Mon, 3rd Aug 2009 (Seminars)
Comparative Smootheology: Workshop on Loops, Strings and Moduli Spaces
Talk given at the Workshop on Loops, Strings and Moduli Spaces held at the Chern Institute, Nankai University, 3rd - 7th August 2009.
Similar in content to the talk give at Ottawa in May 2009, but aimed at a different audience so emphasises different aspects of the theory.
Smooth manifolds are extremely nice spaces. The fact that they have charts means that a vast amount of the theory of Euclidean spaces can be easily transferred to manifolds. This makes for a very useful subject.
However, the charts also make manifolds very fragile: it is easy to do something to a manifold that makes it no longer a manifold. Taking a quotient by a group action is one such, looking at mapping spaces is another. Often, specific operations can be fixed - orbifolds fix the quotienting, infinite dimensional manifolds fix the mapping spaces - but systematic case-by-case fixing is a little unsatisfying. Over the years there have been several attempts to build a suitable category of "smooth objects" generalising smooth manifolds. The general method is to take some property that all manifolds have, which can be defined in a more robust way than charts.
In this talk I shall review some of these attempts, focussing particularly on the similarities between them. I shall try to motivate my own favourite: Frölicher spaces. In addition, it is worth mentioning that the majority of these categories come under the heading of "sets with structure". There have also been attempts to do away with the "sets with" part of this and I shall talk about why one might wish to do this.
This is based partly on Preprints/smthcat
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