John Baez

Lectures at n-Categories: Foundations and Applications

June 7, 2004

Why n-Categories?

Research on n-categories is revolutionizing our concept of mathematics by teaching us how to think of every interesting equation as the summary of an interesting process. Here we sketch how this approach leads to a new understanding of even the simplest mathematical structures: in particular, of the natural numbers. A topological study of the origin of the natural number concept suggests that in the "true natural numbers", addition satisfies an infinite hierarchy of coherence laws for associativity, an infinite hierarchy of coherence laws for commutativity, together with an infinite hierarchy of further hierarchies of coherence laws. All these are built into the concept of the "free k-tuply monoidal n-category on one generator", and this should admit a description as the n-category of "n-braids in codimension k". The objects here are elements of the space of finite subsets of k-dimensional Euclidean space. The morphisms are paths in this space, the 2-morphisms are paths of paths in this space, and so on, with the n-morphisms being homotopy classes of paths of paths of paths... in this space. Similarly, the integers should be related to the n-category of "n-tangles in codimension k".

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What n-Categories Should Be Like

We describe features that any useful theory of n-categories should have. In particular, there should be three specially nice sorts of n-categories: k-tuply monoidal n-categories, n-groupoids and strict n-categories. We describe the effects of imposing these extra conditions separately and in combination, and conjecture the existence of a weakly commutative cube of free and forgetful (n+1)-functors relating the resulting eight classes of n-categories. Among other things, this cube would give an explanation of the relation between n-categories, homotopy theory, stable homotopy theory, and homology theory.

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For more on these subjects try these papers:

Space and State, Spacetime and Process

General relativity may seem very different from quantum theory, but work on quantum gravity has revealed a deep analogy between the two, which is best understood using category theory. General relativity describes space and spacetime in terms of objects and morphisms in nCob, the category of n-dimensional cobordisms. Quantum theory describes states and processes using objects and morphisms in Hilb, the category of Hilbert spaces. The analogy between general relativity and quantum theory is made precise by the fact that both nCob and Hilb are "symmetric monoidal categories with duals". Work on string theory and loop quantum gravity suggests that the analogy goes deeper, in a way that is best understood using n-categories. The "TQFT hypothesis" is a preliminary attempt to make this precise.

Click on this to see the slides:

For more on this subject try these papers:

You can also see photos of the workshop where I gave these talks!

I also wrote an issue of This Week's Finds about this workshop: week209.

© 2004 John Baez