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Monoidal adjunctions and abelian envelopes

Johannes Flake, Robert Laugwitz, Sebastian Posur

TL;DR

This work develops a systematic approach to constructing monoidal abelian envelopes for pseudo-tensor categories via monoidal adjunctions, with a focus on diagrammatic and combinatorial descriptions. By isolating pseudo-diagrammatic categories and proving practical exactness criteria, the authors show how splitting objects can be transported along adjunctions to yield envelopes with the quotient property, and often with enough projectives. The framework yields new abelian envelopes for interpolation categories associated with the hyperoctahedral and modified symmetric groups, demonstrated through combinatorial proofs that avoid heavy algebraic machinery. Overall, the paper suggests a uniform, diagrammatic route to abelian envelopes across broad families of interpolation categories.

Abstract

We show how monoidal adjunctions can be used to prove the existence of monoidal abelian envelopes of pseudo-tensor categories, in particular, those admitting a combinatorial description with certain properties. We derive concrete general criteria that we demonstrate by giving relatively simple combinatorial proofs of the existence of new abelian envelopes for interpolation categories of the hyperoctahedral and of the modified symmetric groups.

Monoidal adjunctions and abelian envelopes

TL;DR

This work develops a systematic approach to constructing monoidal abelian envelopes for pseudo-tensor categories via monoidal adjunctions, with a focus on diagrammatic and combinatorial descriptions. By isolating pseudo-diagrammatic categories and proving practical exactness criteria, the authors show how splitting objects can be transported along adjunctions to yield envelopes with the quotient property, and often with enough projectives. The framework yields new abelian envelopes for interpolation categories associated with the hyperoctahedral and modified symmetric groups, demonstrated through combinatorial proofs that avoid heavy algebraic machinery. Overall, the paper suggests a uniform, diagrammatic route to abelian envelopes across broad families of interpolation categories.

Abstract

We show how monoidal adjunctions can be used to prove the existence of monoidal abelian envelopes of pseudo-tensor categories, in particular, those admitting a combinatorial description with certain properties. We derive concrete general criteria that we demonstrate by giving relatively simple combinatorial proofs of the existence of new abelian envelopes for interpolation categories of the hyperoctahedral and of the modified symmetric groups.
Paper Structure (24 sections, 53 theorems, 88 equations)

This paper contains 24 sections, 53 theorems, 88 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Pseudo-tensor categories
  4. Categories defined using cobordisms
  5. Categories defined using partition diagrams
  6. Pseudo-diagrammatic categories
  7. Exactness conditions
  8. Abelian envelopes for pseudo-diagrammatic categories
  9. Families of pseudo-diagrammatic categories
  10. Categories based on cobordisms or partitions
  11. Knop's tensor envelopes
  12. Harman--Snowden's categories from pro-oligomorphic groups
  13. Monoidal adjunctions
  14. Existence of adjoints
  15. Adjunctions and ind-completions
  16. ...and 9 more sections

Key Result

Theorem A

Any pseudo-diagrammatic linear monoidal category $\mathcal{C}$ satisfies the necessary exactness condition of CEOP to have an abelian envelope.

Theorems & Definitions (116)

  • Theorem A: [S]prop:U-equals-Uex
  • Theorem B: [S]cor:ab-env-from-functor-semisimple, [S]cor:criterion-enough-proj
  • Theorem C: [S]thm:ab-env-projectives
  • Theorem D: [S]thm:partition-cats
  • Theorem E: [S]cor:main-H, [S]cor:main-S-prime
  • Definition 2.1: CEOP*Definition 2.1.1
  • Definition 2.2
  • Definition 3.1
  • Definition 3.2
  • Remark 3.3
  • ...and 106 more