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Recently, answering a question here, 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. I like to think of $N$-complexes are modules over the quiver $$\cdots\to\bullet\to\bullet\to\bullet\to\bullet\to\cdots$$ bound by the relations that say that the product of $N$ consecutive arrows is zero.

Many-step complexes have appeared in recent work of Mikhail Kapranov (arXiv:q-alg/9611005, which appears not to be published, according no MathSciNet?)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?

Recently, answering a question here, 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Many-step complexes have appeared in recent work of Mikhail Kapranov (arXiv:q-alg/9611005, which appears not to be published, according no MathSciNet?)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?

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# $N$-step simplicial complexes

Recently, answering a question here, Dror Bar-Natan observed that «way too often two-step complexes have a natural extension to become many-step complexes».

Many-step complexes have appeared in recent work of Mikhail Kapranov (arXiv:q-alg/9611005, which appears not to be published, according no MathSciNet?)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?