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:

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

Here is some work that has been done on the homotopy hypothesis: For more details on simplicial methods in homotopy theory, see: For model categories, see: For simplicial localization, see: For simplicial approaches to higher categories, start with this introduction: You can also see more about this workshop!

© 2007 John Baez, except for the figures by Aaron Lauda