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Condensations in higher categories

Davide Gaiotto, Theo Johnson-Freyd

TL;DR

This work develops a higher-categorical generalization of the Karoubi envelope by replacing idempotents with condensations in weak $n$-categories, constructing $ ext{Kar}(\mathcal{C})$ and the suspension $\\Sigma(\mathcal{C})$, which universalize the inclusion of condensates and the formation of images of condensations. It links this higher Karoubi completion to gapped topological phases and fully extended TQFTs via the Cobordism Hypothesis, showing that fully dualizable objects arise precisely as condensation-based constructions and that condensates provide a robust, physically realizable notion of image completion compatible with commuting-projector Hamiltonians. The paper develops the formal theory of $n$-condensations, condensation monads, and bimodules, proves uniqueness and universality results (including the key Theorem on Karoubi completion and the state-operator correspondence), and establishes absolute limits, direct sums, and additive enhancements that connect to familiar algebraic settings (e.g., $ ext{Vec}_{\mathbb{C}}$) and the standard models of $(d+1)$-dimensional phases. Overall, it provides a principled bridge between condensed matter constructions and fully extended TQFTs, with explicit categorical machinery to classify and realize condensed phases as higher-categorical dualizable objects.

Abstract

We present a higher-categorical generalization of the "Karoubi envelope" construction from ordinary category theory, and prove that, like the ordinary Karoubi envelope, our higher Karoubi envelope is the closure for absolute limits. Our construction replaces the idempotents in the ordinary version with a notion that we call "condensations." The name is justified by the direct physical interpretation of the notion of condensation: it encodes a general class of constructions which produce a new topological phase of matter by turning on a commuting projector Hamiltonian on a lattice of defects within a different topological phase, which may be the trivial phase. We also identify our higher Karoubi envelopes with categories of fully-dualizable objects. Together with the Cobordism Hypothesis, we argue that this realizes an equivalence between a very broad class of gapped topological phases of matter and fully extended topological field theories, in any number of dimensions.

Condensations in higher categories

TL;DR

This work develops a higher-categorical generalization of the Karoubi envelope by replacing idempotents with condensations in weak -categories, constructing and the suspension , which universalize the inclusion of condensates and the formation of images of condensations. It links this higher Karoubi completion to gapped topological phases and fully extended TQFTs via the Cobordism Hypothesis, showing that fully dualizable objects arise precisely as condensation-based constructions and that condensates provide a robust, physically realizable notion of image completion compatible with commuting-projector Hamiltonians. The paper develops the formal theory of -condensations, condensation monads, and bimodules, proves uniqueness and universality results (including the key Theorem on Karoubi completion and the state-operator correspondence), and establishes absolute limits, direct sums, and additive enhancements that connect to familiar algebraic settings (e.g., ) and the standard models of -dimensional phases. Overall, it provides a principled bridge between condensed matter constructions and fully extended TQFTs, with explicit categorical machinery to classify and realize condensed phases as higher-categorical dualizable objects.

Abstract

We present a higher-categorical generalization of the "Karoubi envelope" construction from ordinary category theory, and prove that, like the ordinary Karoubi envelope, our higher Karoubi envelope is the closure for absolute limits. Our construction replaces the idempotents in the ordinary version with a notion that we call "condensations." The name is justified by the direct physical interpretation of the notion of condensation: it encodes a general class of constructions which produce a new topological phase of matter by turning on a commuting projector Hamiltonian on a lattice of defects within a different topological phase, which may be the trivial phase. We also identify our higher Karoubi envelopes with categories of fully-dualizable objects. Together with the Cobordism Hypothesis, we argue that this realizes an equivalence between a very broad class of gapped topological phases of matter and fully extended topological field theories, in any number of dimensions.

Paper Structure

This paper contains 17 sections, 18 theorems, 22 equations.

Table of Contents

  1. Introduction
  2. Gapped phases of matter vs extended TQFTs
  3. Layering and condensation
  4. Higher Karoubi envelopes
  5. Condensations and higher Karoubi envelopes
  6. $n$-condensations
  7. Condensation monads
  8. Constructing condensates
  9. Boundary conditions and lattice Hamiltonians
  10. Unitality
  11. Unital condensations
  12. Physical interpretation: the state-operator map
  13. Condensation bimodules vs unital algebra bimodules
  14. Absolute limits and full dualizability
  15. Full dualizability
  16. ...and 2 more sections

Key Result

Theorem 2.3.2

Suppose that $\mathcal{C}$ is an $n$-category such that all hom-$(n-1)$-categories $\hom_\mathcal{C}(X,Y)$ in $\mathcal{C}$ have all condensates, and let $(X,e,\dots) : \clubsuit_n \to \mathcal{C}$ be a condensation monad in $\mathcal{C}$. Then the $n$-category of condensates $(X,e,\dots)$ is either

Theorems & Definitions (82)

  • Remark 1.0.1
  • Remark 1.1.1
  • Remark 1.1.2
  • Conjecture 1.2.1
  • Remark 1.2.2
  • Remark 1.2.3
  • Definition 1.2.4
  • Example 1.2.5
  • Remark 1.3.1
  • Example 1.3.2
  • ...and 72 more