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The blob complex

Scott Morrison, Kevin Walker

TL;DR

The paper introduces the blob complex B_*(M;C), a derived, local-resolution analogue unifying TQFT skein modules, Hochschild homology, and mapping-space models for n-categories with strong duality. It develops two parallel formalisms: (i) a system-of-fields approach yielding blob complexes from local data, and (ii) disk-like and A_infty n-categories enabling homotopy-coherent gluings, products, and modules, culminating in a higher Deligne conjecture. Core contributions include a gluing formula for blob complexes, a product structure for blob complexes on product manifolds, and a reconstruction of mapping spaces via blob homology, plus a higher operadic action on blob cochains. The framework provides a robust, local-to-global method to study derived TQFT invariants, with anticipated applications to contact topology, Khovanov-type theories, and higher categorical structures in field theory. Overall, the work offers a comprehensive, scalable toolkit linking n-categorical topology, Hochschild-type invariants, and higher Deligne-type symmetries.

Abstract

Given an n-manifold M and an n-category C, we define a chain complex (the "blob complex") B_*(M;C). The blob complex can be thought of as a derived category analogue of the Hilbert space of a TQFT, and as a generalization of Hochschild homology to n-categories and n-manifolds. It enjoys a number of nice formal properties, including a higher dimensional generalization of Deligne's conjecture about the action of the little disks operad on Hochschild cochains. Along the way, we give a definition of a weak n-category with strong duality which is particularly well suited for work with TQFTs.

The blob complex

TL;DR

The paper introduces the blob complex B_*(M;C), a derived, local-resolution analogue unifying TQFT skein modules, Hochschild homology, and mapping-space models for n-categories with strong duality. It develops two parallel formalisms: (i) a system-of-fields approach yielding blob complexes from local data, and (ii) disk-like and A_infty n-categories enabling homotopy-coherent gluings, products, and modules, culminating in a higher Deligne conjecture. Core contributions include a gluing formula for blob complexes, a product structure for blob complexes on product manifolds, and a reconstruction of mapping spaces via blob homology, plus a higher operadic action on blob cochains. The framework provides a robust, local-to-global method to study derived TQFT invariants, with anticipated applications to contact topology, Khovanov-type theories, and higher categorical structures in field theory. Overall, the work offers a comprehensive, scalable toolkit linking n-categorical topology, Hochschild-type invariants, and higher Deligne-type symmetries.

Abstract

Given an n-manifold M and an n-category C, we define a chain complex (the "blob complex") B_*(M;C). The blob complex can be thought of as a derived category analogue of the Hilbert space of a TQFT, and as a generalization of Hochschild homology to n-categories and n-manifolds. It enjoys a number of nice formal properties, including a higher dimensional generalization of Deligne's conjecture about the action of the little disks operad on Hochschild cochains. Along the way, we give a definition of a weak n-category with strong duality which is particularly well suited for work with TQFTs.

Paper Structure

This paper contains 42 sections, 38 theorems, 180 equations, 49 figures.

Table of Contents

  1. Introduction
  2. Structure of the paper
  3. Motivation
  4. Formal properties
  5. Specializations
  6. Structure of the blob complex
  7. Applications
  8. n-category terminology
  9. Thanks and acknowledgements
  10. TQFTs via fields
  11. Systems of fields
  12. Systems of fields from n-categories
  13. Local relations
  14. Constructing a TQFT
  15. The blob complex
  16. ...and 27 more sections

Key Result

Proposition 3.1

The skein module $A(X)$ is naturally isomorphic to ${\mathcal{B}}_0(X)/\partial({\mathcal{B}}_1(X))) = H_0({\mathcal{B}}_*(X))$.

Figures (49)

  • Figure 1: The main gadgets and constructions of the paper.
  • Figure 3: A 1-blob diagram.
  • Figure 4: A disjoint 2-blob diagram.
  • Figure 5: A nested 2-blob diagram.
  • Figure 6: The subsets $A$, $B$, $C$ and $D$ from Example \ref{['sin1x-example']}. The pair $\{A, D\}$ is a valid configuration of blobs, even though the complement is not a manifold.
  • ...and 44 more figures

Theorems & Definitions (98)

  • Example 2.1
  • Example 2.2
  • Example 2.3: contd.
  • Example 2.4: contd.
  • Definition 2.3
  • Definition 2.4
  • Proposition 3.1
  • Example 3.2
  • Definition 3.3
  • Definition 3.4
  • ...and 88 more