Grothendieck's Galois Theory today - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T08:20:20Z http://mathoverflow.net/feeds/question/32418 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/32418/grothendiecks-galois-theory-today Grothendieck's Galois Theory today lambdafunctor 2010-07-18T22:31:22Z 2012-06-25T18:06:37Z <p>I have recently become aware of, and started to study in my free time (abundant in these summer months) Grothendieck's Galois Theory (GGT), as formulated in SGA 1 and later by Grothendieck's contemporaries. I understand there to have been a number of unresolved and open questions relating to GGT upon its formulation, some of which seem to persist. This is a truly gorgeous subject, and I wonder whether it is still studied rigorously or researched at all today? Where/Who produces interesting results regarding things such as Galois and Atomic topoi, applications of the Grothendieck fundamental group, etc., today?</p> <p>Is there a great deal of utility to GGT beyond the foundational work in algebraic geometry later formulated by Grothendieck? </p> http://mathoverflow.net/questions/32418/grothendiecks-galois-theory-today/32455#32455 Answer by David Roberts for Grothendieck's Galois Theory today David Roberts 2010-07-19T07:32:16Z 2010-07-19T07:32:16Z <p>There is a book on a more general approach to Galois theory by Borceux and Janelidze (called, imaginatively, 'Galois theories' <a href="http://books.google.com.au/books?id=fNJTFzYSgMUC&amp;printsec=frontcover&amp;dq=galois+theory+janelidze&amp;source=bl&amp;ots=G-K918BlU7&amp;sig=Y2NzFBEbfLLZwEcuj9AiaClVhG0&amp;hl=en&amp;ei=s_xDTJpRj7i-A9yXvIYN&amp;sa=X&amp;oi=book_result&amp;ct=result&amp;resnum=5&amp;ved=0CCsQ6AEwBA#v=onepage&amp;q=galois%20theory%20janelidze&amp;f=false" rel="nofollow">Google books</a>). A bit more concrete is 'Galois Theory in Symmetric Monoidal Categories' by Janelidze and Street, which uses a Galois theoretic approach to Tannaka duality (a very Grothendieckian study).</p> <p>Marta Bunge has a bunch of stuff on topoi and Galois-type theories, for example 'Galois Groupoids and Covering Morphisms in Topos Theory', 'Constructive Theory of Galois Toposes' (joint with Eduardo Dubuc) and a bunch of others.</p> http://mathoverflow.net/questions/32418/grothendiecks-galois-theory-today/32470#32470 Answer by Charles Siegel for Grothendieck's Galois Theory today Charles Siegel 2010-07-19T11:26:48Z 2010-07-19T11:26:48Z <p>This isn't my research group (though some of the people for whom it is are around), but <a href="http://www.math.upenn.edu/~galois/" rel="nofollow">here</a>'s a website devoted to this stuff, with lots of papers, names, etc. It's a VERY active area, with a lot of interesting problems.</p> http://mathoverflow.net/questions/32418/grothendiecks-galois-theory-today/33494#33494 Answer by Simon Pepin Lehalleur for Grothendieck's Galois Theory today Simon Pepin Lehalleur 2010-07-27T10:48:08Z 2010-07-27T10:48:08Z <p>I don't know much about this topic, but I was recently recommended the paper <em>An extension of the Grothendieck Galois theory of Grothendieck</em> by Joyal and Tierney as an enlightening abstract generalisation in the language of toposes. It seems that it predates some of the other references given above, but might be worth reading.</p> http://mathoverflow.net/questions/32418/grothendiecks-galois-theory-today/33532#33532 Answer by Anweshi for Grothendieck's Galois Theory today Anweshi 2010-07-27T15:28:37Z 2010-07-27T15:28:37Z <p>I suggest that you read Deligne's wonderful paper “Le Groupe Fondamental de la Droite Projective Moins Trois Points”. I read a little bit of it and was astonished. Please do take it up and read it without further loss of time, notwithstanding the French.</p> <p>Jordan Ellenberg's and Matthew Emerton's opinion is available <a href="http://quomodocumque.wordpress.com/2009/07/24/le-groupe-fondamental-de-la-droite-projective-moins-trois-points-is-now-online/" rel="nofollow">here</a>.</p> http://mathoverflow.net/questions/32418/grothendiecks-galois-theory-today/100615#100615 Answer by Eduardo Dubuc for Grothendieck's Galois Theory today Eduardo Dubuc 2012-06-25T18:06:37Z 2012-06-25T18:06:37Z <p>The basic Grothendieck's assumptions means we are dealing with an connected atomic site $\mathcal{C}$ with a point, whose inverse image is the fiber functor $F: \mathcal{C} \to \mathcal{S}et$:</p> <p>(i) Every arrow $X \to Y$ in $\mathcal{C}$ is an strict epimorphism.</p> <p>(ii) For every $X \in \mathcal{C}$ $F(X) \neq \emptyset$.</p> <p>(iii) $F$ preseves strict epimorphisms.</p> <p>(iv) The diagram of $F$, $\Gamma_F$ is a cofiltered category.</p> <p>Let $G = Aut(F)$ be the localic group of automorphisms of $F$.</p> <p>Let $F: \widetilde{\mathcal{C}} \to \mathcal{S}et$ the pointed atomic topos of sheaves for the canonical topology on $\mathcal{C}$. We can assume that $\mathcal{C}$ are the connected objects of $\widetilde{\mathcal{C}}$.</p> <p>(i) means that the objects are connected, (ii) means that the topos is connected, (iii) that $F$ is continous, and (iv) that it is flat. </p> <p>By considering stonger finite limit preserving conditions (iv) on $F$ (corresponding to stronger cofiltering conditions on $\Gamma_F$) we obtain different Grothendieck-Galois situations (for details and full proofs see [1]):</p> <p>S1) F preserves all inverse limits in $\widetilde{\mathcal{C}}$ of objets in $\mathcal{C}$, that is $F$ is essential. In this case $\Gamma_F$ has an initial object $(a,A)$ (we have a "universal covering"), $F$ is representable, $a: [A, -] \cong F$, and $G = Aut(A)^{op}$ is a discrete group. </p> <p>S2) F preserves arbritrary products in $\widetilde{\mathcal{C}}$ of a same $X \in \mathcal{C}$ (we introduce the name "proessential for such a point [1]). In this case there exists galois closures (which is a cofiltering-type property of $\Gamma_F)$, and $G$ is a prodiscrete localic group, inverse limit in the category of localic groups of the discrete groups $Aut(A)^{op}$, $A$ running over all the galois objects in $\mathcal{C}$.</p> <p>S2-finite) F takes values on finite sets. This is the original situation in SGA1. In this case the condition "F preserves finite products in $\widetilde{\mathcal{C}}$ of a same $X \in \mathcal{C}$ holds automatically by condition (iv) ($F$ preserves finite limits), thus there exists galois closures, the groups $Aut(A)^{op}$ are finite, and $G$ is a profinite group, inverse limit in the category of topological groups of the finite groups $Aut(A)^{op}$.</p> <p>NOTE. The projections of a inverse limit of finite groups are surjective. This is a key property. The projection of a inverse limit of groups are not necessarily surjective, but if the limit is taken in the category of localic groups, they are indeed surjective (proved by Joyal-Tierney). This is the reason we have to take a localic group in 2). Grothendieck follows an equivalent approach in SGA4 by taking the limit in the category of Pro-groups.</p> <p>S3) No condition on $F$ other than preservation of finite limits (iv). This is the case of a general pointed atomic topos. The development of this case we call "Localic galois theory" see [2], its fundamental theorem first proved by Joyal-Tierney.</p> <p>[1] "On the representation theory of Galois and Atomic topoi", JPAA 186 (2004)</p> <p>[2] "Localic galois theory", Advances in mathematics", 175/1 (2003).</p>