Universal homotopy theories

TL;DR

This paper introduces a universal method for expanding small categories into model categories by formally adjoining homotopy colimits, providing a flexible framework for various applications in homotopy theory.

Contribution

It develops a formalism for universal homotopy theories and demonstrates how localization imposes relations within these structures.

Findings

Provides a universal construction for model categories from small categories

Shows how localization imposes relations in homotopy theories

Applies the formalism to homotopy colimits, framings, and schemes

Abstract

Given a small category C, we show that there is a universal way of expanding C into a model category, essentially by formally adjoining homotopy colimits. The technique of localization becomes a method for imposing `relations' into these universal gadgets. The paper develops this formalism and discusses various applications, for instance to the study of homotopy colimits, the Dwyer-Kan theory of framings, and to the homotopy theory of schemes.