ([arXiv:q-alg/9611005](http://arxiv.org/abs/q-alg/9611005), which appears not to be published, according no MathSciNet?) and [Kassel, C.; Wambst, M. Algèbre homologique des N-complexes et homologie de Hochschild aux racines de l’unité. Publ. Res. Inst. Math. Sci. 34, (1998), 91–114], for example, but they certainly predate that: they appear already in [Mayer, W. A new homology theory. I, II. Ann. of Math. 43, (1942). 370–380, 594–605] and in [Spanier, E.H. The Mayer homology theory, Bull. Amer. Math. Soc. 55 (1949), 102- 112], and maybe earlier than that.

> Is there a reasonable analogue for what an $N$-step simplicial complex might be?