Higher categories, colimits, and the blob complex

TL;DR

This paper introduces the blob complex, a chain complex associated with n-manifolds using higher categories, generalizing Hochschild homology and connecting to topological quantum field theories.

Contribution

It defines the blob complex for higher categories, relates it to TQFT invariants, and extends Deligne's conjecture to higher dimensions.

Findings

The 0-th homology recovers TQFT invariants.

Higher homology generalizes Hochschild homology.

Provides a framework for higher-dimensional Deligne's conjecture.

Abstract

We summarize our axioms for higher categories, and describe the blob complex. Fixing an n-category C, the blob complex associates a chain complex B_*(W;C)$ to any n-manifold W. The 0-th homology of this chain complex recovers the usual topological quantum field theory invariants of W. The higher homology groups should be viewed as generalizations of Hochschild homology (indeed, when W=S^1 they coincide). The blob complex has a very natural definition in terms of homotopy colimits along decompositions of the manifold W. We outline the important properties of the blob complex, and sketch the proof of a generalization of Deligne's conjecture on Hochschild cohomology and the little discs operad to higher dimensions.