The blob complex

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.

💡 Research Summary

The paper introduces the blob complex, a chain complex B_*(M; C) associated to an n‑dimensional manifold M and an n‑category C. The construction is motivated by two parallel lines of thought. First, in topological quantum field theory (TQFT) the Hilbert space assigned to a closed (n‑1)‑manifold is traditionally a vector space; the authors propose a derived‑categorical analogue that captures higher homotopical information. Second, Hochschild homology provides a powerful invariant for associative algebras; the authors aim to generalize this invariant to the setting of higher categories and arbitrary n‑manifolds.

1. A model of weak n‑categories with strong duality.
The authors begin by defining a version of weak n‑categories that is particularly suited for TQFT. Each k‑morphism (0 ≤ k ≤ n) is required to possess explicit left and right duals, and the duality data must satisfy strict coherence conditions under composition. This “strong duality” ensures that every object is fully dualizable in the sense of the Cobordism Hypothesis, making the category ready to receive cobordism data. Compared with existing models (e.g., Segal n‑categories, Barwick’s n‑fold complete Segal spaces), the present definition is more concrete: duals are given by explicit adjunction‑like 1‑morphisms together with higher evaluation and coevaluation cells. The authors prove that their model is equivalent (up to appropriate homotopy) to the usual notion of fully dualizable objects in a symmetric monoidal n‑category.

2. Construction of the blob complex.
Given a CW‑decomposition (or any suitable cell structure) of M, one decorates each cell with data from C: objects on 0‑cells, 1‑morphisms on 1‑cells, and so on up to n‑morphisms on n‑cells. A blob is a small embedded n‑ball inside M together with a choice of such decorations on its interior cells. A configuration of k mutually nested blobs determines a generator of degree k in B_k(M; C). The differential is defined by “forgetting” one blob at a time, together with the appropriate composition of the associated morphisms; this mirrors the usual cellular boundary operator. In this way the complex records all possible ways of inserting and removing higher‑dimensional “defects” labeled by C.

3. Formal properties.
The blob complex satisfies several key axioms:

• Hochschild reduction. When M is the n‑disk Dⁿ, B_*(Dⁿ; C) is canonically quasi‑isomorphic to the Hochschild complex of C (viewed as an En‑algebra). Thus the construction genuinely extends classical Hochschild homology.

• Derived Hilbert space. For a closed n‑manifold M, B_*(M; C) behaves like the derived version of the Hilbert space assigned by a fully extended TQFT based on C. In particular, its homology groups can be interpreted as “states” up to higher homotopies.

• Operadic action. The little n‑disks operad Eₙ acts on B_*(M; C) by inserting configurations of disjoint disks (blobs) and using the monoidal structure of C. This yields a higher‑dimensional analogue of Deligne’s conjecture: the Hochschild cochains of C acquire an En‑algebra structure, and the blob complex provides a concrete chain‑level model for that action.

• Gluing formula. If M = M₁ ∪_∂ M₂ is obtained by gluing along a common (n‑1)‑dimensional boundary ∂, then there is a homotopy‑equivalence B_(M; C) ≃ B_(M₁; C) ⊗_{B_(∂; C)} B_(M₂; C). This mirrors the gluing axiom of extended TQFTs and shows that the blob complex is compositional.

4. Computations and examples.
The authors work out several illustrative cases:

• 1‑dimensional circle. For M = S¹ and C an ordinary associative algebra A (viewed as a 1‑category), B_*(S¹; A) recovers the cyclic Hochschild complex of A, confirming the reduction to known invariants.

• Surfaces. For a closed surface Σ_g of genus g, the homology of B_*(Σ_g; C) reproduces the known “string topology” operations when C is the category of chain complexes of a space, linking the construction to Chas‑Sullivan theory.

• 3‑manifolds. Using a handle decomposition of a 3‑manifold, the authors compare B_*(M; C) with the state spaces of Reshetikhin‑Turaev or Chern‑Simons TQFTs, showing that the derived invariants capture the same modular data together with higher extensions.

5. Outlook.
The paper concludes with several directions for future work. One is to relate the blob complex to factorization algebras and En‑algebras arising in quantum field theory, thereby bridging the algebraic and geometric approaches to locality. Another is to explore the role of the blob complex in categorified traces and centers of higher categories, which could lead to new invariants of manifolds and knots. Finally, the authors suggest that the operadic action might be leveraged to construct explicit higher‑categorical versions of the Deligne‑Kontsevich formality theorem.

In summary, the blob complex provides a robust, homotopy‑coherent framework that simultaneously generalizes Hochschild homology, supplies a derived model for TQFT state spaces, and fulfills a higher‑dimensional Deligne conjecture. Its blend of categorical duality, operadic symmetry, and gluing compatibility makes it a promising tool for both mathematicians studying higher category theory and physicists interested in the algebraic structures underlying topological quantum field theories.