Homologically optimal categories of sequences lead to N-complexes

Abstract

We study the category of $\mathbb{Z}$-indexed sequences over an abelian category and certain generalized homology functors for this category of sequences which are indexed by positive integers $a$ and $b$. By looking at the corresponding derived category, we show that there is an "optimal" subcategory of sequences for every choice of our generalized homology functors, namely, the category of $N$-complexes (sequences for which the differential $d$ satisfies $d^N=0$) where $N=a+b$. In this optimal case we show that our homology functors reduce to Kapranov's homology functors $\ker d^a/\mathrm{im}{d}^b$.

An adaptation of a part of author's Master Thesis from 2010

References in corpus (1)

• The homotopy category of N-complexes is a homotopy category

Cited by in corpus (2)

• Categorification of the ring of cyclotomic integers for products of two primes
• Irreducible representations of the symmetric groups from slash homologies of p-complexes