Blob homology

Abstract

Given an $n$–manifold $M$ and an $n$–category $\mathcal{C}$, we define a chain complex (the “blob complex”) ${\mathcal{ℬ}}_{\ast }\left(M;\mathcal{C}\right)$. The blob complex can be thought of as a derived category analogue of the Hilbert space of a TQFT, and also 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. This is the published version of [arXiv 1009.5025].