Tannaka Theory for Topos (original) (raw)
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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.
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.
A Tannakian Context for Galois
arXiv (Cornell University), 2011
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.
Homotopical algebraic geometry I: topos theory
Advances in Mathematics, 2005
This is the first of a series of papers devoted to lay the foundations of Algebraic Geometry in homotopical and higher categorical contexts. In this first part we investigate a notion of higher topos. For this, we use S-categories (i.e. simplicially enriched categories) as models for certain kind of ∞-categories, and we develop the notions of S-topologies, S-sites and stacks over them. We prove in particular, that for an S-category T endowed with an S-topology, there exists a model category of stacks over T , generalizing the model category structure on simplicial presheaves over a Grothendieck site of A. Joyal and R. Jardine. We also prove some analogs of the relations between topologies and localizing subcategories of the categories of presheaves, by proving that there exists a one-to-one correspodence between S-topologies on an S-category T , and certain left exact Bousfield localizations of the model category of pre-stacks on T. Based on the above results, we study the notion of model topos introduced by C. Rezk, and we relate it to our model categories of stacks over S-sites. In the second part of the paper, we present a parallel theory where S-categories, S-topologies and S-sites are replaced by model categories, model topologies and model sites. We prove that a canonical way to pass from the theory of stacks over model sites to the theory of stacks over S-sites is provided by the simplicial localization construction of Dwyer and Kan. As an example of application, we propose a definition ofétale K-theory of ring spectra, extending theétale K-theory of commutative rings.
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...
Advances in Mathematics, 2003
In this article we prove the following: A topos with a point is connected atomic if and only if it is the classifying topos of a localic group, and this group can be taken to be the locale of automorphisms of the point.
Essentially algebraic theories and localizations in toposes and abelian categories
[Thesis]. Manchester, UK: The University of Manchester; 2012., 2012
To see this, suppose we have an object X and arrows x 1 , x 2 : X → A, x 3 : X → K 2 , satisfying s x 1 = s x 2 = g 1 x 3 = g 2 x 3. Since s x 1 = s x 2 , we have bs x 1 = bs x 2 , and so sax 1 = sax 2. Since s is monic, this means ax 1 = ax 2. Since (f 1 , f 2) is the kernel pair of a, there is a map x : X → K 1 with f 1 x = x 1 , f 2 x = x 2. It suffices to prove that tx = x 3. But this is true because both of these maps are factorizations of the map s x 1 = s x 2 through the kernel pair (g 1 , g 2), so they are equal by the uniqueness of this factorization. Thus K 1 is the limit of the diagram described. But this limit can be constructed by taking three pullbacks, one after the other-that is, K 1 is isomorphic to the composition of pullbacks ((K 2 × (g 1 ,g 2) K 2) × B A) × B A. Since K 2 and A are each finitely generated, and B is coherent, it follows that these pullbacks, and therefore K 1 , are finitely generated. Thus S is finitely presentable. The notions of finite presentability introduced in this section can be generalized to infinite cardinals. Let λ be a regular cardinal. A partially ordered set (I, ≤) is said
Journal of Algebra, 1976
Any injective module F' E R-Mod defines a torsion radical (torsion theory) and an associated quotient category R-Mod/V, which is a hli reflective Grothendieck subcategory of R-Mod with exact reflector (quotient functor). In this case, V can be chosen so that the ring of quotients it determines can be realized as the bicammutator (double centralizer) of I/-. Morita ji3, 16j has generalized this construction by considering certain modules for which the reflector into the quotient category need not be exact, although he requires that the generalized ring of quotients must coincide with the bicommutator of the module. It is possible to omit the latter condition, as is shown b!; Theorem 1.8 and this enlarges the class of modules,that can be considered to include, in particular, am module that is injective modulo its annihilator. The new conditions obtained are not only more general, but appear to simplify the proofs as well. Heinicke [7] has characterized localization functors as idempotent, ieft exact monads, and quotient categories as their categories of algebras. Given any monad T of R-Nod, there is an associated monad 0, studied by Lambek and Rattray [IO], which is idempotent and left exact if T is left esact and which gives the usual localization functor if T = Hom(Hom(-, F), EV) for R V injective. It is shown in Theorem 2.10 that if T is left exact den restricted to Q,(R)-Mod, then the category of &,-algebras is a generalized quotient category in Morita's sense, thus extending Morita's theory of noncommutative localization. (This gives another proof of part of Theorem 1.8, in which, as in all of Section 1, the categorical language has been wppressed so as to provide easier access to the theory.) The final section gives some applications. Nor&a's characterization of balanced modules [I51 is modified in Theorem 3.1, while Theorem 3.2 extends % Partially supported by NSF Grant So. GP20414.