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: