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What is Simplicial Structure on Complexes about?

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.

Author

Mirmohades, Djalal

Published

2014

Language

EN