000168654 001__ 168654
000168654 005__ 20190509132411.0
000168654 0247_ $$2doi$$a10.5075/epfl-thesis-5200
000168654 02470 $$2urn$$aurn:nbn:ch:bel-epfl-thesis5200-2
000168654 02471 $$2nebis$$a6603019
000168654 037__ $$aTHESIS
000168654 041__ $$aeng
000168654 088__ $$a5200
000168654 245__ $$aHomotopic Descent over Monoidal Model Categories
000168654 269__ $$a2011
000168654 260__ $$bEPFL$$c2011$$aLausanne
000168654 300__ $$a110
000168654 336__ $$aTheses
000168654 520__ $$aThe starting point for this project is the article of  Kathryn Hess [11]. In this article, a homotopic version of  monadic descent is developed. In the classical setting, one  constructs a category D(𝕋) of coalgebras in the  Eilenberg-Moore category of algebras  D𝕋 for a given monad 𝕋 on a  category D. There is a canonical functor  Can𝕃𝕋 from D to  D(𝕋), and if  Can𝕃𝕋 is fully  faithful, then 𝕋 satisfies descent, while if  Can𝕃𝕋 is an equivalence  of categories, then 𝕋 satisfies effective descent  [19]. In [11], these two conditions are replaced by a weaker  one, that these hold only up to homotopy. This is achieved by  working with model categories that are enriched over  simplicial sets. Homotopic descent is then defined by  demanding that each component in  (Can𝕃𝕋)A,B  : MapD(A,B) →  MapD(𝕋)  (Can𝕃𝕋(A),  Can𝕃𝕋 (B)) be a weak equivalence of simplicial sets. A similar but  stronger condition involving the path components in  D(𝕋) expresses effective homotopic descent. The first goal of this project is to develop a framework  of homotopic descent for model categories that are enriched  over model categories other than simplicial sets. The most  important examples we have in mind are chain complexes and  spectra. In order to achieve this goal, we tried to determine  the most general conditions that are sufficient and necessary  to make the theory work. To ease the formulation, let us say that we are working  with a model category D that is enriched over a  monoidal model category V. The crucial constructions  we need are realization, respectively totalization, of  (co)simplicial objects in D. These functors have to be  Quillen functors to ensure that they have the correct  homotopical behaviour. This implies that there must exist a  Quillen adjunction between V and simplicial sets.  Furthermore, we need to be able to transfer the enrichment  and (co)tensoring over V to an enrichment and  (co)tensoring over simplicial sets. This forces the Quillen  adjunction to be monoidal. Another main point that has to be adressed is the  question, of whether the enrichment of D carries over  to an enrichment of D𝕋 and  D(𝕋) and how this enrichment behaves. It turns out  that this works well under mild assumptions on V. This  leads then to the definition of homotopic descent by  requiring that each component in  (Can𝕃𝕋)A,B  : MapD(A,B) →  MapD(𝕋)  (Can𝕃𝕋(A),  Can𝕃𝕋 (B)) be a weak equivalence in M and similarly for  effective homotopic descent. Using this definition, the  theorems in [11] carry over to this more general context.  Although the conditions on V are rather constraining  regarding the relation with simplicial sets, the cases of  chain complexes and spectra are included. For the time being  we do not see how the constraints on V could be  weakened. The second goal of this project is to apply the theory of  homotopic descent to concrete examples. A good source of  examples is homotopic Grothendieck descent in the category of  spectra, i.e., S-modules. Classical Grothendieck  descent deals with the adjunction induced by a morphism φ  : B → A of monoids in a monoidal category  (M,Λ, S), – BΛ A :  ModB ⇄ ModA :  φ*, which in turn induces a monad 𝕋φ :=  φ*(– ΛB A) on  ModB. We consider in particular the case  when the morphism in question is the unit of an  S-algebra E, η : S → E There is a close relationship between comodules over a  Hopf algebroid and objects in D(𝕋η).  Associated to η we have the canonical co-ring  Wη := E  ΛS E and an isomorphism  between D(𝕋η) and the category of  comodules over Wη in the category of  S-modules. This relationship is explored in an  analysis of the stable Adams spectral sequence, the  construction of which heavily relies on the monadic  properties of the functor η*(E  ΛS –) and can therefore be  expressed in terms of D(𝕋η). We  construct a spectral sequence that generalizes the stable  Adams spectral sequence to any stable pointed model category  such as unbounded chain complexes. One can give a description  of the E2-term as an Ext in  D(𝕋η), E2s,t =  ExtD(𝕋η) (Can(A),  Can(B)). If the spectral sequences converges, it abuts to  π⁎MapD(A,B  η^), where Bη^ is the derived  𝕋η-completion of B, which  agrees with the usual derived completion in well-known  special cases. Furthermore, Bη^ := Tot B^•, and B^• is kind of a  fibrant cosimplicial resolution of B. Furthermore, the language of relative homological algebra  for modules and comodules generalizes to definitions for  algebras in D𝕋η and  coalgebras in D(𝕋η). This shows that  the construction of the Adams spectral sequence works in a  more general setting, where one applies a functor to an  abelian category, for example π⁎, only at  the end, to be able to do computations in homological  algebra. This general Adams spectral sequence is closely  related to the descent spectral sequence of [11], and we have  clarified this relationship.
000168654 6531_ $$aHomotopic descent
000168654 6531_ $$aEnriched categories
000168654 6531_ $$aMonoidal model categories
000168654 6531_ $$aHomotopy spectral sequence
000168654 6531_ $$aAdams spectral sequence
000168654 6531_ $$aDerived completion
000168654 6531_ $$aGrothendieck descent
000168654 6531_ $$aDescente Homotopique
000168654 6531_ $$aCatégories enrichies
000168654 6531_ $$aCatégories modèles monoïdales
000168654 6531_ $$aSuite spéctrale homotopique
000168654 6531_ $$aSuite spectrale de Adams
000168654 6531_ $$aComplétion dérivée
000168654 6531_ $$aDéscente de Grothendieck
000168654 700__ $$0243122$$g176653$$aMüller, Patrick
000168654 720_2 $$aHess-Bellwald, Kathryn$$edir.$$g105396$$0240499
000168654 8564_ $$uhttps://infoscience.epfl.ch/record/168654/files/EPFL_TH5200.pdf$$zn/a$$s935276$$yn/a
000168654 909C0 $$xU10968$$0252139$$pUPHESS
000168654 909CO $$pthesis$$pthesis-bn2018$$pDOI$$ooai:infoscience.tind.io:168654$$qDOI2$$qGLOBAL_SET$$pSV
000168654 917Z8 $$x108898
000168654 918__ $$dEDMA$$cMATHGEOM$$aSB
000168654 919__ $$aGR-HE
000168654 920__ $$b2011
000168654 970__ $$a5200/THESES
000168654 973__ $$sPUBLISHED$$aEPFL
000168654 980__ $$aTHESIS