The blob complex

TL;DR

The paper introduces the blob complex, a new chain complex for n-manifolds and n-categories, generalizing Hochschild homology and relating to TQFTs, with formal properties and a new weak n-category concept.

Contribution

It defines the blob complex as a derived category analogue of TQFT Hilbert spaces and generalizes Hochschild homology to higher dimensions, also proposing a new weak n-category with strong duality.

Findings

The blob complex generalizes Hochschild homology to n-manifolds and n-categories.

It exhibits a higher-dimensional Deligne's conjecture analogue.

Introduces a weak n-category with strong duality suited for TQFTs.

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.