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

TL;DR

This paper advances the understanding of cubical set homology with coefficients, establishing new invariance properties, spectral sequences, and connections to classical homology theories, broadening the theoretical framework in algebraic topology.

Contribution

The paper proves that cubical set homology with coefficients in contravariant systems is isomorphic to left satellites of a colimit functor, introducing new invariance and spectral sequence results.

Findings

Homology is invariant under certain morphisms of cubical sets.

Established spectral sequences for colimit homologies.

Connected cubical homology with classical theories like singular homology.

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.