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## Filters

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2-Filteredness and The Point of Every Galois Topos

2008
*
Applied Categorical Structures
*

Then we show that the assignment, to each

doi:10.1007/s10485-008-9145-4
fatcat:zec7lz544jad3ap32maudirehq
*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 = ...##
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2-filteredness and the point of every Galois topos
[article]

2007
*
arXiv
*
pre-print

We show that the full subcategory

arXiv:0801.0010v1
fatcat:7st47bbvdzahxgtwsreys436aa
*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 = ...##
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The Étale Fundamental Groupoid as a Terminal Costack
[article]

2016
*
arXiv
*
pre-print

Let X be

arXiv:1412.5473v2
fatcat:x57m26nblnferarbfuejaiofge
*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*. ...##
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Galois connections categorically

1990
*
Journal of Pure and Applied Algebra
*

If partially ordered classes are interpreted in the standard way as concrete

doi:10.1016/0022-4049(90)90141-4
fatcat:z6fw53docvcallrjpmqt3hnlgu
*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*. ...##
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Pre-Galois Connection on Coalgebras for Generic Component Refinement

2008
*
Electronical Notes in Theoretical Computer Science
*

The technique

doi:10.1016/j.entcs.2008.03.094
fatcat:5nhvcyhng5ewhiz3arckoyksbi
*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 ...##
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Galois and universal universal coverings of linear categories and fibre products
[article]

2010
*
arXiv
*
pre-print

Let k be

arXiv:1012.3434v1
fatcat:ky5pdyma4zdrbdkvgxsdicc2be
*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. ...##
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A simplified categorical approach to several Galois theories
[article]

2020
*
arXiv
*
pre-print

Namely,

arXiv:1805.11062v4
fatcat:f3ookavunfh7vjh3c6fm2p2yvq
*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. ...##
###
Gradings, smash products and Galois coverings of a small category
[article]

2012
*
arXiv
*
pre-print

In this paper we develop the theory

arXiv:1203.5905v1
fatcat:dcdlyswjnfeqzmr6eju7727ffm
*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. ...##
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The Intrinsic Fundamental Group of a Linear Category

2010
*
Algebras and Representation Theory
*

We provide an intrinsic definition

doi:10.1007/s10468-010-9263-1
fatcat:7cxb22sjk5hdvg3lgdnzjs2bjm
*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. ...##
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On the representation theory of Galois and atomic topoi

2004
*
Journal of Pure and Applied Algebra
*

Grothendieck conceived

doi:10.1016/s0022-4049(03)00141-5
fatcat:s6te2ekvqbcj3ofqpfpvosbdca
*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*. ...##
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Formal Contexts, Formal Concept Analysis, and Galois Connections

2013
*
Electronic Proceedings in Theoretical Computer Science
*

Formal concept analysis (FCA) is built on

doi:10.4204/eptcs.129.8
fatcat:qjrolph3e5ay5ba5kxkyn5ezym
*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 ...##
###
Full and convex linear subcategories are incompressible

2013
*
Proceedings of the American Mathematical Society
*

Consider the intrinsic fundamental group \'

doi:10.1090/s0002-9939-2013-11470-x
fatcat:2lw3pwgofbdcbeleb7lstchjxq
*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. ...##
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THH and base-change for Galois extensions of ring spectra

2017
*
Algebraic and Geometric Topology
*

Given

doi:10.2140/agt.2017.17.693
fatcat:bcj3yaf2vvhqpkuoyyucdoadnm
*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. ...##
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On the representation theory of Galois and Atomic Topoi
[article]

2003
*
arXiv
*
pre-print

We establish

arXiv:math/0208222v3
fatcat:d6pbohpyxfai3jxnwhwlmnjrjq
*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. ...##
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Separable morphisms of simplicial sets
[article]

2006
*
arXiv
*
pre-print

We show that the class

arXiv:math/0606634v1
fatcat:qfnfhv57xnegtaewqg6einiyse
*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. ...
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