Homology Groups of Cubical Sets

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 the 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. The use of the main result leads to an isomorphism between homology groups of small category with coefficients in the diagram of abelian groups and normalized homology groups of its cubical nerve with coefficients in the corresponding contravariant system.