## FUNDAMENTAL PUSHOUT TOPOSES

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@MISC{Bunge_fundamentalpushout,

author = {Marta Bunge},

title = {FUNDAMENTAL PUSHOUT TOPOSES},

year = {}

}

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### Abstract

Abstract. The author [2, 5] introduced and employed certain ‘fundamental pushout toposes ’ in the construction of the coverings fundamental groupoid of a locally connected topos. Our main purpose in this paper is to generalize this construction without the local connectedness assumption. In the spirit of [16, 10, 8] we replace connected components by constructively complemented, or definable, monomorphisms [1]. Unlike the locally connected case, where the fundamental groupoid is localic prodiscrete and its classifying topos is a Galois topos, in the general case our version of the fundamental groupoid is a locally discrete progroupoid and there is no intrinsic Galois theory in the sense of [19]. We also discuss covering projections, locally trivial, and branched coverings without local connectedness by analogy with, but also necessarily departing from, the locally connected case [13, 11, 7]. Throughout, we work abstractly in a setting given axiomatically by a category V of locally discrete locales that has as examples the categories D of discrete locales, and Z of zero-dimensional locales [9]. In this fashion we are led to give unified and often simpler proofs of old theorems in the locally connected case, as well as new ones without that assumption.

### Citations

57 |
Equality in hyperdoctrines and comprehension schema as an adjoint functor
- Lawvere
- 1970
(Show Context)
Citation Context ...aid to be a V-fibration if its domain F is a V-determined topos and furthermore ' equals the support of the V-distribution u = F!A~* . The following is the analogue of the comprehensive factorization =-=[22, 29, 7]-=- in the V-setting. It was established in [9]. 4.5. Theorem. Any geometric morphism A~ : F // E with a V-determined domain F admits a factorization F E A~ ##GGGG GGGGF X t^ // X E '--wwwwww ww (13) int... |

54 | Lehrbuch der Topologie - Threlfall - 1934 |

33 | Abstract families and the adjoint functor theorem - Paré, Schumacher - 1978 |

28 |
Pure Galois theory in categories
- Janelidze
- 1990
(Show Context)
Citation Context ...rete and its classifying topos is a Galois topos, in the general case our version of the fundamental groupoid is a locally discrete progroupoid and there is no intrinsic Galois theory in the sense of =-=[19]-=-. We also discuss covering projections, locally trivial, and branched coverings without local connectedness by analogy with, but also necessarily departing from, the locally connected case [13, 11, 7]... |

27 |
and Radu Diaconescu. Atomic toposes
- Barr
- 1980
(Show Context)
Citation Context ...is to generalize this construction without the local connectedness assumption. In the spirit of [16, 10, 8] we replace connected components by constructively complemented, or definable, monomorphisms =-=[1]-=-. Unlike the locally connected case, where the fundamental groupoid is localic prodiscrete and its classifying topos is a Galois topos, in the general case our version of the fundamental groupoid is a... |

22 |
The classifying topos of a continuous groupoid
- Moerdijk
- 1988
(Show Context)
Citation Context ...nnected surjection, it is of effective descent. In other words, \Phi U is an equivalence. Consider now a similar diagram for the morphism pU : ShS (E!U ) // GU (E ). By methods of classifying toposes =-=[24]-=-, we extract from it an etale complete groupoid GU in GU :s196 MARTA BUNGE G2 p // m // q // G1 d // c // G0 ioo Denote by D (pU )the descent category for Ker(pU ) and by PU : D (pU ) // GU (E ) the c... |

12 |
Exponentiability and Single Universes
- Bunge, Niefeld
- 1966
(Show Context)
Citation Context ...nse of [19]. We also discuss covering projections, locally trivial, and branched coverings without local connectedness by analogy with, but also necessarily departing from, the locally connected case =-=[13, 11, 7]-=-. Throughout, we work abstractly in a setting given axiomatically by a category V of locally discrete locales that has as examples the categories D of discrete locales, and Z of zero-dimensional local... |

10 |
H.: Covering spaces with singularities
- Fox
- 1957
(Show Context)
Citation Context ...tion of the coverings fundamental groupoid of a locally connected topos. Our main purpose in this paper is to generalize this construction without the local connectedness assumption. In the spirit of =-=[16, 10, 8]-=- we replace connected components by constructively complemented, or definable, monomorphisms [1]. Unlike the locally connected case, where the fundamental groupoid is localic prodiscrete and its class... |

9 | On the constructin of the Grothendieck fundamental group of a topos by paths - Bunge, Moerdijk - 1997 |

7 |
Classifying toposes and fundamental localic groupoids
- Bunge
(Show Context)
Citation Context ...al locales [9]. In this fashion we are led to give unified and often simpler proofs of old theorems in the locally connected case, as well as new ones without that assumption. Introduction The author =-=[2, 5]-=- introduced certain ‘fundamental pushout toposes’ in the construction and study of the prodiscrete fundamental groupoid of a locally connected topos. The main purpose of this paper is to generalize th... |

7 | Factorization Theorems for Geometric Morphisms, II, in ”Categorical Aspects of Topology and Analysis - Johnstone - 1989 |

6 |
Singular coverings of toposes
- Bunge, Funk
(Show Context)
Citation Context ...nse of [19]. We also discuss covering projections, locally trivial, and branched coverings without local connectedness by analogy with, but also necessarily departing from, the locally connected case =-=[13, 11, 7]-=-. Throughout, we work abstractly in a setting given axiomatically by a category V of locally discrete locales that has as examples the categories D of discrete locales, and Z of zero-dimensional local... |

6 |
Intensive and extensive quantities, Notes for the lectures given at the workshop
- Lawvere
- 1983
(Show Context)
Citation Context ...into a V-initial geometric morphism followed by a V-fibration. The V-fibrations are an abstraction of the complete spreads [8], and are the geometric counterpart of the V-valued Lawvere distributions =-=[23]-=-. We also establish new closure and stability theorems for the factors of the comprehensive V-factorization. In $ 5 we introduce notions of locally trivial V-coverings and branched V-coverings suitabl... |

4 |
The fundamental localic groupoid of a topos
- Kennison
- 1992
(Show Context)
Citation Context ...nitions and results of [2, 5] to the V-context. For a Vdetermined topos E , and a cover U in E , the localic V-reflection of E /U determines a `fundamental V-pushout' topos GU (E ). We use results of =-=[21, 15]-=- in order to obtain, from the system of toposes GU (E ) indexed by a generating category of covers in E , a limit topos. Our results show that this topos (or its corresponding progroupoid) represents ... |

3 | Galois Groupoids and Covering Morphisms in Topos Theory, in: G. Janelidze et al
- Bunge
- 2004
(Show Context)
Citation Context ...al locales [9]. In this fashion we are led to give unified and often simpler proofs of old theorems in the locally connected case, as well as new ones without that assumption. Introduction The author =-=[2, 5]-=- introduced certain ‘fundamental pushout toposes’ in the construction and study of the prodiscrete fundamental groupoid of a locally connected topos. The main purpose of this paper is to generalize th... |

3 |
Quasicomponents in topos theory: the hyperpure, complete spread factorization
- Bunge, Funk
- 2007
(Show Context)
Citation Context ...actorization theorem [9], modeled on the pure, complete spread factorization in case of geometric morphisms with a locally connected domain [7], and on the hyperpure, complete spread factorization of =-=[8]-=- for geometric morphisms with a definable dominance domain. The following is a generalization of the notion of a Lawvere distribution on a topos [23]. 4.1. Definition. A V–distribution on a topos E is... |

3 |
Indexed categories and generated topologies
- Par'e
- 1980
(Show Context)
Citation Context ...)1 (E /S) = lim{GU (E /S) | U 2 Cov(E )}, for the pushout toposes GU (E /S). We may always consider the largest topology in any Grothendieck topos E for which certain objects of the topos are sheaves =-=[27]-=-. A monomorphism m : A // // B is dense for this largest topology iff B*A // B*Am (transpose of the projection) is an isomorphism. We apply this idea, as in [7], to the class of objects of the form E*... |

2 | An application of descent to a classification theorem for toposes - Bunge - 1990 |

2 |
Universal covering localic toposes
- Bunge
- 1992
(Show Context)
Citation Context ...inition of the coverings fundamental groupoid of a locally connected topos [2] became a useful tool in the comparison theorems between the coverings and the paths versions of the fundamental groupoid =-=[4, 12]-=-. Such a comparison could not be attempted in this context without a prior generalization of the paths fundamental groupoid to the non-locally connected case. The latter might prove to be an interesti... |

2 |
Quasi locally connected toposes, Theory and Applications of Categories 18:08
- Bunge, Funk
- 2007
(Show Context)
Citation Context ...hroughout, we work abstractly in a setting given axiomatically by a category V of locally discrete locales that has as examples the categories D of discrete locales, and Z of zero-dimensional locales =-=[9]-=-. In this fashion we are led to give unified and often simpler proofs of old theorems in the locally connected case, as well as new ones without that assumption. Introduction The author [2, 5] introdu... |

2 | A construction of 2-filtered bicolimits of categories
- Dubuc, Street
- 2006
(Show Context)
Citation Context ...initions and results of [2, 5] to the V-context. For a Vdetermined topos E , and a cover U in E , the localic V-reflection of E /U determines a ‘fundamental V-pushout’ topos GU(E ). We use results of =-=[21, 15]-=- in order to obtain, from the system of toposes GU(E ) indexed by a generating category of covers in E , a limit topos. Our results show that this topos (or its corresponding progroupoid) represents f... |

2 |
Shape theory and covering spaces
- Fox
- 1974
(Show Context)
Citation Context ...rings. In connection with the locally constant coverings, we comment on an alternative construction of the fundamental progroupoid of a (Grothendieck) topos given in [14] and inspired by shape theory =-=[17, 18]-=-. In § 4 we first review the V-comprehensive factorization [9] of a geometric morphism. with a V-determined domain into a V-initial geometric morphism followed by a V-fibration. The V-fibrations are a... |

2 |
Fundamental pro-groupoids and covering projections. Fundamenta Mathematicae 156
- Hernández-Paricio
- 1998
(Show Context)
Citation Context ...rings. In connection with the locally constant coverings, we comment on an alternative construction of the fundamental progroupoid of a (Grothendieck) topos given in [14] and inspired by shape theory =-=[17, 18]-=-. In § 4 we first review the V-comprehensive factorization [9] of a geometric morphism. with a V-determined domain into a V-initial geometric morphism followed by a V-fibration. The V-fibrations are a... |

2 |
Completing a spread (in the sense of Fox) without local connectedness
- Michael
- 1963
(Show Context)
Citation Context ...pos. The main purpose of this paper is to generalize this construction without the local connectedness assumption. The key to this program is the theory of spreads in topos theory [7], originating in =-=[16, 25]-=-, and motivated therein by branched coverings. Our main tools are the factorization theorems [10] and [8] for geometric morphisms whose domains are definable dominances, a notion which states that the... |

2 | General Topology. Addison-Wesley Series in Mathematics - Willard - 1970 |

1 | Branched coverings over a quasi locally connected topos - Bunge - 2007 |

1 | The Michael completion of a topos spread - Bunge, Funk, et al. - 2002 |

1 |
Kampen theorems for toposes
- Van
- 2003
(Show Context)
Citation Context ...nse of [19]. We also discuss covering projections, locally trivial, and branched coverings without local connectedness by analogy with, but also necessarily departing from, the locally connected case =-=[13, 11, 7]-=-. Throughout, we work abstractly in a setting given axiomatically by a category V of locally discrete locales that has as examples the categories D of discrete locales, and Z of zero-dimensional local... |

1 | The fundamental progroupoid of a general topos
- Dubuc
(Show Context)
Citation Context ... of [11] for locally constant coverings. In connection with the locally constant coverings, we comment on an alternative construction of the fundamental progroupoid of a (Grothendieck) topos given in =-=[14]-=- and inspired by shape theory [17, 18]. In § 4 we first review the V-comprehensive factorization [9] of a geometric morphism. with a V-determined domain into a V-initial geometric morphism followed by... |