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2-Filteredness and The Point of Every Galois Topos
2008
Applied Categorical Structures
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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]
2007
arXiv
pre-print
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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]
2016
arXiv
pre-print
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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
1990
Journal of Pure and Applied Algebra
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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
2008
Electronical Notes in Theoretical Computer Science
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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]
2010
arXiv
pre-print
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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]
2020
arXiv
pre-print
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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]
2012
arXiv
pre-print
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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
2010
Algebras and Representation Theory
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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
2004
Journal of Pure and Applied Algebra
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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
2013
Electronic Proceedings in Theoretical Computer Science
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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
2013
Proceedings of the American Mathematical Society
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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
2017
Algebraic and Geometric Topology
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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]
2003
arXiv
pre-print
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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]
2006
arXiv
pre-print
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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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