Homotopical Galois theory of coverings - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T14:07:32Z http://mathoverflow.net/feeds/question/111386 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/111386/homotopical-galois-theory-of-coverings Homotopical Galois theory of coverings tetrapharmakon 2012-11-03T15:39:10Z 2012-11-03T15:39:10Z <p>In the hope this won't turn into a trivial problem (I couldn't find a similar discussion here), here's my question.</p> <p>I'm studying a little homotopical algebra in <a href="http://www.math.uni-hamburg.de/home/schreiber/Abstract%2520homotopy%2520theory%2520and%2520generalized%2520sheaf%2520cohomology.pdf" rel="nofollow">this</a> article by Brown. You can easily notice that Theorem 3 (page 430) and Proposition 3 (in the following page) imply that one can internalize the notion of "$\pi_1$ acting on the fibers of a covering", idea which dates back, if I'm not wrong, to Quillen's "Homotopical Algebra".</p> <p>This could be the starting point for some natural (?) questions: the action of $\pi_1$ on the fibers of a covering is worth to be studied because of Galois' theory of coverings (in fact the philosophy is that of Grothendieck's Galois Theory: Galois groups "are" homotopy groups).</p> <p>Now allow me to state the 64 thousand dollar question: </p> <blockquote> <p>can we recover Galois' theory of coverings in a suitable model/fibrant category? </p> </blockquote> <p>I.e., can we classify subgroups(#) of the fundamental group(#) of the base space of a fiber space(#), finding an (anti-)monotone bijection(#) between the lattice of intermediate objects between the base and a suitable "universal"(##) covering?</p> <p>My two cents: classically, we know very well what to do and how do do it. Here we certainly have enough informations about how to internalize each ingredient (at least those marked with "#"):</p> <ol> <li>Subgroups of a group object are (iso classes of) group mono to that object;</li> <li>$\pi_1(B)=\Omega B$ = pullback obtained exploiting a path object for B, which is doing externally what $\pi_1(B)$ did internally (it is a group which acts on the fibers of a fibration, Omega(Omega(B)) is abelian, ...);</li> <li>A fiber/coverng space is a fibration (here and in (2) one needs a pointed fibrant/model category);</li> <li>Antimonotone bijections are Galois' equivalences: here one looks to the subobjects poset of Omega(B), and to the posetal category C_B, having as objects fibrations with base B (the order is defined by: X &lt; Y iff one fibers over the other - a choice we are rather forced to, just because classically it is so).</li> </ol> <p>The problem seems to be that we lack something forcing C_B to admit a top element.</p> <p>Another question which is still in the handwaving zone: In studying classical Galois theory, I found really bothering that the splitting field of a field is only a weak limit (any two splitting fields are isomorphic, but not with a unique iso). All the same, it is really annoying to notice that the universal covering of (even a good) space is a weak limit. What if the localization functor killed this ambiguity "contracting" the groupoid of isomorphisms between different universal coverings, in passing to the homotopy category? Is there a way to write it down without using theology?</p> <p>Try to meet up this challenge: example 1.1.1.1 in Higher Topos Theory by Lurie suggests (not so coincidentally?) that "being homotopic" in Grp means to be conjugate; now, any two splitting fields are conjugate, am I wrong?</p>