Definable functors between triangulated categories

TL;DR

This paper develops a comprehensive theory of definable functors between compactly generated triangulated categories, characterizing their properties and applications in algebraic and topological contexts.

Contribution

It introduces the concept of coherent functors, characterizes definable functors via preservation of colimits and products, and explores their role in the Ziegler spectrum.

Findings

Coherent functors are characterized as purity-preserving functors.

Definable functors are those extending uniquely along the restricted Yoneda embedding.

Applications to the functoriality of the Ziegler spectrum in pure homological algebra.

Abstract

We systematically develop the theory of definable functors between compactly generated triangulated categories. Such functors preserve pure triangles, pure injective objects, and definable subcategories, and as such appear in a wide range of algebraic and topological settings. Firstly we investigate and characterise purity preserving functors from a triangulated category into a finitely accessible category with products, which we term coherent functors. This yields a new property for the restricted Yoneda embedding as the universal coherent functor. We build upon the utility of coherent functors to provide several equivalent conditions for an additive, not necessarily triangulated, functor between triangulated categories to be definable: a functor is definable if and only if it preserves filtered homology colimits and products, if and only if it uniquely extends along the restricted Yoneda embedding to a definable functor between the corresponding module categories. We apply these results to the functoriality of the Ziegler spectrum, an object of study in pure homological algebra and representation theory.