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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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:

- John Baez and James Dolan, Categorification
- Eugenia Cheng and Aaron Lauda, Higher-Dimensional Categories: an illustrated guide book
- Tom Leinster, A Survey of Definitions of n-Category
- Tom Leinster, Higher Operads, Higher Categories
- Michael Makkai, On Comparing Definitions of "Weak n-Category"

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:

- John Baez, Higher-Dimensional Algebra and Planck-Scale Physics
- John Baez, Quantum Quandaries: a Category-Theoretic Perspective
- John Baez and James Dolan, Higher-Dimensional Algebra and Topological Quantum Field Theory

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

baez@math.removethis.ucr.andthis.edu