## Galois Groupoids and Covering Morphisms in Topos Theory

Citations: | 2 - 2 self |

### BibTeX

@MISC{Bunge_galoisgroupoids,

author = {Marta Bunge},

title = {Galois Groupoids and Covering Morphisms in Topos Theory},

year = {}

}

### OpenURL

### Abstract

The goals of this paper are (1) to compare the Galois groupoid that appears naturally in the construction of the fundamental groupoid of a topos E bounded over an arbitrary base topos S given by Bunge (1992), with the formal Galois groupoid defined by Janelidze (1990) in a very general setting given by a pair of adjoint functors, and (2) to discuss a good notion of covering morphism of a topos E over S which is general enough to include, in addition to the covering projections determined by the locally constant objects, also the unramified morphisms of topos theory given by those local homeomorphisms which are at the same time complete spreads in the sense of Bunge-Funk (1996, 1998).

### Citations

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(Show Context)
Citation Context ...equivalent to the (Galois) groupoid of automorphisms of the universal cover. We will now consider an unpointed version of a notion of a Galois topos introduced by Moerdijk [27] following Grothendieck =-=[-=-1]. Denition 4.7 A topos E bounded over S will be said to be a S - Galois topos if E is locally connected and has an internal site C of denition determined by objects A of E for which there exists a m... |

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Citation Context ... the base (elementary) topos S satises an \axiom of (small) stack completions", which was suggested by Lawvere in 1974 and which is known to hold at least of all Grothendieck toposes S , as shown=-= in [16, 23]-=-. We prove in section 4 (Galois groupoids and Galois toposes) that there is indeed an equivalence (not just a Morita equivalence) between the Galois groupoid of automorphisms of a universal cover of a... |

134 |
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(Show Context)
Citation Context ...e bag of points (\paquet des points"). Pointed connected Galois Groupoids and Covering Morphisms in Topos Theory 3 Galois toposes over S have been investigated by Moerdijk [27] following Grothend=-=ieck [18]-=- (see also [23]). We obtain here characterization theorems in a manner parallel to [27]; in particular we show that Galois toposes in our sense (and which correspond, modulo the intervention of locale... |

114 |
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Citation Context ...rating for GU since p is faithful. Hence GU is generated by a single S -Galois family. Finally, since p is a local homeomorphism hence open, and since it is surjective, it is of eective descent by [2=-=-=-4]. It follows that there is an equivalencesGU ' B(GU ) where GU = Aut(p) is a localic groupoid, discrete since p is (not just open but) a local homeomorphism. From the fact that represents p now fo... |

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24 |
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(Show Context)
Citation Context ...he Galois groupoid of automorphisms of the canonical \point" (in eect, a \bag of points") of the coverings fundamental group topos of E as dened in [8], with the formal Galois groupoid of Ja=-=nelidze [21]-=- in this setting. The notion of a locally constant object is central to these considerations; we begin then by investigating in section 2 (Locally constant objects in toposes) the connection between t... |

20 |
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Citation Context ...e classifying toposes of prodiscrete (localic) groupoids in S . In section 5 (Locally paths simply connected toposes over an arbitrary base) we recall the paths version of the fundamental group topos =-=[29, 30]-=-, give a constructive version of the existence of a comparison map from the paths to the coverings fundamental group toposes, and then we prove the equivalence of the comparison map under an assumptio... |

19 |
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Citation Context ...d of the pure theory associated with thesrst pair of adjoints in the 3-tuple e ! a e a e given by the (locally connected) geometric morphism e. We also show that, unlike the pure Galois theory of [2=-=1, 5]-=- in the general case, the Galois theory for toposes is implicit in the construction of the Galois groupoid, and that this is so on account of the presence of the third adjoint in the sequel of three d... |

14 |
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(Show Context)
Citation Context ...nd each geometric morphism ' : F ! E induces, by composition with the S - cocontinuous functor A (') = ' : E ! F , a functor Dist(') : Dist(F ) ! Dist(E ). For each E 2 TopS there is a topos M (E ) [=-=9] (th-=-e \symmetric topos") and an S -essential geometric morphisms: E ! M (E ) such that the pair hM (E ) ; i classies distributions on E . If ' : F ! E is S -essential, then composition with the S -co... |

9 |
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(Show Context)
Citation Context ...damental groupoid of a topos) we revisit the construction of the coverings fundamental groupoid of a locally connected topos E over a base topos S given in [8], an account of which is brie y given in =-=[13]-=- under the implicit assumption that the base topos S is Set. A new ingredient is the observation, not previously made explicit in either [8] or [13], that there are two groupoids involved, to wit, for... |

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6 |
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Citation Context ...ects in toposes) the connection between the notion of a locally constant object given in [12] (inspired by [8]), with that of Janelidze [21, 22] and, in passing, also with that of Barr and Diaconescu =-=[3]-=-, with which both agree if the topos E is connected and the base topos S is Set, but not in general. In section 3 (Stack completions and the fundamental groupoid of a topos) we revisit the constructio... |

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5 |
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Citation Context ...tack completions in the case of groupoids in S . Stacks of category objects (for the regular epimorpisms topology of S ) are discussed in [14], and stack completions shown to exist (and described) in =-=[15]-=-. We are here interested only in groupoids. For G a groupoid in S (regarded as a category object), its stack completion is given (up to equivalence) by any pairsA ; F >, with A an 8 Marta Bunge S -ind... |

4 |
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Citation Context ... the base (elementary) topos S satises an \axiom of (small) stack completions", which was suggested by Lawvere in 1974 and which is known to hold at least of all Grothendieck toposes S , as shown=-= in [16, 23]-=-. We prove in section 4 (Galois groupoids and Galois toposes) that there is indeed an equivalence (not just a Morita equivalence) between the Galois groupoid of automorphisms of a universal cover of a... |

4 |
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Citation Context ... ' up to iso 2-cells, also depend on the up to unique iso 2-cells. This and the conality of Cov(E ) implies that the 2-system is biltered (and biordered) in the sense of the following denition [25]. Denition 3.5 A 2-system fG i g of discrete groupoids, groupoid homomorphisms and 2-cells between them, indexed by a category C , is said to be biltered and biordered if 1. For any two groupoids G ... |

4 |
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(Show Context)
Citation Context ...ne ought work with a suitable bag of points (\paquet des points"). Pointed connected Galois Groupoids and Covering Morphisms in Topos Theory 3 Galois toposes over S have been investigated by Moer=-=dijk [27]-=- following Grothendieck [18] (see also [23]). We obtain here characterization theorems in a manner parallel to [27]; in particular we show that Galois toposes in our sense (and which correspond, modul... |

3 | A note on Barr-Diaconescu covering theory - Janelidze - 1992 |

3 |
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Citation Context ...ry products if for each pair of objects E and F in K , the canonical functors C (E + F ) ! C (E) C (F ) induced by the injections i : E ! E+F and j : F ! E+F are equivalences. In the terminology of [=-=26]-=-, C is an intensive quantity on K . Notice that the 2-extensivity of K (for K = TopS , or K = LTopS ) says that L preserves binary products. Notice also that there is a fully faithful pseudonatural tr... |

2 |
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(Show Context)
Citation Context ...[I ; G], given by a morphism f : I ! G 1 in S such that d 0 f = x and d 1 f = y, the natural transformation f : p Gx ! p G y. This assignment is easily checked to be S -indexed. It follows just as in =-=[7-=-] (Proposition 3.1) that Points S (BG) is a stack, using for this now that the S -essential surjections (a particular case of open surjections) are of eective descent in TopS [24]. It remains to verif... |

2 |
longue marche a travers la theorie de Galois, Tome I, Transcription d'un manuscrit inedit, par
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Citation Context ...the one hand, and its stack completion U which classies U-split torsors on the other. When the base topos is Set, this distinction dissapears and the two groupoids are usually identied in practice [19=-=, 13]. Th-=-is brings into consideration the desirability of assuming that the base (elementary) topos S satises an \axiom of (small) stack completions", which was suggested by Lawvere in 1974 and which is k... |

1 |
Internal Presheaf Toposes, Cahiers de
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(Show Context)
Citation Context ...! U) and is the classifying topos of the discrete localic groupoid Aut( ). Proof It follows from Lemma 2.5 that p is locally connected therefore S -essential; it is also surjective. Therefore, by [6] Proposition 1.2, its inverse image p is represented by a family : A ! g e ! U in the sense that there is a natural isomorphism p ' hom G U (E )=g e ! U ( ; e ! (U) ( )) where e ! (U) (... |

1 |
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(Show Context)
Citation Context ...note by L : K op ! CAT the pseudofunctor which assigns to a topos E the slice category LTopS =E and to a geometric morphism ' : F ! E the functor given by pulling back along '. We will say (following =-=[1-=-4]) that a subpseudofunctor C of L : K op ! CAT is a stack for a class of morphisms of eective descent in TopS , assumed to be closed under composition and pullbacks, if for any object E of K , given... |

1 |
Esquisse d'un programme, in: L. Scnepps and P. Lochak, Geometric Galois Actions. 1. Around Grothendieck's Esquisse d'un
- Grothendieck
- 1997
(Show Context)
Citation Context ...nnected or pointed, they come naturally equipped with a bag of points indexed by the connected components of a (non-connected) universal cover; this is in line with the view advocated by Grothendieck =-=[20] that-=- rather thansxing a single base point, one ought work with a suitable bag of points (\paquet des points"). Pointed connected Galois Groupoids and Covering Morphisms in Topos Theory 3 Galois topos... |

1 |
Continuous and inverse limits of toposes, Compositio Math
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- 1986
(Show Context)
Citation Context ... groupoid homomorphisms g : GU ! G V , for each . Furthermore, since the geometric morphisms ' are connected (and locally connected), the homomorphisms g are full and essentially surjective by [28]. From the above diagram follows, by virtue of the connectedness of U : E ! GU , that the geometric morphisms ' depend on up to unique iso 2cell. In particular, the groupoid homomorphisms g , w... |