151499
20190717172521.0
doi
10.5075/epfl-thesis-4862
urn
urn:nbn:ch:bel-epfl-thesis4862-2
nebis
6141118
THESIS
fre
4862
Catégories simpliciales enrichies et K-théorie de Waldhausen
Lausanne
EPFL
2010
2010
109
Theses
This 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).
enriched category
model category
stable model category
Top – Cat
Top – sCat
Algebraic K-theory
catégorie enrichie
catégorie modèle
catégorie modèle stable
Top – Cat
Top – sCat
K-théorie algébrique
243124
Amrani, Ilias
129064
240499
Hess-Bellwald, Kathryn
dir.
105396
854771
http://infoscience.epfl.ch/record/151499/files/EPFL_TH4862.pdf
Texte intégral / Full text
Texte intégral / Full text
252139
UPHESS
U10968
oai:infoscience.tind.io:151499
DOI
thesis-bn2018
thesis
SV
GLOBAL_SET
DOI2
SB
IGAT
EDMA
GR-HE
2010
4862/THESES
EPFL
PUBLISHED
THESIS