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Duals and adjoints in higher Morita categories

Owen Gwilliam, Claudia Scheimbauer

TL;DR

The paper develops a higher Morita theory framework by modeling $E_n$-algebras in a symmetric monoidal $( ext{∞},N)$-category via constructible factorization algebras, and proves that the resulting $( ext{∞},n+N)$-category $ ext{Alg}_n( ext{S})$ is fully $n$-dualizable. This is established first in dimension two by explicit geometric constructions of objects and adjoints, then extended to arbitrary dimensions using affine-flag stratifications and fold/collapse techniques. A central insight is that the factorization-algebra model yields concrete duals and adjoints, aligning with Lurie’s conjectures and clarifying connections to fully extended TFTs; the work also analyzes $(n+1)$-dualizability and shows pointedness obstructs it, motivating unpointed variants. Together, these results provide a robust, geometrically flavored toolkit for understanding duals and adjoints in higher Morita categories and their TFT interpretations, with potential applications to twisted and defect field theories. The paper thus advances the Cobordism-Hypothesis program in a higher-dimensional, constructive setting and lays groundwork for future exploration of unpointed higher Morita categories and their field-theoretic realizations.

Abstract

We study duals for objects and adjoints for $k$-morphisms in $\operatorname{Alg}_n(\mathcal{S})$, an $(\infty,n+N)$-category that models a higher Morita category for $E_n$ algebra objects in a symmetric monoidal $(\infty,N)$-category $\mathcal{S}$. Our model of $\operatorname{Alg}(\mathcal{S})$ uses the geometrically convenient framework of factorization algebras. The main result is that $\operatorname{Alg}_n(\mathcal{S})$ is fully $n$-dualizable, verifying a conjecture of Lurie. Moreover, we unpack the consequences for a natural class of fully extended topological field theories and explore $(n+1)$-dualizability.

Duals and adjoints in higher Morita categories

TL;DR

The paper develops a higher Morita theory framework by modeling -algebras in a symmetric monoidal -category via constructible factorization algebras, and proves that the resulting -category is fully -dualizable. This is established first in dimension two by explicit geometric constructions of objects and adjoints, then extended to arbitrary dimensions using affine-flag stratifications and fold/collapse techniques. A central insight is that the factorization-algebra model yields concrete duals and adjoints, aligning with Lurie’s conjectures and clarifying connections to fully extended TFTs; the work also analyzes -dualizability and shows pointedness obstructs it, motivating unpointed variants. Together, these results provide a robust, geometrically flavored toolkit for understanding duals and adjoints in higher Morita categories and their TFT interpretations, with potential applications to twisted and defect field theories. The paper thus advances the Cobordism-Hypothesis program in a higher-dimensional, constructive setting and lays groundwork for future exploration of unpointed higher Morita categories and their field-theoretic realizations.

Abstract

We study duals for objects and adjoints for -morphisms in , an -category that models a higher Morita category for algebra objects in a symmetric monoidal -category . Our model of uses the geometrically convenient framework of factorization algebras. The main result is that is fully -dualizable, verifying a conjecture of Lurie. Moreover, we unpack the consequences for a natural class of fully extended topological field theories and explore -dualizability.

Paper Structure

This paper contains 38 sections, 18 theorems, 73 equations.

Table of Contents

  1. Introduction
  2. Generalizing the Morita 2-category
  3. Dualizability and our main result
  4. Relationship with topological field theory
  5. Pointed comment
  6. Connections with other results on duals and adjoints
  7. Future directions
  8. Guide to reader
  9. Acknowledgements
  10. Recollection: dualizability
  11. Recollection: the factorization higher Morita category Algn(S)
  12. The factorization model for the Morita 2-category Alg1(S)
  13. Constructible one-dimensional factorization algebras
  14. Collapse-and-rescale maps
  15. The factorization model for the Morita 2-category
  16. ...and 23 more sections

Key Result

Theorem 1

The symmetric monoidal $(\infty,n)$-category $\tau_{(\infty,n)} \mathop{\mathrm{Alg}}\nolimits_n(\mathcal{S})$ underlying the factorization higher Morita category $\mathop{\mathrm{Alg}}\nolimits_n(\mathcal{S})$ is fully $n$-dualizable, namely

Theorems & Definitions (61)

  • Theorem
  • Theorem
  • Remark 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Remark 2.6
  • Lemma 2.7
  • ...and 51 more