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Higher extension closure and $d$-exact categories

Sondre Kvamme

TL;DR

The paper develops a comprehensive framework linking higher exact structures to $d$-cluster tilting subcategories. By proving that weakly idempotent complete $d$-exact categories embed as $d$-cluster tilting subcategories of ambient exact categories, and that algebraic $(d+2)$-angulated categories arise from such tilting inside algebraic triangulated categories with $d$-shifts, it provides a canonical bridge between axiomatizations and ambient realizations. A central universal property shows the ambient exact category is unique up to exact equivalence, enabling seamless translation between $d$-exact and $d$-cluster tilting perspectives. The results yield natural hypotheses and constructions for higher torsion and wide subcategories, and extend the theory to non-weakly idempotent complete settings by clarifying the correct $d$-exact structure via homotopy closures. Overall, the work advances higher AR theory by clarifying when and how $d$-cluster tilting subcategories capture the intrinsic higher-extension phenomena of their ambient categories, with broad implications for representation theory and related areas.

Abstract

We prove that any weakly idempotent complete $d$-exact category is equivalent to a $d$-cluster tilting subcategory of a weakly idempotent complete exact category, and that any weakly idempotent complete algebraic $(d+2)$-angulated category is equivalent to a $d$-cluster tilting subcategory of an algebraic triangulated category closed under $d$-shifts. Furthermore, we show that the ambient exact category of a $d$-cluster tilting subcategory is unique up to exact equivalence, assuming it is weakly idempotent complete. This follows from the inclusion of the $d$-cluster tilting subcategory satisfying a universal property. As a consequence of our theory we also get that any $d$-torsion class is $d$-cluster tilting in an extension-closed subcategory, and we recover the fact that any $d$-wide subcategory is $d$-cluster tilting in a unique wide subcategory. In the last part of the paper we rectify the description of the $d$-exact structure of a $d$-cluster tilting subcategory of a non-weakly idempotent complete exact category.

Higher extension closure and $d$-exact categories

TL;DR

The paper develops a comprehensive framework linking higher exact structures to -cluster tilting subcategories. By proving that weakly idempotent complete -exact categories embed as -cluster tilting subcategories of ambient exact categories, and that algebraic -angulated categories arise from such tilting inside algebraic triangulated categories with -shifts, it provides a canonical bridge between axiomatizations and ambient realizations. A central universal property shows the ambient exact category is unique up to exact equivalence, enabling seamless translation between -exact and -cluster tilting perspectives. The results yield natural hypotheses and constructions for higher torsion and wide subcategories, and extend the theory to non-weakly idempotent complete settings by clarifying the correct -exact structure via homotopy closures. Overall, the work advances higher AR theory by clarifying when and how -cluster tilting subcategories capture the intrinsic higher-extension phenomena of their ambient categories, with broad implications for representation theory and related areas.

Abstract

We prove that any weakly idempotent complete -exact category is equivalent to a -cluster tilting subcategory of a weakly idempotent complete exact category, and that any weakly idempotent complete algebraic -angulated category is equivalent to a -cluster tilting subcategory of an algebraic triangulated category closed under -shifts. Furthermore, we show that the ambient exact category of a -cluster tilting subcategory is unique up to exact equivalence, assuming it is weakly idempotent complete. This follows from the inclusion of the -cluster tilting subcategory satisfying a universal property. As a consequence of our theory we also get that any -torsion class is -cluster tilting in an extension-closed subcategory, and we recover the fact that any -wide subcategory is -cluster tilting in a unique wide subcategory. In the last part of the paper we rectify the description of the -exact structure of a -cluster tilting subcategory of a non-weakly idempotent complete exact category.

Paper Structure

This paper contains 18 sections, 50 theorems, 204 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Exact categories
  4. Cluster tilting subcategories
  5. Higher exact categories
  6. Left exact functors
  7. Maximal rigid subcategories
  8. Properties and examples
  9. Realizing rigid subcategories as maximal rigid
  10. Higher extension closure
  11. Preservation of higher extensions
  12. Higher extension closure implies cluster tilting
  13. Higher torsion classes
  14. Higher wide subcategories
  15. The embedding theorem
  16. ...and 3 more sections

Key Result

Theorem A

If $\mathcal{M}$ is a weakly idempotent complete $d$-exact category, then there exists an equivalence $\mathcal{M}\cong \mathcal{N}$ of $d$-exact categories where $\mathcal{N}$ is a $d$-cluster tilting subcategory of a weakly idempotent complete exact category. Furthermore, the exact category is uni

Theorems & Definitions (115)

  • Theorem A: \ref{['Cor:WeaklyIdemdExactdCT']} and \ref{['Theorem:UniquenessAmbientExact']}
  • Theorem B: \ref{['Theorem:Algd+2AngulatedisdCT']}
  • Definition 1
  • Theorem C
  • Theorem D
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • ...and 105 more