On perfectly generating projective classes in triangulated categories

TL;DR

This paper investigates perfect projective classes in triangulated categories, demonstrating that objects can be constructed as homotopy colimits of cellular towers, and provides a new proof of Brown's representability theorem.

Contribution

It introduces the concept of perfect projective classes and shows how they enable a new proof of Brown's theorem using cellular towers and homotopy colimits.

Findings

Objects are isomorphic to homotopy colimits of cellular towers

Perfect projective classes are closed under coproducts of maps

New proof of Brown's representability theorem

Abstract

We say that a projective class in a triangulated category with coproducts is perfect if the corresponding ideal is closed under coproducts of maps. We study perfect projective classes and the associated phantom and cellular towers. Given a perfect generating projective class, we show that every object is isomorphic to the homotopy colimit of a cellular tower associated to that object. Using this result and the Neeman's Freyd--style representability theorem we give a new proof of Brown Representability Theorem.