Simplicial Structure on Complexes

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 décalage 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.

Cited by in corpus (1)

• Categorification of the ring of cyclotomic integers for products of two primes