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 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.

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

“…In [35] it is proved that the homology groups H n (X, G) of the cubical set X with coefficients in the contravariant system of Abelian groups G : (✷/X) op → Ab are isomorphic to values on G of the left nth derived of the colimit functor lim − →…”

“…Recall that a functor S : C → D is said to be aspherical if its left fibers S/d are contractible [39]. In this case lim − → This assertion was proved in [35,Corollary 24] for the case A = Ab.…”

“…Proof. In [35] a proof is given that contains gaps in the assertion that the section of pr I n mappings is natural. In order to fix this, we construct a natural transformation r : Zh I k → ZD k , inverse from the left to the embedding ZD k ⊆ Zh I k .…”

“…The work is devoted to the homology of cubical sets with coefficients in contravariant systems of objects in Abelian category with exact coproduct (shortly AB4-category). This paper was preceded by the author's papers on the homology of semicubical sets [45] and cubical sets [35] with coefficients in contravariant systems of abelian groups.…”

Non-Abelian homology and homotopy colimit of classifying spaces for a diagram of groups

“…• lim − → C n : Ab C → Ab (∀n 0) are Abelian homology functors defined in [16, Application II] as left satellites of the colimit functor, and in [21] as homology of small categories with coefficients in object diagrams of the abelian category.…”