Homology groups of cubical sets

TL;DR

This paper investigates the homology groups of cubical sets with coefficients in contravariant Abelian group systems, establishing isomorphisms with derived functors and constructing spectral sequences for their analysis.

Contribution

It introduces a new isomorphism criterion for homology groups of cubical sets and develops spectral sequences for their coverings and morphisms.

Findings

Homology groups are isomorphic to derived functors of colimits.

Established isomorphism criterion for homology groups.

Constructed spectral sequences for cubical set coverings.

Abstract

The paper is devoted to homology groups of cubical sets with coefficients in contravariant systems of Abelian groups. The study is based on the proof of the assertion that the homology groups of the category of cubes with coefficients in the diagram of Abelian groups are isomorphic to the homology groups of normalized complex of the cubical Abelian group corresponding to this diagram. The main result shows that the homology groups of a cubical set with coefficients in a contravariant system of Abelian groups are isomorphic to the values of left derived functors of the colimit functor on this contravariant system. This is used to obtain the isomorphism criterion for homology groups of cubical sets with coefficients in contravariant systems, and also to construct spectral sequences for the covering of a cubical set and for a morphism between cubical sets. This version of the preprint contains a refinement of the proof of Lemma 3.2 corresponding to Lemma 4 from the paper of the same name published in the journal Appl. Category. Struct. 27, 199-216 (2019).