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the entry algebraic fundamental group used to be a puny stub – and still is. I have now at least cross-linked it with étale homotopy.
(copy from another thread).I have added stuff to algebraic fundamental group adapted from some notes of Tomás Szamuely, which I have given as reference. The entry is still stubby and I need to work out several points that are obscure (like what is $\Omega$, but thought it better to get something down rather than to leave the entry very empty (I had promised myself 18 months ago to do something about this subject … better late than never. :-)) I also started a web page on Szamuely but this is just a link to his homepage (He looks to have some good sets of notes and other interesting stuff.)
(copy from another thread). I started a page on what Borceux and Janelidze call the Chevalley fundamental group. This is the algebraic / Grothendieck approach to the usual fundamental group, at least on spaces having a universal covering. They gave Chevalley’s book on Lie groups as the source.
Has anyone read / looked at Szamuely’s book in detail? I heard that it was good and the contents look good. (There is a link to an electronic copy that I have added, but I am not sure as to whether it is the final version. I trust it is legal.)
“The algebraic fundamental group, $\pi_1 (S,\overline{s})$ is defined to be the automorphism group of this functor.” When unpacked properly this is certainly accurate, but I think people usually say something like “It prorepresents the fiber functor,” since the fiber functor is usually not representable. Or it is defined to be the colim over all automorphism groups of all Galois covers. There is a subtlety about working with systems of covers that isn’t conveyed when it is expressed as “the automorphism group of this functor.”
Szamuely is a very readable book – at least up to Chapter 6, which I have not read.
@5 Matt I agree that there is more to write; it is still a stub. My problem is that the SGA1 deals with the case of finite covers in general and that is a good one for explaining things, so perhaps the exposition that you are suggesting would fit better in an entry on Galois categories. The question of pro-representability is not quite as you state it. The fibre functor is pro-representable and the pro-representing object that you get can be replaced by one made up of Galois objects and then the automorphism group (or its opposite) of this pro-object is the colimit of finite groups and so is a pro-finite group which is isomorphic to the algebraic fundamental group in Grothendieck’s definition. (I’ve just written up an exposition of this for another project.) The geometric content of the technical pro-representability stuff is, I think I would claim, that it is the automorphism group of the fibre functor. That is perfectly useless without the translation to the pro-representing aspect. I am suggesting that the right place to work may be the entry on Grothendieck’s Galois Theory. That still needs a lot of work to bring out all the points that need making. Feel free to edit it.
Have you seen Quick’s paper on Profinite Homotopy Theory. (There are some minor errors in it which are annoying but they do not impinge on the main ideas.) He looks at simplicial profinite spaces as a category and then working with an obvious notion of covering map gets a Galois category. The pro-representing object of the fibre functor is a pro-object in the category of simplicial pro-finite spaces and it is then not hard to see that its limit is an object in that category which is a universal covering in an obvious sense. In this case the fibre functor is representable and its automorphism group is exactly the automorphism group of that universal covering. This is already hinted at in some results back in SGA1 but Grothendieck did not work in a suitable category to push the final bit through.
Your explanation is what I had in mind, but attempting to say it in as few words as possible I guess it came out confusing. The main thing I was thinking of adding was the worked example of $\mathbb{A}^1 \setminus \{0\}$ which despite being easy enough to write down completely what is going on provides a good example of why passing to the pro-category is needed (no algebraic exponential map). Would this be a good “Motivation” section to compare the topological universal cover $\mathbb{C}\to\mathbb{C}^*$ with what needs to be done algebraically, or maybe as an after the definition “Example” section?
I haven’t seen that paper, but that’s probably because I shy away from homotopy things.
at algebraic fundamental group (still a stub) I added mentioning of the words “Galois group” and “arithmetic fundamental group” (which I also made a redirect) and added a pointer to
have been adding further pointers and cross-links following the first pages of Minhyong Kim’s 09 lecture notes to entries such as section conjecture, algebraic fundamental group, Diophantine geometry, Abel-Jacobi map and maybe others.
I notice that the present state of the entry section conjecture gives no indication whatsoever of the perspective that MK provides, but I am not touching it yet.
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