## On the representation theory of Galois and atomic topoi

Venue: | J. Pure Appl. Algebra |

Citations: | 5 - 2 self |

### BibTeX

@ARTICLE{Dubuc_onthe,

author = {Eduardo J. Dubuc},

title = {On the representation theory of Galois and atomic topoi},

journal = {J. Pure Appl. Algebra},

year = {},

pages = {233--275}

}

### OpenURL

### Abstract

introduction

### Citations

194 |
Topos Theory
- Johnstone
- 1977
(Show Context)
Citation Context ...s yields a prodiscrete localic group as the localic group of automorphisms of the point. This is the method introduced and developed by Grothendieck in SGA1 [12] to treat the profinite case (see also =-=[14]-=-, and [10] for a detailed and elementary description of all this). Later, in a series of commented exercises in SGA4 [1] he gave guidelines to treat the general prodiscrete case by means of locally co... |

119 |
An extension of the Galois theory of Grothendieck
- Joyal, Tierney
- 1984
(Show Context)
Citation Context ...npointed theory also applies in the presence of points, but it yields a different groupoid that the pointed theory. We compare these groupoids in section 3.4. In their seminal paper on Galois Theory, =-=[15]-=- (after Grothendieck’s [12]), Joyal and Tierney develop an unpointed theory of representation for a completely arbitrary topos in terms of localic groupoids, which culminates with their fundamental th... |

27 |
and Radu Diaconescu. Atomic toposes
- Barr
- 1980
(Show Context)
Citation Context ...subcategory such that with the canonical topology is a pointed site C F −→ S for E, E = C ∼ , F = f ∗ |C. Then: 1) If E is a pointed connected atomic topos, a site as above can be chosen so that (see =-=[3]-=-): i) Every arrow Y −→ X in C is an strict epimorphism. ii) For every X ∈ C FX ̸= ∅. iii) F preserves strict epimorphisms. iv) The diagram of F, ΓF, is a cofiltered category. 2) Given any pointed site... |

21 |
The classifying topos of a continuous groupoid
- Moerdijk
- 1988
(Show Context)
Citation Context ... µ ∗ [<x | y>] ≤ µ ∗ [<fx | fy>]. Given any localic groupoid G, the category of discrete G spaces is defined in an standard way in [15] VIII, 3, and proved therein to be a topos, denoted BG (see also =-=[18]-=- 5.2). Consider the enrichment lS of the category of sets over the category of localic spaces, 1.2.3. It is straightforward to check the following: Proposition 1.3.6. Given any localic groupoid with d... |

9 |
On the constructin of the Grothendieck fundamental group of a topos by paths
- Bunge, Moerdijk
- 1997
(Show Context)
Citation Context ... by the same reasons that they loose their arrows (as we explain explicitly in this paper). It seems natural then to develop a pointless theory also at the level of objects. M. Bunge in [6] (see also =-=[8]-=-) develop an unpointed theory for galois topoi following the inverse limit techniques implicit in [12] and [1] and made explicit in [19]. Around the same time, J. Kennison [16] also developed an equiv... |

7 |
Classifying toposes and fundamental localic groupoids
- Bunge
(Show Context)
Citation Context ...ization of pointed Galois topoi as the classifying topoi of prodiscrete localic groups. We develop the theory of locally constant objects as defined in [1] Ex IX in an apendix-section 5. We take from =-=[6]-=- the idea of presenting the topos of objects split by a cover as a push-out topos, and show how the existence of galois closures follows automatically. Connected groupoids are considered already in [1... |

6 |
On locally simply connected toposes and their fundamental groups
- Barr, Diaconescu
- 1981
(Show Context)
Citation Context ...orphic to the the discrete group Aut(A) op , where A is any representing object. In this case the object A is a universal covering and the topos E in 3.0.8 is said to be locally simple connected (see =-=[5]-=-, were this notion was first investigated in detail in the topos setting). Proposition 3.1.1. If A is a representing object of F, every arrow X f −→ A is an isomorphism. In particular, every endomorph... |

4 |
On the Galois theory of Grothendieck
- Dubuc, Vega
- 1998
(Show Context)
Citation Context ...s theory is conceived as the axiomatic characterization of the classifying pretopos of a profinite group G. The fundamental theorem takes the form of a representation theorem for Galois pretopos (see =-=[10]-=- for the explicit interpretation of this work in terms of filtered unions of categories - the link to filtered inverse limits of topoi - and its relation to classical Galois’s galois theory). An impor... |

4 | Galois Theory. Graduate Texts - Edwards - 1984 |

4 |
The fundamental localic groupoid of a topos
- Kennison
- 1992
(Show Context)
Citation Context ...osé III, 4) that the arrow C ∼ −→ BG is an equivalence. We comment that the corresponding theorem for filtered inverse limits of discrete groupoids has been stated and proved with do care by Kennison =-=[16]-=- 4.8. In the statement of the result it is necessary to assume that the transition morphisms are composable onto, (see [16]). This takes care of the necessary surjectivity of the system at the level o... |

4 |
Prodiscrete groups and Galois toposes
- Moerdijk
- 1989
(Show Context)
Citation Context ...s an isomorphism (where γ is the morphism E −→ S). Clearly, a galois object A is A-split, thus locally constant. After Grothendieck’s “Categories Galoisiennes” of [12] and Moerdiejk “Galois Topos” of =-=[19]-=-, we state the following definition: Definition 5.2.1. A Galois Topos is a connected locally connected topos generated by its galois objects. Notice that unlike [12] and [19] we do not require the top... |

3 | Localic Galois theory - Dubuc - 2003 |

2 |
Continuous Fibrations and Inverse
- Moerdijk
- 1986
(Show Context)
Citation Context ... said to be a locally connected morphism if the topos E considered as an F-Topos is locally connected. This relative version was introduced in [4] under the name F-essential, see also the appendix of =-=[17]-=-. Recall that a connected atomic topos is a connected, locally connected and boolean topos. For atomic topoi and atomic sites the reference is [3], see also [15]. A covering of a topos E is a geometri... |

2 |
Localic Groups, Cahiers de Top. et Geom. Diff. Vol XXII-1
- Wraith
- 1981
(Show Context)
Citation Context ...responding object in the inf-lattice and in the locale. Remark that this object is the finite infimun of the (xi, yi) (see [9] for details, there for the case X = Y ). G. Wraith in an inspiring paper =-=[20]-=- defines the locales of functions and of bijections between two sets X and Y by considering the appropriate generators and relations. In our context these relations become covers in the free inf-latti... |

1 |
Abstract Galois Theory II
- Barr
- 1982
(Show Context)
Citation Context ...). If the topos is not connected this “single set” concept is clearly not equivalent (we can have a different set for each connected component of 1), and not the right one, as it has been observed in =-=[2]-=-. We comment to the reader interested in the relative theory over an arbitrary base topos that the connectivity argument depends on the excluded middle. Based on this, even when the topos is connected... |

1 | A Van Kampen theorem for toposes
- Bunge, Lack
- 2001
(Show Context)
Citation Context ...of the form E/X −→ E, with X a locally constant object. 5.1. locally constant objects. We recall now the definition of locally constant object in an arbitrary topos given in SGA4, Exposé IX (see also =-=[7]-=- where this definition is considered over an arbitrary base topos). γ Definition 5.1.1. An object X of a topos E −→ S is said to be U-split, for a cover U = {Ui}i∈I (i.e. epimorphic family Ui → 1), if... |

1 |
Longe Marche a travers la theorie de Galois (1980-81), edited by Jean Malgoire
- Grothendieck, La
- 1995
(Show Context)
Citation Context ...es. Because of this in general it is enough to prove results for the connected case. In [12] V 9. the non connected theory is left to the reader (“Nous en laissons le detail au lecteur”). However, in =-=[13]-=-, locally connected (but not connected) Galois topoi are considered under the name “Topos Multigaloisiennes”. There the topos are supposed to have enough points. Definition 4.0.5. A Multigalois topos ... |