Comparison of Cubical and Simplicial Derived Functors

TL;DR

This paper proves that simplicial and cubical derived functors are naturally isomorphic, generalizing the known equivalence of their singular homologies in topology.

Contribution

It establishes a natural isomorphism between simplicial and cubical derived functors, extending the classical equivalence in singular homology.

Findings

Simplicial and cubical derived functors are naturally isomorphic.

Generalizes the equivalence of simplicial and cubical singular homologies.

Provides a unified perspective on derived functors in different cubical and simplicial frameworks.

Abstract

In this note we prove that the simplicial derived functors introduced by Tierney and Vogel [TV69] are naturally isomorphic to the cubical derived functors introduced by the author in [P09]. We also explain how this result generalizes the well-known fact that the simplicial and cubical singular homologies of a topological space are naturally isomorphic.