## 2-FILTEREDNESS AND THE POINT OF EVERY GALOIS TOPOS (801)

### BibTeX

@MISC{Dubuc8012-filterednessand,

author = {Eduardo J. Dubuc},

title = {2-FILTEREDNESS AND THE POINT OF EVERY GALOIS TOPOS},

year = {801}

}

### OpenURL

### Abstract

Abstract. A locally connected topos is a Galois topos if the Galois objects generate the topos. We show that the full subcategory of Galois objects in any connected locally connected topos is an inversely 2-filtered 2-category, and as an application of the construction of 2-filtered bilimits of topoi, we show that every Galois topos has a point. introduction. Galois topoi (definition 1.5) arise in Grothendieck’s Galois theory of locally connected topoi. They are an special kind of atomic topoi. It is well known that atomic topoi may be pointless [6], however, in this paper we show that any Galois topos has points. We show how the full subcategory of Galois objects (definition 1.2) in any connected locally connected topos E has an structure of 2-filtered 2-category (in the sense of [3]). Then we show that the assignment, to each Galois object A, of the category DA of connected locally constant objects trivialized by

### Citations

4 | On the representation theory of Galois and atomic topoi
- Dubuc
(Show Context)
Citation Context ...s A, B, in a connected locally connected topos E, any connected component of the product A × B is a connected locally constant object. It follows from the existence of Galois closure (see for example =-=[2]-=- A.1.4) that there is a Galois object C and morphisms C → A, C → B. The full subcategory A of Galois objects fails to be (inversely) filtered because, clearly, different morphisms A s u u v � B betwee... |

4 |
Prodiscrete groups and Galois toposes
- Moerdijk
- 1989
(Show Context)
Citation Context ... 2-category given in [3] and verify the assertion. □ f�� � �� 2-FILTEREDNESS AND THE POINT OF EVERY GALOIS TOPOS 3 After Grothendieck “Categories Galoisiennes” of [5] and Moerdiejk “Galois Topos” of =-=[7]-=-, we state the following definition: 1.5. Definition. A Galois Topos is a connected locally connected topos generated by its Galois objects, or, equivalently, such that any connected object is covered... |

2 | A construction of 2-filtered bicolimits of categories
- Dubuc, Street
- 2006
(Show Context)
Citation Context ... any Galois topos has points. We show how the full subcategory of Galois objects (definition 1.2) in any connected locally connected topos E has an structure of 2-filtered 2-category (in the sense of =-=[3]-=-). Then we show that the assignment, to each Galois object A, of the category DA of connected locally constant objects trivialized by A (definition 3.1), determines a 2-functor into the category of ca... |

1 |
A construction of 2-filtered bi-limits of Topoi
- Dubuc, Yuhjtman
(Show Context)
Citation Context ...of categories. Furthermore, this 2-system becomes a pointed 2-system of pointed sites (considering the topology in which each single arrow is a cover). By the results on 2-filtered bi-limits of topoi =-=[4]-=-, it follows that, if E is a Galois topos, then it is the bi-limit of this system, and thus, it has a point. context. Throughout this paper S = Sets denotes the topos of sets. All topoi E are assumed ... |

1 |
Full continuous embeddings of toposes, Trans
- Makkai
- 1982
(Show Context)
Citation Context ...oduction. Galois topoi (definition 1.5) arise in Grothendieck’s Galois theory of locally connected topoi. They are an special kind of atomic topoi. It is well known that atomic topoi may be pointless =-=[6]-=-, however, in this paper we show that any Galois topos has points. We show how the full subcategory of Galois objects (definition 1.2) in any connected locally connected topos E has an structure of 2-... |