Homology groups of semicubical sets

Abstract

We study the homology groups of semicubical sets with coefficients in the homological systems of abelian groups. The main theorem states that the groups under consideration are isomorphic to the homology groups of the category of singular cubes. This yields an isomorphism criterion for the homology groups of semicubical sets, the spectral sequence of a locally directed covering, and the spectral sequence of a morphism of semicubical sets.

Homology Groups of Cubical Sets

“…In this section it is shown that in many cases Theorem 2 allows us to solve this problem. The results analogous to those obtained in [17] for precubical sets are proved. 2.…”

“…It is easy to see that the sequence of γ k , k 0, is an isomorphism of the complexes C * → K * . According to [17,Lemma 4.1], the homology groups of C * are equal 0 in dimensions k > 0 and H 0 (C * ) = Z. Consequently, the same is true for K * .…”

“…Hence, the colimit functor has left derived functors lim − → D op n : Ab D op → Ab. The description of the chain complex whose homology groups are naturally isomorphic to lim − → D op n F is given in [15,Application 2] and in [17,Definition 3.1]. For the definition of the derived functors of the colimit, we can take the following assertion:…”

“…Proof: We want to prove that the complex K * is isomorphic to the complex C * constructed in [17,Lemma 4.1] and consisting of the Abelian groups and homomorphisms…”

“…It easy to see that Lan Q op X F is isomorphic to the cubical Abelian group (C n (X, F ), d n,τ i , s n i ). See the proof in [17,Proposition 3.7] for the general case of a small category D (instead ✷) and for a functor F : (D/X) op → Ab. It follows from Proposition 4 that the normalized complexes C N * (X, F ) and C N * (Lan Q op X F ) are isomorphic.…”