Homology and cohomology of cubical sets with coefficients in systems of objects

Abstract

This paper continues the research of the author on the homology of cubical and semi-cubical sets with coefficients in systems of objects. The main result is the theorem that the homology of cubical sets with coefficients in contravariant systems in an Abelian category with exact coproducts is isomorphic to the left satellites of a colimit functor. This made it possible to prove a number of the following new assertions, presented in the paper, about the homology and cohomology of cubical sets with coefficients in systems of objects. These homology are invariant under morphism between cubical sets when passing to the direct image of the system of coefficients. There is a criterion for the invariance of these homologies when passing to the inverse image. These homology generalize the singular cubical homology with local coefficients and the homology of semi-cubical sets with coefficients in contravariant systems. There is a spectral sequence for colimit homologies of cubical sets with coefficients in contravariant systems. The weak equivalence of cubical sets induces an isomorphism of homology with local systems. For a morphism of cubical sets whose inverse fiber morphisms are weak equivalences, there exists a spectral sequence for homology with local systems converging to the homology of the domain of this morphism. The homology of small category with coefficients in a diagram can be calculated as cubical homology. The Baues-Wirsching cohomologies with coefficients in natural systems are isomorphic to cubical cohomologies with coefficients in covariant systems.