# A realization functor for abelian model categories

## TL;DR

This paper constructs a realization functor linking the derived category of a Grothendieck abelian category with a hereditary abelian model structure to its homotopy category, facilitating better understanding of their relationship.

## Contribution

It introduces a realization functor that connects derived and homotopy categories in the context of abelian model categories, extending existing frameworks.

## Key findings

- Established a functor from derived to homotopy category
- Applied to Grothendieck abelian categories with hereditary model structures
- Provides a new tool for studying abelian model categories

## Abstract

We study liftings of abelian model structures to categories of chain complexes and construct a realization functor from the derived category of a Grothendieck abelian category equipped with a cofibrantly generated, hereditary abelian model structure to the homotopy category of that model structure.