  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 $\operatorname{ker} d^a /
\operatorname{im} d^b$.
