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Higher Galois theory. (English) Zbl 1403.18003

Classical Galois theory is well known to correspond, conceptually, to the theory of covering spaces and, in [A. Grothendieck (ed.) and M. Raynaud, Séminaire de géométrie algébrique du Bois Marie 1960/61 (SGA 1), dirigé par Alexander Grothendieck. Augmenté de deux exposés de M. Raynaud. Revêtements étales et groupe fondamental. Exposés I à XIII. (Seminar on algebraic geometry at Bois Marie 1960/61 (SGA 1), directed by Alexander Grothendieck. Enlarged by two reports of M. Raynaud. Ètale coverings and fundamental group). Berlin-Heidelberg-New York: Springer-Verlag (1971; Zbl 0234.14002)], Grothendieck showed how that could be used to give a rich theory of fundamental group(oid)s, actions, etc. in an algebraic geometric context, linking analogues of covering spaces, principal bundles / torsors, fibred categories and schemes. Later Grothendieck, in his ‘letter to Quillen’ extended this rich theory to a sketched theory of \(n\)-stacks. replacing covering spaces (interpreted as locally constant sheaves of sets) by locally constant stacks of \(n\)-types. (The link between these and (non-Abelian) cohomology had already been sketched out in his letter to Larry Breen some years earlier.) The resulting outline of a possible theory, based on an \(\infty\)-category theoretic analogue of sheaves, actions etc., and given in the long manuscript ‘Pursuing Stacks’, gave a basis for a lot of explorative research, and, in particular, for J. Lurie [Higher topos theory. Princeton, NJ: Princeton University Press (2009; Zbl 1175.18001)] who gave what has become the main source for ideas and methods for the study of \(\infty\)-categorical analogues of lots of the classical theory. (Note that a working knowledge of the early parts of that source is needed for a full appreciation of the paper under review here.)
From this point of view, classical Galois theory can be seen to state that the étale topos, \(\mathfrak{X}\), of a field, \(k\), is equivalent to the topos of \(G\)-sets for \(G\) the absolute Galois group, \(\mathrm{Gal}(k)\), of \(k\), and thus to the classifying topos of that group. In general, \(\mathrm{Gal}(k)\) is, however, not just a group, but rather is a pro-finite group and can be better viewed as an inverse system / pro-object in the category of (finite) groups, so the classifying ‘space’, \(\mathrm{BGal}(k^s/k)\), is equivalent, as a pro-groupoid, to the fundamental pro-groupoid, \(\Pi_1\mathfrak{X}\), of \(\mathfrak{X}\).
Working with Lurie’s higher topos theory, in this paper the author generalises the above theory to arbitrary dimensions, giving interpretations of \(\Pi_\infty\mathfrak{X}\), the fundamental \(\infty\)-groupoid of a topos \(\mathfrak{X}\). It is shown that locally constant sheaves in a locally \((n-1)\)-connected \(n\)-topos are equivalent to representations / actions of its fundamental pro-\(n\)-groupoid, all this for suitable interpretations of the terms here left undefined.
The paper makes the link between this theory and the now classical theories of étale homotopy (Artin-Mazur and Friedlander), plus in a topological context, the theory of shape, as given by Mardešić and Segal, and this reviewer.

MSC:

18B25 Topoi
54C56 Shape theory in general topology
55P55 Shape theory
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References:

[1] Lurie, J., Higher Topos Theory, Annals of Mathematical Studies, vol. 170 (2009), Princeton University Press · Zbl 1175.18001
[2] B. Toën, G. Vezzosi, Segal topoi and stacks over Segal categories, 2003.; B. Toën, G. Vezzosi, Segal topoi and stacks over Segal categories, 2003.
[3] Moerdijk, I., Prodiscrete groups and Galois toposes, Indag. Math. (Proc.), 92, 2, 219-234 (1989) · Zbl 0687.18004
[4] Kennison, J. F., The fundamental localic groupoid of a topos, J. Pure Appl. Algebra, 77, 1, 67-86 (1991) · Zbl 0760.18006
[5] Grothendieck, A. (1975), letter to L. Breen
[6] Artin, M.; Mazur, B., Étale Homotopy, Lecture Notes in Mathematics, vol. 100 (1969), Springer · Zbl 0182.26001
[7] Friedlander, E. M., Étale Homotopy of Simplicial Schemes, Annals of Mathematical Studies, vol. 104 (1982), Princeton University Press · Zbl 0538.55001
[8] Lurie, J., Derived algebraic geometry XIII: rational and \(p\)-adic homotopy theory (2011)
[9] Bousfield, A. K.; Kan, D. M.; Limits, Homotopy, Completions and Localizations, Lecture Notes in Mathematics, vol. 304 (1972), Springer · Zbl 0259.55004
[10] J. Lurie, Higher Algebra, http://www.math.harvard.edu/ lurie/papers/HA.pdf; J. Lurie, Higher Algebra, http://www.math.harvard.edu/ lurie/papers/HA.pdf
[11] Artin, M.; Grothendieck, A.; Verdier, J.-L., Théorie des topos et cohomologie étale des schémas, Lecture Notes in Mathematics, vol. 269 (1972), Springer · Zbl 0234.00007
[12] Isaksen, D. C., Strict model structures for pro-categories, (Categorical Decomposition Techniques in Algebraic Topology. Categorical Decomposition Techniques in Algebraic Topology, Progress in Mathematics, vol. 215 (2007), Birkhauser), 179-198 · Zbl 1049.18008
[13] Mardešić, S.; Segal, J., Shapes of compact and ANR-systems, Fundam. Math., 72, 41-59 (1971) · Zbl 0222.55017
[14] E.J. Dubuc, Spans and simplicial families, 2010.; E.J. Dubuc, Spans and simplicial families, 2010.
[15] Bunge, M., Classifying toposes and fundamental localic groupoids, (Seely, R. A.G., Category Theory 1991. Category Theory 1991, CMS Conf. Proc., vol. 13 (1992), AMS) · Zbl 0787.18005
[16] Barr, M.; Diaconescu, R., On locally simply connected toposes and their fundamental groups, Cah. Topol. Géom. Différ. Catég., 22, 301-314 (1981) · Zbl 0472.18002
[17] Dugger, D.; Hollander, S.; Isaksen, D. C., Hypercovers and simplicial presheaves, Math. Proc. Camb. Philos. Soc., 136, 1, 9-51 (2004) · Zbl 1045.55007
[18] Dubuc, E. J., The fundamental progroupoid of a general topos, J. Pure Appl. Algebra, 212, 11, 2479-2492 (2008) · Zbl 1155.18001
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