There are numerous contenders for the successor to the category of smooth manifolds, such as diffeological spaces, Frolicher spaces, Chen spaces, etc. It has become clear that we need to generalize from smooth manifolds to a larger category of spaces containing the former, just as we generalized from Euclidean space to manifolds to describe general relativity. Infinite dimensional manifolds allow you to perform some of these constructions, but they do not address all the problems. The important part of the text for the average physicist is the first part, which is only about a hundred pages or so, so it isn't a very long exposition.įor a theorist interested in the long-term future of mathematical and theoretical physics: the category of smooth manifolds is very badly behaved, since many operations and constructions (such as exponential objects) are often difficult or impossible to perform with just smooth manifolds.
It gives you a different way of thinking about differential geometry, allowing the rigorous use of infinitesimal arguments, which are hard or clumsy to justify classically. It's worth a read if you've ever wondered whether the infinitesimal arguments invoked by physicists had any rigorous foundation (as I did when I was a physics undergrad), or if you're interested in seeing a more intuitive presentation of the basics of differential geometry than you would find in a typical differential geometry text.įor most physicists, this would probably be at best an interesting supplement to a classical differential geometry text, since so much of current theory is formulated in the classical context. I hope my work will serve to bring justification to the synthetic method besides the analytical one." After long vacillations, I have decided to use a half synthetic, half analytic form. But I soon realized that, as expedient the synthetic method is for discovery, as difficult it is to give a clear exposition on synthetic investigations, which deal with objects that till now have almost exclusively been considered analytically. I found these theories originally by synthetic considerations. “The reason why I have postponed for so long these investigations, which are basic to my other work in this field, is essentially the following. The theory is also very much inspired by the thought process and work of Sophus Lie (who developed the theory of Lie algebras and Lie groups). Lawvere's longterm goal has been to develop a more suitable mathematical language for physics, and synthetic differential geometry emerged from his categorical dynamics program. It is a theory with a very physical and geometric spirit that rigorously captures the way physicists work with infinitesimals. In case you haven't heard of it, synthetic differential geometry is a synthetic (as opposed to analytic) approach to calculus and differential geometry developed by Bill Lawvere, Anders Kock, and several other prominent category theorists which heavily relies on infinitesimals. I recently found out that the synthetic differential geometry text by Anders Kock is freely available online.
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