Constructive Galois toposes

Marta Bunge – 27 June 2001

The notion of a Galois topos was origin ally considered implicitly by Grothendieck [AGV,AM] and explicitly by Moerdijk [M], as that of a pointed connected atomic Grothendieck topos E that is generated by its Galois objects (or normal atoms). A Grothendieck topos E is a Galois topos iff it is pointed connected locally connected and generated by its locally constant objects. Galois toposes are alternatively described as the classifying toposes BG for G a prodiscrete localic group (in Set). In joint work with Dubuc [CTGT] we ask what is the appropriate notion of a Galois topos in the unpointed case, where also the base category Set is replaced by an arbitrary base topos S. We base our discussion on a construction in [B] which associates, with any connected locally connected (possibly pointless) topos E, bounded over S, a prodiscrete localic groupoid G in S which plays the role of the fundamental group of E (i.e., which represents first-degree cohomology of E with coefficients in discrete groups in S). The classifying topos BE is a connected atomic topos denoted by Pi1cov(E), obtained as the limit of a filtered system of connected atomic toposes GU and connected locally connected transition maps, where the U run over a cofinal small system of coverings in E. By an S-Galois topos we mean (in [CTGT]) a connected atomic S-bounded topos E that is generated by its ``Galois families''. The S-Galois toposes are then shown to correspond precisely to the classifying toposes BG, for G a prodiscrete localic groupoid in S. Alternatively, S-Galois toposes are shown to correspond to those connected locally connected toposes bounded over S that are generated by their ``locally componentwise constant objects''. Recall that Barr and Diaconescu [BD] show that for each covering U in a connected locally connected topos E over S, an object X of E is U-split iff X is U-componentwise constant. Thus, in the case of a locally simply connected topos E over S, the associated topos GU is pointed. For U connected, there is even a canonical point. However, for the general coverings U, a consistent choice of these points cannot (a priori) be made if the base topos S is arbitrary. Moreover, the connected coverings U in E do not form a cofinal system of coverings in E. Either one of these reasons explains why is it that the limit topos Pi1cov(E)= limUGU need not be pointed when the base topos S is arbitrary and so, that a (prodiscrete localic) groupoid is needed. However, as shown in [CTGT], if S is taken to be Set, then the resulting notion of Set-Galois topos agrees with the the original notion. In particular, using non-constructive arguments, we show that Set-Galois toposes are always pointed, so that this assumption in [M] is superfluous. As shown in [BM] in the case where the base topos is Set, there is given a ``comparison map'' Pi1cov(E)--> Pi1path(E) from the coverings to the paths fundamental group constructions associated with a connected locally connected topos E, where the ``paths fundamental group'' of E is that of [MW]. This is easily seen using that locally constant objects have the unique path-lifting property stated in terms of a pullback condition. The existence of the comparison map in the case of an arbitrary base topos S can be shown as well, but it is based on the observations that the locally componentwise constant coverings (are complete spread objects in the sense of [BF] and) satisfy the (weaker) path-lifting property stated in terms of open surjections. In particular, if E is a Galois topos, then the notions of (i) a locally componentwise object, (ii) a complete spread object, and (iii) an object with an action by paths, are all equivalent for objects in E. A question that is still open concerns the equivalence of (just) the notions (i) and (ii) (given above) under suitable conditions on the topos E (more general than the ``locally paths simply connected as in [BM], which identifies all three), in particular, the question of when is the full subcategory U(E) of complete spread objects in E (called ``unramified maps'' in [FT]) is itself a topos bounded over S. As shown in [FT], U(E) is always, in some sense, a homotopy invariant of E. A sufficient condition for a complete spread object to be locally (componentwise) constant is a certain condition $\nabla$ given in [BF]. As stated in [BF], we ignore at this point if this condition is also necessary for the equivalence of U(E) with Pi1cov(E). [AGV] M. Artin, A. Grothendieck and J.-L. Verdier, Théorie des Topos et Cohomologie Etale des Schemas (SGA4), Lecture Notes in Mathematics 269, Springer-Verlag, Berlin-Heidelberg-New York, 1972. [AM] M. Artin and B. Mazur, Etale Homotopy, Lecture Notes in Mathematics 100, Springer-Verlag, Berlin-Heidelberg-New York, 1969. [BD] M. Barr and R. Diaconescu, On locally simply conected toposes and their fundamental groups. Cahiers Top. Geo. Diff. cat. 22-3 (1981)301-304. [B] M. Bunge, Classifying toposes and fundamental localic groupoids, in: R.A.G. Seely, ed., Category Theory '91, CMS Conf. Proc. 13 (1992) 75-96. [CTGT] M. Bunge and E. Dubuc, Constructive Theory of Galois Toposes (dvi, ps). [BF] M. Bunge and J. Funk, Spreads and the Symmetric Topos II, J. P ure Appl. Alg. 130 (1998) 49-84. [BM] M. Bunge and I. Moerdijk, On the construction of the Grothendieck fundamental group of a topos by paths, J. Pure Appl. Alg. 116 (1997) 99-113. [FT] J. Funk and E. Tymchatyn, Unramified Maps, Preprint April 2001. [M] I. Moerdijk, Prodiscrete groups and Galois toposes. Proc. Kon. Nederl. Akad. van Wetens. Series A, 92-2 (1989) 219-234. [MW] I. Moerdijk and G. Wraith, Connected locally connected toposes are path-connected, Trans. Amer. Math. Soc. 295 (1986) 849-859.