John Baez

Lecture at Higher Categories and Their Applications

January 10, 2007

The Homotopy Hypothesis

Crudely speaking, the Homotopy Hypothesis says that n-groupoids are the same as "homotopy n-types" — nice spaces whose homotopy groups above the nth vanish for every basepoint. We summarize the evidence for this hypothesis. Naively, one might imagine this hypothesis allows us to reduce the problem of computing homotopy groups to a purely algebraic problem. While true in principle, in practice information flows the other way: established techniques of homotopy theory can be used to study coherence laws for n-groupoids, and a bit more speculatively, n-categories in general.

Click on this to see the transparencies of my talk:

• The Homotopy Hypothesis - in PDF or Postscript.

For a gentle introduction to n-categories and the homotopy hypothesis, try these:

• John Baez and James Dolan, Higher-dimensional algebra and topological quantum field theory.
• John Baez and James Dolan, Categorification.
• John Baez and Michael Shulman, Lectures on n-categories and cohomology.

• Zouhair Tamsamani, Equivalence de la théorie homotopique des n-groupoïdes et celle des espaces topologiques n-tronqués.
• Michael Batanin, Monoidal globular categories as a natural environment for the theory of weak n-categories.
• Carlos Simpson, Homotopy types of strict 3-groupoids.
• Michael Batanin, The combinatorics of iterated loop spaces.
• Clemens Berger, Double loop spaces, braided monoidal categories and algebraic 3-type of space.
• Clemens Berger, A cellular nerve for higher categories.
• Denis-Charles Cisinski, Batanin higher groupoids and homotopy types.
• Simona Paoli, Semistrict models of connected 3-types and Tamsamani's weak 3-groupoids.
• Simona Paoli, Semistrict Tamsamani n-groupoids and connected n-types.

• Paul G. Goerss and J. F. Jardine, Simplicial Homotopy Theory, Birkhäuser, Basel, 1999.

• Mark Hovey, Model Categories, American Mathematical Society, Providence, Rhode Island, 1999.

• William G. Dwyer and Daniel M. Kan, Simplicial localizations of categories, J. Pure Appl. Algebra 17 (1980), 267-284.
• William G. Dwyer and Daniel M. Kan, Calculating simplicial localizations, J. Pure Appl. Algebra 18 (1980), 17-35.
• William G. Dwyer and Daniel M. Kan, Function complexes in homotopical algebra, Topology 19 (1980), 427-440.

• Bertrand Toen, Tamsamani categories, Segal categories, and applications.

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