Cubical Resolutions and Derived Functors

TL;DR

This paper develops a cubical framework for derived functors using pseudocubical objects with pseudoconnections, generalizing classical simplicial derived functor theory to an arbitrary category.

Contribution

It introduces pseudocubical objects with pseudoconnections and constructs a cubical analog of the Tierney-Vogel derived functor theory, extending classical results to broader categorical contexts.

Findings

Pseudocubical resolutions possess pseudodegeneracies and pseudoconnections.

The cubical derived functors coincide with classical derived functors in abelian categories.

The theory generalizes the Tierney-Vogel approach to arbitrary categories.

Abstract

We introduce pseudocubical objects with pseudoconnections in an arbitrary category, obtained from the Brown-Higgins structure of a cubical object with connections by suitably relaxing their identities, and construct a cubical analog of the Tierney-Vogel theory of simplicial derived functors. The crucial point in the construction is that projective precubical resolutions which are naturally used to define our cubical derived functors possess pseudodegeneracies and pseudoconnections. The same fact is essentially used for proving that in the case of an additive functor between abelian categories, our theory coincides with the classical relative theory of derived functors by Eilenberg-Moore.