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2-Filteredness and The Point of Every Galois Topos

Eduardo J. Dubuc
2008 Applied Categorical Structures  
Then we show that the assignment, to each Galois object A, of the category D A of connected locally constant objects trivialized by A (definition 3.1), determines a 2-functor into the category of categories  ...  A locally connected topos is a Galois topos if the Galois objects generate the topos.  ...  The category A of Galois objects in a connected locally connected topos becomes a (inversely) 2-filtered 2-category (in the sense of θ uu = id, θ wv • θ vu = θ wu , (thus θ uv = θ −1 vu ), θ vu θ sr =  ... 
doi:10.1007/s10485-008-9145-4 fatcat:zec7lz544jad3ap32maudirehq

2-filteredness and the point of every Galois topos [article]

Eduardo J. Dubuc
2007 arXiv   pre-print
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 bi-limits of  ...  A locally connected topos is a Galois topos if the Galois objects generate the topos.  ...  The category A of Galois objects in a connected locally connected topos becomes a (inversely) 2-filtered 2-category (in the sense of θ uu = id, θ wv • θ vu = θ wu , (thus θ uv = θ −1 vu ), θ vu θ sr =  ... 
arXiv:0801.0010v1 fatcat:7st47bbvdzahxgtwsreys436aa

The Étale Fundamental Groupoid as a Terminal Costack [article]

Ilia Pirashvili
2016 arXiv   pre-print
Let X be a noetherian scheme. We denote by Π_1(X) the fundamental groupoid. In this paper we prove that the assignments UΠ_1(U) is the 2-terminal costack over the site of étale coverings of X.  ...  Let C be a finitely connected Galois category.  ...  Let C be a finitely-connected Galois category.  ... 
arXiv:1412.5473v2 fatcat:x57m26nblnferarbfuejaiofge

Galois connections categorically

H. Herrlich, M. Hušek
1990 Journal of Pure and Applied Algebra  
If partially ordered classes are interpreted in the standard way as concrete categories over a category T having precisely one morphism, then Galois connections of the second kind are just Galois connections  ...  A,B be concrete categories over a category X with small fibers, B be cocomplete or A be complete. Then every pair of adjoint concrete jiinctors forms a Galois connection of the fourth kind.  ...  This last result demonstrates how difficult it is to provide natural examples of adjoint situations for concrete functors which are not Galois connections.  ... 
doi:10.1016/0022-4049(90)90141-4 fatcat:z6fw53docvcallrjpmqt3hnlgu

Pre-Galois Connection on Coalgebras for Generic Component Refinement

Meng Sun
2008 Electronical Notes in Theoretical Computer Science  
The technique of Galois connections has been applied successfully in many areas of computer science.  ...  We propose a notion of pre-Galois connection in the context of coalgebras for refinement of state-based software components which takes into consideration not only the refinement ordering but also the  ...  Acknowledgement The work reported in this paper is supported by the National Natural Science Foundation of China under Grant No. 60473056, and a grant from the GLANCE funding program of the Dutch National  ... 
doi:10.1016/j.entcs.2008.03.094 fatcat:5nhvcyhng5ewhiz3arckoyksbi

Galois and universal universal coverings of linear categories and fibre products [article]

Claude Cibils, Maria Julia Redondo, Andrea Solotar
2010 arXiv   pre-print
Let k be a commutative ring. We study the behaviour of coverings of k-categories through fibre products and find a criterion for a covering to be Galois or universal.  ...  a universal covering since there exist at least two connected components of the category of connected gradings.  ...  Next we recall the definition of a Galois covering: Definition 2.10. A covering F : C −→ B of k-categories is a Galois covering if C is connected and if Aut 1 (F ) acts transitively on some fibre.  ... 
arXiv:1012.3434v1 fatcat:ky5pdyma4zdrbdkvgxsdicc2be

A simplified categorical approach to several Galois theories [article]

David Blázquez-Sanz, Carlos A. Marín Arango, Juan Felipe Ruiz Castrillon
2020 arXiv   pre-print
Namely, a Galois structure for an epimorphism π M→ B in some category C is the action of a group object that gives to M the structure of principal homogeneous space in the relative category C_B.  ...  We discuss the concept of Galois structure and Galois epimorphism in a general setting.  ...  Dubuc and Yuri Poveda who gently spent some time discussing the topic with the first author during a pleasant brief encounter in Pereira, Colombia.  ... 
arXiv:1805.11062v4 fatcat:f3ookavunfh7vjh3c6fm2p2yvq

Gradings, smash products and Galois coverings of a small category [article]

Claude Cibils, John MacQuarrie
2012 arXiv   pre-print
In this paper we develop the theory of coverings of a small connected category B.  ...  We show that the category of Galois coverings of B is equivalent to the category of Galois coverings of its fundamental groupoid.  ...  Introduction In this paper we consider a small connected category B and its category of Galois coverings.  ... 
arXiv:1203.5905v1 fatcat:dcdlyswjnfeqzmr6eju7727ffm

The Intrinsic Fundamental Group of a Linear Category

Claude Cibils, María Julia Redondo, Andrea Solotar
2010 Algebras and Representation Theory  
We provide an intrinsic definition of the fundamental group of a linear category over a ring as the automorphism group of the fibre functor on Galois coverings.  ...  If the universal covering exists, we prove that this group is isomorphic to the Galois group of the universal covering.  ...  A covering F : C −→ B of k-categories is a Galois covering if C is connected and if Aut 1 F acts transitively on some fibre.  ... 
doi:10.1007/s10468-010-9263-1 fatcat:7cxb22sjk5hdvg3lgdnzjs2bjm

On the representation theory of Galois and atomic topoi

Eduardo J. Dubuc
2004 Journal of Pure and Applied Algebra  
Grothendieck conceived Galois theory as the axiomatic characterization of the classifying topos of a progroup in terms of a representation theorem for pointed Galois Topoi.  ...  Classical Galois theory corresponds to discrete groups (the point is essential), and the representation theorem can be proved by elementary category-theory.  ...  Bunge for many fruitful and motivating discussions on Galois topoi and its points or lack of them, and to A.  ... 
doi:10.1016/s0022-4049(03)00141-5 fatcat:s6te2ekvqbcj3ofqpfpvosbdca

Formal Contexts, Formal Concept Analysis, and Galois Connections

Jeffrey T. Denniston, Austin Melton, Stephen E. Rodabaugh
2013 Electronic Proceedings in Theoretical Computer Science  
Formal concept analysis (FCA) is built on a special type of Galois connections called polarities.  ...  We present new results in formal concept analysis and in Galois connections by presenting new Galois connection results and then applying these to formal concept analysis.  ...  Acknowledgements The second author wishes to thank Dave Schmidt for his friendship, encouragement, and research advice and collaboration, especially, when the second author was making a career change from  ... 
doi:10.4204/eptcs.129.8 fatcat:qjrolph3e5ay5ba5kxkyn5ezym

Full and convex linear subcategories are incompressible

Claude Cibils, Maria Julia Redondo, Andrea Solotar
2013 Proceedings of the American Mathematical Society  
Consider the intrinsic fundamental group \'a la Grothendieck of a linear category using connected gradings.  ...  We start by proving the functoriality of the intrinsic fundamental group with respect to full subcategories, based on the study of the restriction of connected gradings.  ...  Note that the evident functor F X : B#X −→ B is a Galois covering. Let B be a connected k-category with a fixed object b 0 . The category Gal(B, b 0 ) has as objects the Galois coverings of B.  ... 
doi:10.1090/s0002-9939-2013-11470-x fatcat:2lw3pwgofbdcbeleb7lstchjxq

THH and base-change for Galois extensions of ring spectra

Akhil Mathew
2017 Algebraic and Geometric Topology  
Given a faithful Galois extension A → B of ring spectra, we consider whether the map THH(A) ⊗_A B → THH(B) is an equivalence.  ...  We treat the question of base-change in THH for faithful Galois extensions of ring spectra in the sense of Rognes.  ...  Let X be a simply connected, pointed space, and let A → B be a faithful G-Galois extension of E ∞ -rings. In this case, the map of E ∞ -rings (X ⊗ A) ⊗ A B → X ⊗ B, is an equivalence.  ... 
doi:10.2140/agt.2017.17.693 fatcat:bcj3yaf2vvhqpkuoyyucdoadnm

On the representation theory of Galois and Atomic Topoi [article]

Eduardo J. Dubuc
2003 arXiv   pre-print
We establish a hierarchy of connected topoi: [1. essentially pointed Atomic = locally simply connected], [2. proessentially pointed Atomic = pointed Galois], [3. pointed Atomic].  ...  We introduce the concept of "proessential point" and use it to give a new characterization of pointed Galois topoi.  ...  Any filtered inverse limit of Galois topoi and connected locally connected geometrical morphisms is a Galois topos. Proof.  ... 
arXiv:math/0208222v3 fatcat:d6pbohpyxfai3jxnwhwlmnjrjq

Separable morphisms of simplicial sets [article]

Dimitri Chikhladze
2006 arXiv   pre-print
We show that the class of separable morphisms in the sense of G. Janelidze and W. Tholen in the case of Galois structure of second order coverings of simplicial sets due to R. Brown and G.  ...  Janelidze coincides with the class of covering maps of simplicial sets.  ...  Note that this structure is a special case of the example from the section 1, when A is the category of connected simplicial sets.  ... 
arXiv:math/0606634v1 fatcat:qfnfhv57xnegtaewqg6einiyse
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