Simplicial Structure on Complexes

TL;DR

This paper uncovers a new simplicial structure on the category of chain complexes, linking it to N-complexes and providing a categorification of the triangulated homotopy category.

Contribution

It demonstrates a novel equivalence between the lax nerve of chain complexes and the dcalage of N-complexes, revealing a simplicial origin of triangulated categories.

Findings

Simplicial structure on the lax nerve of chain complexes

Equivalence between the lax nerve and dcalage of N-complexes

Evidence for simplicial axioms underlying triangulated categories

Abstract

While chain complexes are equipped with a differential $d$ satisfying $d^2 = 0$, their generalizations called $N$-complexes have a differential $d$ satisfying $d^N = 0$. In this paper we show that the lax nerve of the category of chain complexes is pointwise adjoint equivalent to the d\'ecalage of the simplicial category of $N$-complexes. This reveals additional simplicial structure on the lax nerve of the category of chain complexes which provides a categorfication of the triangulated homotopy category of chain complexes. We study this phenomena in general and present evidence that the axioms of triangulated categories have simplicial origin.