Cofibrations in Homotopy Theory

TL;DR

This paper introduces ABC cofibration categories, constructs homotopy colimits and limits within them, and develops associated derivators, providing a comprehensive framework for homotopy theory in these categories.

Contribution

It defines ABC cofibration categories, constructs homotopy colimits and limits, and links them to derivators, extending homotopy theory tools beyond Quillen model categories.

Findings

Homotopy colimits constructed for ABC cofibration categories.

Homotopy limits developed for ABC fibration categories.

Establishment of left and right Heller derivators for these categories.

Abstract

We define Anderson-Brown-Cisinski (ABC) cofibration categories, and construct homotopy colimits of diagrams of objects in ABC cofibration categories. Homotopy colimits for Quillen model categories are obtained as a particular case. We attach to each ABC cofibration category a left Heller derivator. A dual theory is developed for homotopy limits in ABC fibration categories and for right Heller derivators. These constructions provide a natural framework for 'doing homotopy theory' in ABC (co)fibration categories.