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Definable functors between triangulated categories

Isaac Bird, Jordan Williamson

TL;DR

This work introduces definable functors between compactly generated triangulated categories as purity-preserving morphisms, and establishes the restricted Yoneda embedding as the universal coherent functor into finitely accessible categories. It develops coherent functors, proves a universal lifting property to $ ext{Flat}( op^c)$, and extends lifts to module categories, thereby categorifying the transfer of the pure structure. The framework yields robust criteria for definability, preserves endofinite objects, and clarifies interactions with definable subcategories and the Ziegler spectrum. These results unify and extend known constructions (Beligiannis, Krause) in a triangulated setting and provide tools for transporting purity across settings, with broad applicability in homotopy theory, representation theory, and tensor-triangular geometry.

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.

Definable functors between triangulated categories

TL;DR

This work introduces definable functors between compactly generated triangulated categories as purity-preserving morphisms, and establishes the restricted Yoneda embedding as the universal coherent functor into finitely accessible categories. It develops coherent functors, proves a universal lifting property to , and extends lifts to module categories, thereby categorifying the transfer of the pure structure. The framework yields robust criteria for definability, preserves endofinite objects, and clarifies interactions with definable subcategories and the Ziegler spectrum. These results unify and extend known constructions (Beligiannis, Krause) in a triangulated setting and provide tools for transporting purity across settings, with broad applicability in homotopy theory, representation theory, and tensor-triangular geometry.

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.
Paper Structure (16 sections, 38 theorems, 59 equations)

This paper contains 16 sections, 38 theorems, 59 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Purity in triangulated categories
  4. Triangles and enhancements
  5. The universal property of coherent functors
  6. The proof of \ref{['uniprop']}
  7. Extension to all modules
  8. Transferring definability and endofinite objects
  9. Examples of coherent functors
  10. Definable functors
  11. Properties of definable functors
  12. Uniqueness of the definable lift
  13. Criteria for definability
  14. Triangulated functors
  15. Examples of definable functors
  16. ...and 1 more sections

Key Result

Theorem A

(uniprop) Let $H\colon\mathsf{T}\to\mathscr{A}$ be a coherent functor. Then there is a unique definable functor $\widehat{H}\colon\mathsf{Flat}(\mathsf{T}^{\mathrm{c}})\to\mathscr{A}$ between finitely accessible categories such that \begin{tikzcd}[row sep=1cm, column sep=1.5cm] \T \arrow[d,swap, "

Theorems & Definitions (79)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Lemma 2.12
  • proof
  • Definition 3.1
  • Theorem 3.2
  • Remark 3.3
  • Proposition 3.4
  • ...and 69 more