A Tannakian Context for Galois (original) (raw)
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A Tannakian context for Galois theory
Advances in Mathematics, 2013
Strong similarities have been long observed between the Galois (Categories Galoisiennes) and the Tannaka (Categories Tannakiennes) theories of representation of groups. In this paper we construct an explicit (neutral) Tannakian context for the Galois theory of atomic topoi, and prove the equivalence between its fundamental theorems. Since the theorem is known for the Galois context, this yields, in particular, a proof of the fundamental (recognition) theorem for a new Tannakian context. This example is different from the additive cases or their generalization, where the theorem is known to hold, and where the unit of the tensor product is always an object of finite presentation, which is not the case in our context.
On the representation theory of Galois and atomic topoi
Journal of Pure and Applied Algebra, 2004
The notion of a (pointed) Galois pretopos ("catégorie Galoisienne") was considered originally by Grothendieck in in connection with the fundamental group of a scheme. In that paper Galois theory is conceived as the axiomatic characterization of the classifying pretopos of a profinite group G. The fundamental theorem takes the form of a representation theorem for Galois pretopos (see for the explicit interpretation of this work in terms of filtered unions of categories -the link to filtered inverse limits of topoi -and its relation to classical Galois's galois theory). An important motivation was pragmatical. The fundamental theorem is tailored to be applied to the category of etal coverings of a connected locally noetherian scheme pointed with a geometric point over an algebraically closed field. We quote: "Cetteéquivalence permet donc de interpréter les opérations courantes sur des revêtements en terms des opérations analogues dans BG, i.e. en terms des opérationsévidentes sur des ensembles finis où G opére". Later, in collaboration with Verdier ([1] Ex IV), he considers the general notion of pointed Galois Topos in a series of commented exercises (specially Ex IV, 2.7.5). There, specific guidelines are given to develop the theory of classifying topoi of progroups. It is stated therein that Galois topoi correspond exactly, as categories, to the full subcategories generated by locally constant objects in connected locally connected topoi (this amounts to the construction of Galois closures), and that they classify progroups. In , Moerdiejk developed this program under the light of the localic group concept. He proves the fundamental theorem (in a rather sketchy way, theorem 3.2 loc.cit.) in the form of a characterization of pointed Galois topoi as the classifying topoi of prodiscrete localic groups.
arXiv (Cornell University), 2015
We consider locales B as algebras in the tensor category sℓ of sup-lattices. We show the equivalence between the Joyal-Tierney descent theorem for open localic surjections sh(B) q −→ E in Galois theory [An extension of the Galois Theory of Grothendieck, AMS Memoirs 151] and a Tannakian recognition theorem over sℓ for the sℓ-functor Rel(E) Rel(q *) −→ Rel(sh(B)) (B-Mod) 0 into the sℓ-category of discrete B-modules. Thus, a new Tannaka recognition theorem is obtained, essentially different from those known so far. This equivalence follows from two independent results. We develop an explicit construction of the localic groupoid G : G
On the Galois Theory of Grothendieck
2000
In this paper we deal with Grothendieck's interpretation of Artin's interpretation of Galois's Galois Theory (and its natural relation with the fundamental group and the theory of coverings) as he developed it in Expose V, section 4, ``Conditions axiomatiques d'une theorie de Galois'' in the SGA1 1960/61. This is a beautiful piece of mathematics very rich in categorical concepts, and goes much beyond the original Galois's scope (just as Galois went much further than the non resubility of the quintic equation). We show explicitly how Grothendieck's abstraction corresponds to Galois work. We introduce some axioms and prove a theorem of characterization of the category (topos) of actions of a discrete group. This theorem corresponds exactly to Galois fundamental result. The theorem of Grothendieck characterizes the category (topos) of continuous actions of a profinite topological group. We develop a proof of this result as a "passage into th...
A simplified categorical approach to several Galois theories
arXiv: Differential Geometry, 2018
We discuss the concept of Galois structure and Galois epimorphism in a general setting. Namely, a Galois structure for an epimorphism picolonMtoB\pi\colon M\to BpicolonMtoB in some category bfCat{\bf Cat}bfCat is the action of a group object that gives to MMM the structure of principal homogeneous space in the relative category bfCatB{\bf Cat}_BbfCatB. We see that this general setting applies to coverings, finite field extensions, strongly normal extensions of differential fields, etc. We also explore Galois structures in the category of foliated manifolds, arriving to a purely geometric and smooth counterpart of differential Galois theory.
An introduction to Tannaka duality and quantum groups
Lecture Notes in Mathematics, 1991
The goal of this paper is to give an account of classical Tannaka duality [C⁄ ] in such a way as to be accessible to the general mathematical reader, and to provide a key for entry to more recent developments [⁄ SR, DM⁄ ] and quantum groups [⁄ D1⁄ ]. Expertise in neither representation theory nor category theory is assumed.
2018
We study the quotient of mathcalTn=Rep(GL(n∣n))\mathcal{T}_n = Rep(GL(n|n))mathcalTn=Rep(GL(n∣n)) by the tensor ideal of negligible morphisms. If we consider the full subcategory mathcalTn+\mathcal{T}_n^+mathcalTn+ of mathcalTn\mathcal{T}_nmathcalTn of indecomposable summands in iterated tensor products of irreducible representations up to parity shifts, its quotient is a semisimple tannakian category Rep(Hn)Rep(H_n)Rep(Hn) where HnH_nHn is a pro-reductive algebraic group. We determine the connected derived subgroup GnsubsetHnG_n \subset H_nGnsubsetHn and the groups Glambda=(Hlambda)der0G_{\lambda} = (H_{\lambda})_{der}^0Glambda=(Hlambda)der0 corresponding to the tannakian subcategory in Rep(Hn)Rep(H_n)Rep(Hn) generated by an irreducible representation L(lambda)L(\lambda)L(lambda). This gives structural information about the tensor category Rep(GL(n∣n))Rep(GL(n|n))Rep(GL(n∣n)), including the decomposition law of a tensor product of irreducible representations up to summands of superdimension zero. Some results are conditional on a hypothesis on 222-torsion in pi0(Hn)\pi_0(H_n)pi0(Hn).
Remarks on abstract Galois theory
Manuscrito, 2011
This paper is a historical companion to a previous one, in which it was studied the so-called abstract Galois theory as formulated by the Portuguese mathematician José Sebastião e Silva (see da Costa, Rodrigues ). Our purpose is to present some applications of abstract Galois theory to higher-order model theory, to discuss Silva's notion of expressibility and to outline a classical Galois theory that can be obtained inside the two versions of the abstract theory, those of Mark Krasner and of Silva. Some comments are made on the universal theory of (set-theoretic) structures.
Atomical Grothendieck categories
International Journal of Mathematics and Mathematical Sciences, 2003
Motivated by the study of Gabriel dimension of a Grothendieck category, we introduce the concept of atomical Grothendieck category, which has only two localizing subcategories, and we give a classification of this type of Grothendieck categories.