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000168654 001__ 168654 000168654 005__ 20190316235212.0 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__ $$a2011 000168654 260__ $$aLausanne$$bEPFL$$c2011 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$$aMüller, Patrick$$g176653 000168654 720_2 $$0240499$$aHess-Bellwald, Kathryn$$edir.$$g105396 000168654 8564_ $$s935276$$uhttps://infoscience.epfl.ch/record/168654/files/EPFL_TH5200.pdf$$yn/a$$zn/a 000168654 909C0 $$0252139$$pUPHESS$$xU10968 000168654 909CO $$ooai:infoscience.tind.io:168654$$pthesis$$pthesis-bn2018$$pDOI$$pSV$$qDOI2$$qGLOBAL_SET 000168654 917Z8 $$x108898 000168654 918__ $$aSB$$cMATHGEOM$$dEDMA 000168654 919__ $$aGR-HE 000168654 920__ $$b2011 000168654 970__ $$a5200/THESES 000168654 973__ $$aEPFL$$sPUBLISHED 000168654 980__ $$aTHESIS