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000151498 001__ 151498 000151498 005__ 20190509132331.0 000151498 0247_ $$2doi$$a10.5075/epfl-thesis-4861 000151498 02470 $$2urn$$aurn:nbn:ch:bel-epfl-thesis4861-5 000151498 02471 $$2nebis$$a6141102 000151498 037__ $$aTHESIS 000151498 041__ $$aeng 000151498 088__ $$a4861 000151498 245__ $$aCategorical Foundations for K-theory 000151498 269__ $$a2010 000151498 260__ $$aLausanne$$bEPFL$$c2010 000151498 300__ $$a184 000151498 336__ $$aTheses 000151498 520__ $$aK-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]. 000151498 6531_ $$aK-theory 000151498 6531_ $$aLocal triviality 000151498 6531_ $$aGrothendieck fibration 000151498 6531_ $$aGrothendieck topology 000151498 6531_ $$aMonoidal fibred category 000151498 6531_ $$aModule 000151498 6531_ $$aK-théorie 000151498 6531_ $$aTrivialité locale 000151498 6531_ $$aFibration de Grothendieck 000151498 6531_ $$aTopologie de Grothendieck 000151498 6531_ $$aCatégorie fibrée monoïdale 000151498 6531_ $$aModule 000151498 700__ $$0243125$$aMichel, Nicolas$$g114710 000151498 720_2 $$0240499$$aHess-Bellwald, Kathryn$$edir.$$g105396 000151498 8564_ $$s1286971$$uhttps://infoscience.epfl.ch/record/151498/files/EPFL_TH4861.pdf$$yTexte intégral / Full text$$zTexte intégral / Full text 000151498 909C0 $$0252139$$pUPHESS$$xU10968 000151498 909CO $$ooai:infoscience.tind.io:151498$$pthesis$$pthesis-bn2018$$pDOI$$pSV$$qDOI2$$qGLOBAL_SET 000151498 919__ $$aGR-HE 000151498 918__ $$aSB$$cIGAT$$dEDMA 000151498 920__ $$b2010 000151498 973__ $$aEPFL$$sPUBLISHED 000151498 970__ $$a4861/THESES 000151498 980__ $$aTHESIS