Enhancement for categories and homotopical algebra

TL;DR

This paper develops a model-independent foundation for abstract homotopy theory using derivators, aiming to unify and extend applications like derived categories without relying on traditional model structures.

Contribution

It introduces a new framework for homotopy theory based on derivators, independent of model categories or simplicial sets, facilitating broader applications.

Findings

Provides a model-independent approach to homotopy theory

Enables enhancements for derived categories of coherent sheaves

Bridges classical category theory with homotopical methods

Abstract

We develop foundations for abstract homotopy theory based on Grothendieck's idea of a "derivator". The theory is model-independent, and does not depend on model categories, nor on simplicial sets. It is designed to accomodate all the usual potential applications, such as e.g. enhancements for derived categories of coherent sheaves, in a way that is as close as possible to usual category theory.