Invited talk at the Workshop on Smooth Structures in Logic, Category Theory, and Physics held at Ottawa University, 1st - 3rd May 2009.

Updated to the version actually given (essentially just added more pictures)

Abstract:

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.

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This is based partly on Preprints/smthcat

This talk is a beamer presentation. I now use lots of my own macros for beamer presentations so it's not practical to make the source available. However, it was the first time I tried using the TikZ package and I found it easiest to devise the pictures separately to the main presentation at first. I don't claim any kind of expertise, but I make it available just in case it's useful to anyone. When run through htlatex (from TeX4ht), TikZ produces SVG output. This isn't perfect and I've had to do some editing to get the SVG to look like the original. This mainly involves editing the text fields (I still don't have it quite right - the superscripts aren't working as they should) because (it seems) the SVG text field is fairly limited unlike the TikZ original.

Please do not print out the beamer or trans versions of this talk. They are intended solely for on-screen viewing (the trans version has just one overlay per frame so is a little simpler to step through). If you really want to print something out (why?) then print out the handout version. Better still, take a look at the original article: Preprints/smthcat.

• Presentation version in PDF. This is what is actually shown on the screen.
• Slides version in PDF. This was the backup version for an OHP in case of computer failure. For the most part, it is a one slide per frame version of the presentation though it skips the fantastic C⁵ joke.
• Handout version in PDF. Note: this uses pgfpages to format it as 4-up. With double-sided printing, this only uses 4 pieces of paper. Neat, huh? This version skips a few "redundant" slides to keep the number of pages down. No actual information is lost, though.

• Pictures: In the presentation, some of these contain overlay commands (demonstrating the fantastic integration of TikZ with beamer; almost as if they were made for each other) which I've left out of the samples.

  • Smooth map of manifolds: PDF SRC SVG. This takes a while to compile. To speed up the process, change the angle step size (as indicated in the source).
  • Smooth map of smooth spaces: PDF SRC SVG. Same speed issue as for manifolds.
  • A smooth space: PDF SRC SVG.
  • Composition failure: PDF SRC SVG.
  • Completed input: PDF SRC SVG. The original had an overlined phi but that didn't display correctly so I changed it to a "varphi".
  • Completed inputs and outputs: PDF SRC SVG. Ditto on the overlines.
  • Diagram of categories: PDF SRC SVG. It took a little time to figure out how to do the double and triple arrows. The nodes are "labelled" and I wanted to use the labels as coordinates (so that I could move them around while experimenting and have the arrows in the right places), however that means that arguments such as [yshift=5pt] don't work. I don't know if the solution I found is the best, but it worked.
  • Quotients in Souriau and Frolicher spaces: PDF SRC SVG. The superscripts don't work in this one, not sure why.

  • Subspaces in Sikorski and Frolicher spaces: PDF SRC SVG. Ditto on the superscripts.

  • In the final version there were more pictures, however they were essentially derivatives of the above. I shall probably add them to this list at some future date for completeness sake.