I'm interested in learning a little smooth infinitesimal analysis. There is a free book by Kock: Smooth Differential Geometry, http://home.imf.au.dk/kock/ . As I dive into it, I feel that I'm not quite sufficiently prepared. He seems to be assuming some category theory and maybe also some knowledge of nonaristotelian logic.

I have copies of Lawvere and Rosebrugh, Sets for Mathematics, and Priest, An Introduction to Non-Classical Logic. It seems like I probably need to read the first $m$ chapters of Lawvere and the first $n$ chapters of Priest. Does this seem about right, and if so, what would be the values of $m$ and $n$? Does $n=0$? Amazon will let you see the table of contents of the two books with their "click to look inside" feature:



I have browsed the beginning of the Priest book and found it enjoyable, but haven't solved any of the problems. Category theory seemed dull and pointless to me, which is why I haven't really tackled much of Lawvere -- but maybe the light will dawn and some point and I'll realize why I should care about the subject.

If anyone wants to suggest replacing Kock, Lawvere, or Priest with some other book, that would be fine. I'm not hoping to become technically adept in smooth infinitesimal analysis, just to understand the basic ideas. Is there much of a distinction between smooth infinitesimal analysis, which is what I want to know about, and the subject of Kock's book, whose title says it's about smooth differential geometry?

Does this subject relate to topos theory?

[EDIT] It sounds like I may want to read Bell, A Primer of Infinitesimal Analysis rather than Kock.

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    $\begingroup$ You might want to take a look at the introductory book A Primer of Infinitesimal Analysis by John L. Bell. And yes, the subject relates to Topos theory. $\endgroup$ – Michael Greinecker Mar 11 '12 at 23:12
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    $\begingroup$ You can learn the main ideas of synthetic differential geometry without going to much into category theory. For a very brief overview look at Mike Shulman's notes www.math.ucsd.edu/~mshulman/papers/sdg/pizza-seminar.pdf and then Bell's book recommended by Michael Greinecker. $\endgroup$ – Omar Antolín-Camarena Mar 12 '12 at 4:27

Sets for Mathematics, nice as it is, is not sufficient preparation for studying the model theory of synthetic differential geometry. You will need to know topos theory to a level closer to, say, Chapter VI of Mac Lane and Moerdijk's Sheaves in Geometry and Logic.

But one does not need to dive in to the model theory straight away: after all, the point of synthetic differential geometry is that it can be presented axiomatically. In terms of logical requirements, you only need to know intuitionistic higher order logic – which is essentially the same as the logic of ordinary mathematics modulo the law of excluded middle. No knowledge of modal logic is required. (The internal logic of a topos is not modal!) It is good to do some exercises in intuitionistic logic to get a feel for what modes of reasoning remain valid. For example, you might want to try these:

  1. Show that $\lnot (p \lor q)$ is equivalent to $\lnot p \land \lnot q$.

  2. Show that $\lnot p \lor \lnot q$ implies $\lnot (p \land q)$.

  3. Show that $\lnot (p \land q)$ does not imply $\lnot p \lor \lnot q$: find a Heyting algebra $\mathfrak{A}$ and a valuation $[-] : \{ p, q \} \to \mathfrak{A}$ so that $[\lnot (p \land q)] \nleq [ \lnot p \lor \lnot q ]$.

In much the same way as real analysis forms the foundations for differential geometry, smooth infinitesimal analysis forms the foundation on which synthetic differential geometry is built. So one might regard it as a branch of synthetic differential geometry.

Now, a word about why we should care about topos theory in synthetic differential geometry: it's all very well to work axiomatically, but at some stage one has to worry about whether the axioms are consistent. Because synthetic differential geometry is a non-classical higher-order theory, one has to look beyond ordinary model theory to find models for the axioms. Fortunately, topos theory provides a ready-made solution: the internal logic of a topos is intuitionistic higher-order logic, and so the axioms can be interpreted in any topos without any work; our problem is then reduced to finding a topos in which the axioms are satisfied. There are several constructions of varying complexity which provide models for synthetic differential geometry, but the most intriguing ones are the models which admit a nice embedding of the category of smooth manifolds: these models are called ‘well-adapted’, and the existence of well-adapted models tells us that reasoning in the framework of synthetic differential geometry is sound for doing ordinary differential geometry!


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