Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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Many-step complexes have appeared in recent work of Mikhail Kapranov (arXiv:q-alg/9611005, which appears not to be published, according no MathSciNet?)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

share|improve this question
    
What precisely is an $N$-step complex? Is it simply a chain complex in which the groups are trivial outside a certain range of dimensions? –  Tom Goodwillie Jun 23 '11 at 23:07
    
@Tom - I think it is a diagram of abelian groups (say - obvious extension to abelian or even pointed categories) $\mathbb{Z} \to Ab$ such that $d_i\circ \ldots\circ d_{i+N-1}$ is the zero map for all $i$, where $d_i:G_i \to G_{i+1}$. –  David Roberts Jun 24 '11 at 1:03
5  
Oh! That's not how I interpreted what Dror wrote. –  Tom Goodwillie Jun 24 '11 at 1:49
    
Yes, I thought by "step" Dror meant the number of chain groups in a chain complex... –  Qiaochu Yuan Jun 24 '11 at 14:20
add comment

2 Answers

One approach would be to take Dold-Kan as a motivating principle, i.e. to say that a good notion of n-step simplicial object will be equivalent to n-step complexes via an equivalence which "normalizes" simplicial objects to get complexes, and which associates simplices to a complex by considering maps from the normalized k-simplex, i.e.

$X_.\mapsto (NX.,d)$

$(C_.,d)\mapsto KC_.$

where $KC_k:=Hom_{n-step~complexes}(N\Delta^k,C)$

In proposition 0.2 of his article, Kapranov gives an assignment of an n-step complex to any simplicial vector space. This seems like a reasonable candidate for the assignment on objects of a normalization functor N, and the test would be whether defining K as above plays well with the homological properties of the n-step complex.

share|improve this answer
add comment

Oh, there are many papers recently devoted to $N$-complexes, in particular in connection with theoretical physics. A lot of these papers have simplicial stuff in it. I don't have access to MathSciNet now so some of my suggestions you have to look up yourself as I sometimes don't know the exact titles.

Here goes:

  • Tikaradze: "Homological cobstructions on $N$-complexes" Journal of pure and applied algebra (I'm pretty sure that there are some simplical stuff in this.)

  • Michel Dubois-Violette has a lot of papers that are very relevant. Especially a paper called $d^N=0$ or something to that effect in $K$-theory.

  • Angel and Diaz: "Differential graded algebras" Journal Pure and Applied Algebra I think.

  • Connes (Alain, that is) et al has some papers on so called Homogenous algebras and Yang-Mills algebras, you can look up.

  • Berger and Marconnet: "Koszul and Gorenstein properties of Homogenous algebras" Algebras and representation theory.

  • Same goes for Richard Kerner and Victor Abramov

  • And, if I may bang my own drum, I ever so briefly dabbled a bit in this area a few years back: Larsson and Silvestrov: "On $q$-differential graded algebras and $N$-complexes". Nothing very deep though ;)

Finally, and maybe most importantly, my friend Goro Kato tried to construct a derived category of $N$-complexes but managed to show instead that in reality one gains almost nothing with $N$-complexes instead of ordinary $2$-complexes, at least not from a homological perspective: From an $N$-complex one can construct, essentially in a unique manner, an ordinary $2$-complex. I don't remember the title but MathSciNet should solve that easily.

share|improve this answer
    
I am familiar with several of these... Do you know if any of them actually has an $N$-step analogue of simplicial sets/complexes? –  Mariano Suárez-Alvarez Jul 8 '11 at 15:55
    
I'm pretty sure Tikaradze's paper has some simplicial complexes in it (my memory could fail me here). Also, I think Dubois-Violette's paper in K-theory has some simplicial complexes appearing. Once again, I could be wrong. –  Daniel Larsson Jul 8 '11 at 16:30
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.