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Poincaré Duality Pairs of $\infty$-Categories

Andrea Bianchi, Kaif Hilman, Dominik Kirstein, Christian Kremer

TL;DR

This work develops a comprehensive framework for Poincaré duality in the realm of ∞-categories by introducing Poincaré ∞-category pairs. It builds a Morita-theoretic foundation for classifying linear functors between parametrised presheaf categories, including a coend formula and a sharp distinction between classifying and dualising data. Central results include a general fibration theorem that extends Klein–Qin–Su to arbitrary ads with coefficients in any presentably symmetric monoidal ∞-category, and a suite of cutting-and-pasting techniques enabling local-to-global verification of Poincaré duality in diagrams indexed by combinatorial manifolds. The paper also develops fibred ambidexterity and a fibrewise dualising-object factorisation, establishing a robust theory for Poincaré cobordisms, isovariant spaces, and string topology, with broad geometric-topology applications. Altogether, the authors provide a powerful, uniform categorical language that encapsulates Wall’s ads and higher Poincaré cobordism constructions, while enabling new interactions across base and fibre data in a highly structured setting.

Abstract

We introduce a notion of Poincaré duality for pairs of $\infty$-categories, extending Poincaré-Lefschetz duality for pairs of spaces. This categorical extension yields an efficient book-keeping device that affords, among other things, a uniform treatment of Wall's Poincaré ads of spaces, iterated Poincaré cobordisms, and in general, diagrams of spaces parametrised by the face poset of a combinatorial manifold. In each of these cases, the theory reduces them to studying a single pair of $\infty$-categories and the properties of a single functor, the relative cohomology functor. Using this formalism, we prove a very general fibration theorem which, in particular, specialises to a generalisation of Klein-Qin-Su's fibration theorem for Poincaré triads to all ads. This theory also lays the foundation for future work by the authors on Poincaré cobordism categories, isovariant Poincaré spaces and string topology.

Poincaré Duality Pairs of $\infty$-Categories

TL;DR

This work develops a comprehensive framework for Poincaré duality in the realm of ∞-categories by introducing Poincaré ∞-category pairs. It builds a Morita-theoretic foundation for classifying linear functors between parametrised presheaf categories, including a coend formula and a sharp distinction between classifying and dualising data. Central results include a general fibration theorem that extends Klein–Qin–Su to arbitrary ads with coefficients in any presentably symmetric monoidal ∞-category, and a suite of cutting-and-pasting techniques enabling local-to-global verification of Poincaré duality in diagrams indexed by combinatorial manifolds. The paper also develops fibred ambidexterity and a fibrewise dualising-object factorisation, establishing a robust theory for Poincaré cobordisms, isovariant spaces, and string topology, with broad geometric-topology applications. Altogether, the authors provide a powerful, uniform categorical language that encapsulates Wall’s ads and higher Poincaré cobordism constructions, while enabling new interactions across base and fibre data in a highly structured setting.

Abstract

We introduce a notion of Poincaré duality for pairs of -categories, extending Poincaré-Lefschetz duality for pairs of spaces. This categorical extension yields an efficient book-keeping device that affords, among other things, a uniform treatment of Wall's Poincaré ads of spaces, iterated Poincaré cobordisms, and in general, diagrams of spaces parametrised by the face poset of a combinatorial manifold. In each of these cases, the theory reduces them to studying a single pair of -categories and the properties of a single functor, the relative cohomology functor. Using this formalism, we prove a very general fibration theorem which, in particular, specialises to a generalisation of Klein-Qin-Su's fibration theorem for Poincaré triads to all ads. This theory also lays the foundation for future work by the authors on Poincaré cobordism categories, isovariant Poincaré spaces and string topology.
Paper Structure (38 sections, 79 theorems, 84 equations)

This paper contains 38 sections, 79 theorems, 84 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Summary of the results
  4. Historical remarks
  5. Overview and acknowledgements
  6. Preliminaries
  7. Recollections on parametrised category theory
  8. Summary of notations
  9. Foundations
  10. Morita theory
  11. Classification of linear functors
  12. A coend formula
  13. Classifying vs dualising systems
  14. Poincaré duality category pairs
  15. Left closed subcategories and category pairs
  16. ...and 23 more sections

Key Result

Theorem A

Let $\mathcal{X}$ and $\mathcal{Y}$ be small $\infty$-categories, let ${\mathcal{C}}\in \mathrm{CAlg}(\mathrm{Pr}^L_{\mathrm{st}})$, and let $F\in\mathop{\mathrm{\mathrm{Fun}}}\nolimits^L_{{\mathcal{C}}}(\mathscr{P}(\mathcal{X};{\mathcal{C}}),{\mathcal{C}})$ and $G\in\mathop{\mathrm{\mathrm{Fun}}}\n

Theorems & Definitions (220)

  • Theorem A: Full versions in \ref{['prop:formula_with_twisted_arrow', 'prop:categorical_poincare_lefschetz_duality', 'lem:naturality_classification_C_linear_functors', 'cor:formula_in_terms_of_dualising_system_for_dualish']}
  • Theorem B: Full version in \ref{['prop:categorical_poincare_lefschetz_duality']}
  • Theorem C: cf. \ref{['cor:seven_equivalent_properties', 'prop:wall_vs_us_ads']}
  • Theorem D: Full and precise version in \ref{['prop:combinatorial_manifold_Poincare']}
  • Theorem E: Precise version in \ref{['cor:kleinQinSu_integration']}
  • Example 2.1.1: Presheaf topoi
  • Remark 2.1.2: Checking equivalences on objects
  • Example 3.0.1
  • Proposition 3.1.1
  • Lemma 3.1.2
  • ...and 210 more