Higher categories, colimits, and the blob complex

Details Higher categories, colimits, and the blob complex Higher categories, colimits, and the blob complex

2011-08-26

arxiv.org/abs/1108.5386v1

Proceedings of the National Academy of Sciences, May 17, 2011, vol. 108 no. 20, pp. 8139-8145

Mathematics

Category Theory

7 pages

Scientific paper

10.1073/pnas.1018168108

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.

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