# Derived category of weak chain 𝒰-complexes

## TL;DR

This paper introduces and characterizes the derived category of weak chain U-complexes and establishes a natural isomorphism with homology functors.

## Contribution

The paper introduces a new derived category for weak chain U-complexes and establishes a novel natural isomorphism via an adjoint pair.

## Key findings

- A derived category of weak chain U-complexes is defined and characterized.
- An adjoint pair is constructed between chain complexes and weak chain U-complexes.
- The homology functor is shown to be a natural isomorphism.

## Abstract

In this paper, we define the derived category of weak chain U-complexes, and we give a characterization of any weak chain U-complex as an object in the right bounded homotopy category of weak chain U-complexes of projective modules. Then, we show that the homology functor between the category of chain complexes and weak chain U-complexes is a natural isomorphism by constructing an adjoint pair between these two categories.