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Higher categories, colimits, and the blob complex.

Scott Morrison1, Kevin Walker

  • 1Miller Institute for Basic Research, University of California, Berkeley, CA 94704, USA. scott@tqft.net

Proceedings of the National Academy of Sciences of the United States of America
|May 4, 2011
PubMed
Summary
This summary is machine-generated.

We introduce the blob complex, a novel chain complex for higher categories. Its higher homology groups generalize Hochschild homology, offering new tools for topological quantum field theory and algebraic topology research.

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Area of Science:

  • Algebraic Topology
  • Category Theory
  • Mathematical Physics

Background:

  • Axioms for higher categories provide a foundational framework.
  • Topological quantum field theory (TQFT) invariants are crucial in mathematical physics.
  • Hochschild homology is a key invariant in algebraic topology and number theory.

Purpose of the Study:

  • To introduce and describe the "blob complex," a new mathematical construct.
  • To demonstrate how the blob complex generalizes existing concepts like Hochschild homology.
  • To establish the blob complex as a tool for TQFT and higher category theory.

Main Methods:

  • Associating a chain complex B(*)(W;C) to any n-manifold W, based on an n-category.
  • Utilizing homotopy colimits along manifold decompositions for the blob complex's definition.
  • Generalizing Deligne's conjecture on Hochschild cohomology and the little discs operad to higher dimensions.

Main Results:

  • The zeroth homology of the blob complex recovers standard TQFT invariants for W.
  • Higher homology groups of the blob complex generalize Hochschild homology.
  • The blob complex offers a natural definition via homotopy colimits.

Conclusions:

  • The blob complex provides a powerful new framework for studying higher categories and related invariants.
  • This work extends the understanding of Hochschild homology and its connections to TQFT.
  • The generalization of Deligne's conjecture opens new avenues for research in higher dimensional algebra.