Sunday, September 29, 2013

On Topos Theory

I hesitate to write about Topos Theory since I am not an expert in this area. However it seems to me a subject of great importance which is being presented in a distorted fashion of late.

Grothendieck introduced toposes as a notion both of space and of categories of space, a notion which permits spatial intuition to be applied to widely different parts of mathematics. It is a huge conceptual advance on Hausdorff's notion of topological space.

Lawvere and Tierney extracted elementary axioms of toposes which also served as an algebra of higher-order predicate logic, as boolean algebras are an algebra of propositional logic through Lawvere's discovery that both higher-order and quantifiers arise as adjoints.

Grothendieck's work lead, for example, to Deligne's proof of the Weil conjectures used in such concrete results as Zhang's theorem on gaps in the primes.

Lawvere and Tierney's work lead to a revolution in logic, but also to the possibility of a further development of Grothendieck's notion of space for teaching and progress in geometry.


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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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Saturday, September 21, 2013

Isle of Thorns 1976

The rest of the abstracts will be put shortly on the Como Category Theory Archive.

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As others see us

Here a drawing by some students in Milan,  before the moustache became white:


Friday, September 20, 2013

Evaluation of research

The Italian government has recently engaged in a procedure for the evaluation of research. The committees set up to do the evaluation have considered thousands (or tens of thousands) of publications. I have just seen the (private) evaluations of my recent papers. Poppycock!


Memories of primary school

I lived in Brisbane for 24 years, in Canberra for 3 years, in Sydney for 29 years, and now for almost 15 years in Italy. My memories of the early years are not very clear. However recently some have come back to mind as a result of a visit from a classmate in primary school at Eagle Junction State School, Queensland. His name is Ross and he has been visiting Italy for years with his wife. Last year there was a reunion of the class and he found out from my childhood friend David my whereabouts and visited us.

The visit was a pleasure because names, faces, incidents started coming back  to me from the long forgotten past, cut off from me by my travels.

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Thursday, September 19, 2013

Collocamento a riposo

A few days ago I  received from the Rector of the University of Insubria my letter of "collocamento a riposo" - that is, on the 1st November 2013 I will no longer be a professor at the the university.

I admit to a certain relief.

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