Thursday, September 26, 2013

Ross Street's Orientals

I recently had a look at one of Jacob Lurie's expositions (in 111 pages) on the clasification of topological field theories - very briefly and with little comprehension. I find it difficult to read mathematics in which not even formal definitions are given: Lurie himself says "In many instances, we have not attempted to give precise definitions, let alone careful proofs".

I have always been interested in lower category theory (up to symmetric monoidal bicategories) but only once have ventured beyond. Always we had the idea that natural transformations were like homotopies, and that one could consider homotopies between homotopies and so on, and so one would get higher  categories. Thinking about axiomatizing higher categories, in particular higher associative laws lead Ross Street to considering the free n-category on an n-simplex. Mike Johnson and I also worked on this problem.

Following this link there is an early picture produced by Ross of an oriental.

This seems to me still an interesting problem but I don't see where it has gone in Joyal and Lurie's work.

In any case infinity-category theory seems to be a distinct subject from category theory. Category theory should resist the attempted takeover.

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Anonymous Anonymous said...

I agree and disagree.
While it is true that the infinity category crowd tends to sweep unpleasant details under its well funded rug, sometimes this way of thinking actually leads to good ideas and ideals.
From my experience with Lurie at least, his statements tend more to be true and meaningful than not. I think the fact that he is surrounded by sometimes blindly admiring and immitating followers shouldn't detract from the actual work he has done.

5:12 PM  

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