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Concept# K-théorie

Résumé

En mathématiques, la K-théorie est un outil utilisé dans plusieurs disciplines. En topologie algébrique, la sert de théorie de cohomologie. Une variante est utilisée en algèbre sous le nom de K-théorie algébrique.
Les premiers résultats de la K-théorie ont été dans le cadre de la topologie algébrique, comme une théorie de cohomologie extraordinaire (elle ne vérifie pas l'axiome de dimension). Par la suite, ces méthodes ont été utilisées dans beaucoup d'autres domaines comme la géométrie algébrique, l'algèbre, la théorie des nombres, la théorie des opérateurs, etc.
Histoire
C'est Alexandre Grothendieck qui a fait la première construction d'un groupe de K-théorie dans son travail sur le théorème maintenant connu comme le théorème de Grothendieck-Riemann-Roch. Il a introduit la complétion de la catégorie additive des (classes d'isomorphisme de) faisceaux de groupes abéliens (munie de la somme directe) en utilisant des inverses formels. Cette idée a été reprise par Michael

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Cours associés (4)

MATH-311: Rings and modules

The students are going to solidify their knowledge of ring and module theory with a major emphasis on commutative algebra and a minor emphasis on homological algebra.

MATH-497: Homotopy theory

We propose an introduction to homotopy theory for topological spaces. We define higher homotopy groups and relate them to homology groups. We introduce (co)fibration sequences, loop spaces, and suspensions. We study long exact sequences. We construct Eilenberg-Mac Lane spaces.

MATH-726(2): Working group in Topology II

The theme of the working group varies from year to year. Examples of recent topics studied include: Galois theory of ring spectra, duality in algebra and topology, topological algebraic geometry and twisted K-theory

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In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined f

K-théorie algébrique

En mathématiques, la K-théorie algébrique est une branche importante de l'algèbre homologique. Son objet est de définir et d'appliquer une suite de foncteurs K de la catégorie des anneaux dans celle

Topologie algébrique

La topologie algébrique, anciennement appelée topologie combinatoire, est la branche des mathématiques appliquant les outils de l'algèbre dans l'étude des espaces topologiques. Plus exactement, elle

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In nature, one observes that a K-theory of an object is defined in two steps. First a “structured” category is associated to the object. Second, a K-theory machine is applied to the latter category that produces an infinite loop space. We develop a general framework that deals with the first step of this process. The K-theory of an object is defined via a category of “locally trivial” objects with respect to a pretopology. We study conditions ensuring an exact structure on such categories. We also consider morphisms in K-theory that such contexts naturally provide. We end by defining various K-theories of schemes and morphisms between them.

2013This thesis, which presents a new approach to the algebraic K-theory, is divided into two parts. The first one is devoted to the category of small simplicial categories. First, we construct a new model structure on sCat = [Δop,Cat] which is called the diagonal model structure, in reference to the diagonal model structure of Moerdijk on bisimplicial sets sSet2. Then we show that the new structure is proper and cellular. Note that this new model structure is not tensored and cotensored over the category of simplicial sets sSet in a manner consistent with the model structure. To remedy this, we use another model structure on sSet2 defined in the article of Cegarra and Remedios [3], which is equivalent to the Moerdijk structure. So we build a second new model structure on [Δop,Cat], which is cofibrantly generated, left proper, cellular and (co)tensored on sSet in a compatible way. Based on the work of [13], we construct the stable category of spectra (not symmetric) SpN(sCat*, Σ). It garantees the existence of Ω-spectra, which allows us to define thenotion of "weak Waldhausen category". The calculation of the simplicial enrichment map of the model category SpN(sCat*, Σ), leads to our new definition of algebraic K-theory of weak Waldhausen categories . The second part of this thesis is an attempt to generalize the previous results for enriched categories. First we begin by recalling the theory of ∞-categories and ∞-groupoids, based on the work of Joyal [14] and Lurie [18]. Then we make comparisons of ∞-categories with the category of simplicial sets equipped with the usual model structure. Our first result is the construction of a model structure on Top – Cat , the category of small categories enriched over the category of topological spaces Top, based on the work of Bergner [1] . The category Top – Cat is Quillen equivalent to sSet – Cat. Note that all objects in Top – Cat are fibrant ; this remark will play an important role in this theory. Our second result is the construction of a new model structure on the category of small simplicial categories enriched over Top, denoted by Top – sCat = [Δop,Top – Cat]. We show that this structure is proper and cellular. The fact that Top – sCat is not (co)tensored over sSet poses a barrier to defining the category of spectra SpN(sCat*, Σ).

K-Theory was originally defined by Grothendieck as a contravariant functor from a subcategory of schemes to abelian groups, known today as K0. The same kind of construction was then applied to other fields of mathematics, like spaces and (not necessarily commutative) rings. In all these cases, it consists of some process applied, not directly to the object one wants to study, but to some category related to it: the category of vector bundles over a space, of finitely generated projective modules over a ring, of locally free modules over a scheme, for instance. Later, Quillen extracted axioms that all these categories satisfy and that allow the Grothendieck construction of K0. The categorical structure he discovered is called today a Quillen-exact category. It led him not only to broaden the domain of application of K-theory, but also to define a whole K-theory spectrum associated to such a category. Waldhausen next generalized Quillen's notion of an exact category by introducing categories with weak equivalences and cofibrations, which one nowadays calls Waldhausen categories. K-theory has since been studied as a functor from the category of suitably structured (Quillen-exact, Waldhausen, symmetric monoidal) small categories to some category of spectra1. This has given rise to a huge field of research, so much so that there is a whole journal devoted to the subject. In this thesis, we want to take advantage of these tools to begin studying K-theory from another perspective. Indeed, we have the impression that, in the generalization of topological and algebraic K-theory that has been started by Quillen, something important has been left aside. K-theory was initiated as a (contravariant) functor from the various categories of spaces, rings, schemes, …, not from the category of Waldhausen small categories. Of course, one obtains information about a ring by studying its Quillen-exact category of (finitely generated projective) modules, but still, the final goal is the study of the ring, and, more globally, of the category of rings. Thus, in a general theory, one should describe a way to associate not only a spectrum to a structured category, but also a structured category to an object. Moreover, this process should take the morphisms of these objects into account. This gives rise to two fundamental questions. What kind of mathematical objects should K-theory be applied to? Given such an object, what category "over it" should one consider and how does it vary over morphisms? Considering examples, we have made the following observations. Suppose C is the category that is to be investigated by means of K-theory, like the category of topological spaces or of schemes, for instance. The category associated to an object of C is a sub-category of the category of modules over some monoid in a monoidal category with additional structure (topological, symmetric, abelian, model). The situation is highly "fibred": not only morphisms of C induce (structured) functors between these sub-categories of modules, but the monoidal category in which theses modules take place might vary from one object of C to another. In important cases, the sub-categories of modules considered are full sub-categories of "locally trivial" modules with respect to some (possibly weakened notion of) Grothendieck topology on C . That is, there are some specific modules that are considered sufficiently simple to be called trivial and locally trivial modules are those that are, locally over a covering of the Grothendieck topology, isomorphic to these. In this thesis, we explore, with K-theory in view, a categorical framework that encodes these kind of data. We also study these structures for their own sake, and give examples in other fields. We do not mention in this abstract set-theoretical issues, but they are handled with care in the discussion. Moreover, an appendix is devoted to the subject. After recalling classical facts of Grothendieck fibrations (and their associated indexed categories), we provide new insights into the concept of a bifibration. We prove that there is a 2-equivalence between the 2-category of bifibrations over a category ℬ and a 2-category of pseudo double functors from ℬ into the double category of adjunctions in CAT. We next turn our attention to composable pairs of fibrations , as they happen to be fundamental objects of the theory. We give a characterization of these objects in terms of pseudo-functors ℬop → FIBc into the 2-category of fibrations and Cartesian functors. We next turn to a short survey about Grothendieck (pre-)topologies. We start with the basic notion of covering function, that associate to each object of a category a family of coverings of the object. We study separately the saturation of a covering function with respect to sieves and to refinements. The Grothendieck topology generated by a pretopology is shown to be the result of these two steps. We define then, inspired by Street [89], the notion of (locally) trivial objects in a fibred category P : ℰ → ℬ equipped with some notion of covering of objects of the base ℬ. The trivial objects are objects chosen in some fibres. An object E in the fibre over B ∈ ℬ is locally trivial if there exists a covering {fi : Bi → B}i ∈ I such the inverse image of E along fi is isomorphic to a trivial object. Among examples are torsors, principal bundles, vector bundles, schemes, locally constant sheaves, quasi-coherent and locally free sheaves of modules, finitely generated projective modules over commutative rings, topological manifolds, … We give conditions under which locally trivial objects form a subfibration of P and describe the relationship between locally trivial objects with respect to subordinated covering functions. We then go into the algebraic part of the theory. We give a definition of monoidal fibred categories and show a 2-equivalence with monoidal indexed categories. We develop algebra (monoids and modules) in these two settings. Modules and monoids in a monoidal fibred category ℰ → ℬ happen to form a pair of fibrations . We end this thesis by explaining how to apply this categorical framework to K-theory and by proposing some prospects of research. ______________________________ 1 Works of Lurie, Toën and Vezzosi have shown that K-theory really depends on the (∞, 1)-category associated to a Waldhausen category [94]. Moreover, topological K-theory of spaces and Banach algebras takes the fact that the Waldhausen category is topological in account [62, 70].