000151499 001__ 151499
000151499 005__ 20190205040512.0
000151499 0247_ $$2doi$$a10.5075/epfl-thesis-4862
000151499 02470 $$2urn$$aurn:nbn:ch:bel-epfl-thesis4862-2
000151499 02471 $$2nebis$$a6141118
000151499 037__ $$aTHESIS
000151499 041__ $$afre
000151499 088__ $$a4862
000151499 245__ $$aCatégories simpliciales enrichies et K-théorie de Waldhausen
000151499 269__ $$a2010
000151499 260__ $$aLausanne$$bEPFL$$c2010
000151499 300__ $$a109
000151499 336__ $$aTheses
000151499 520__ $$aThis 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).
000151499 6531_ $$aenriched category
000151499 6531_ $$amodel category
000151499 6531_ $$astable model category
000151499 6531_ $$aTop – Cat
000151499 6531_ $$aTop – sCat
000151499 6531_ $$aAlgebraic K-theory
000151499 6531_ $$acatégorie enrichie
000151499 6531_ $$acatégorie modèle
000151499 6531_ $$acatégorie modèle stable
000151499 6531_ $$aTop – Cat
000151499 6531_ $$aTop – sCat
000151499 6531_ $$aK-théorie algébrique
000151499 700__ $$0243124$$aAmrani, Ilias$$g129064
000151499 720_2 $$0240499$$aHess-Bellwald, Kathryn$$edir.$$g105396
000151499 8564_ $$s854771$$uhttps://infoscience.epfl.ch/record/151499/files/EPFL_TH4862.pdf$$yTexte intégral / Full text$$zTexte intégral / Full text
000151499 909C0 $$0252139$$pUPHESS$$xU10968
000151499 909CO $$ooai:infoscience.tind.io:151499$$pthesis$$pthesis-bn2018$$pDOI$$pSV$$qDOI2$$qGLOBAL_SET
000151499 919__ $$aGR-HE
000151499 918__ $$aSB$$cIGAT$$dEDMA
000151499 920__ $$b2010
000151499 973__ $$aEPFL$$sPUBLISHED
000151499 970__ $$a4862/THESES
000151499 980__ $$aTHESIS