Auswahl der wissenschaftlichen Literatur zum Thema „Cofibrant resolutions“

AbstractWe prove that a weak equivalence between two cofibrant (colored) props in chain complexes induces a Dwyer–Kan equivalence between the simplicial localisations of the associated categories of algebras. This homotopy invariance under base change implies that the homotopy category of homotopy algebras over a propPdoes not depend on the choice of a cofibrant resolution ofP, and gives thus a coherence to the notion of algebra up to homotopy in this setting. The result is established more generally for algebras in combinatorial monoidal dg categories.

AbstractThe homotopy coherent nerve from simplicial categories to simplicial sets and its left adjoint are important to the study of (∞, 1)-categories because they provide a means for comparing two models of their respective homotopy theories, giving a Quillen equivalence between the model structures for quasi-categories and simplicial categories. The functor also gives a cofibrant replacement for ordinary categories, regarded as trivial simplicial categories. However, the hom-spaces of the simplicial category X arising from a quasi-category X are not well understood. We show that when X is a quasi-category, all Λ21 horns in the hom-spaces of its simplicial category can be filled. We prove, unexpectedly, that for any simplicial set X, the hom-spaces of X are 3-coskeletal. We characterize the quasi-categories whose simplicial categories are locally quasi, finding explicit examples of 3-dimensional horns that cannot be filled in all other cases. Finally, we show that when X is the nerve of an ordinary category, X is isomorphic to the simplicial category obtained from the standard free simplicial resolution, showing that the two known cofibrant “simplicial thickenings” of ordinary categories coincide, and furthermore its hom-spaces are 2-coskeletal.

AbstractLet k be a field with resolution of singularities, and X a separated k-scheme of finite type with structure map g. We show that the slice filtration in the motivic stable homotopy category commutes with pullback along g. Restricting the field further to the case of characteristic zero, we are able to compute the slices of Weibel's homotopy invariant K-theory [24] extending the result of Levine [10], and also the zero slice of the sphere spectrum extending the result of Levine [10] and Voevodsky [23]. We also show that the zero slice of the sphere spectrum is a strict cofibrant ring spectrum HZXsf which is stable under pullback and that all the slices have a canonical structure of strict modules over HZXsf. If we consider rational coefficients and assume that X is geometrically unibranch then relying on the work of Cisinski and Déglise [4], we deduce that the zero slice of the sphere spectrum is given by Voevodsky's rational motivic cohomology spectrum HZX ⊗ ℚ and that the slices have transfers. This proves several conjectures of Voevodsky [22, conjectures 1, 7, 10, 11] in characteristic zero.

Le but de cette thèse est de fournir une construction explicite d'une résolution cofibrante des opérades de Balteanu-Fiedorowicz-Schwänzl-Vogt M_n, qui régissent les catégories monoidales itérées.Dans une première partie de la thèse, nous examinons en détail la définition des structures monoïdales dans les 2-catégories, ainsi que la définition des opérades dans les 2-catégories monoïdales, en prenant la 2-catégorie des catégories comme exemple principal. Ensuite, nous démontrons que la catégorie des opérades dans la catégorie des petites catégories hérite d'une structure de modèle par transfert de la structure de modèle folk sur la catégorie des petites catégories. Nous introduisons une notion de présentation polygraphique des opérades dans la catégorie des petites catégories afin de définir des opérades en terme de générateurs et relations à la fois dans la direction opératique et dans la direction catégorique au niveau des morphismes. Nous réexaminons la définition des opérades M_n en termes de présentations polygraphiques, et nous donnons une présentation de l'opérade M_1^infinity qui fournit une résolution cofibrante de l'opérade M_1 dans la structure de modèle folk. Enfin, nous étudions une généralisation du produit tensoriel de Boardman-Vogt dans le contexte des opérades dans la catégorie des catégories. Nous utilisons cette construction pour fournir une résolution cofibrante M_n^infinity de l'opérade M_n à partir de la résolution M_1^infinity de M_1, et ainsi répondre à la question initiale de la thèse
The goal of this thesis is to give an effective construction of a cofibrant resolution of the Balteanu-Fiedorowicz-Schwänzl-Vogt operads M_n, which govern iterated monoidal categories.In a first part of the thesis, we study thoroughly the definition of monoidal structures in 2-categories, and the definition of operads in monoidal 2-categories, with the 2-category of categories as a main motivating example. Then we prove that the category of operads in the category of small categories inherits a model structure by transfer of the folk model structure on the category of small categories. We introduce a notion of polygraphic presentation of operads in the category of small categories in order to define operads with generators and relations in both the operadic direction and the categorical direction at the morphism level. We revisit the definition of the operads M_n in terms of polygraphic presentations, and we gives a presentation of an operad M_1^infinity that provides a cofibrant resolution of the operad M_1 in the folk modelstructure. Eventually, we study a generalization of the Boardman-Vogt tensor product in the context of operads in the category of small categories. We use this construction to provide a cofibrant resolution M_n^infinity of the operad M_n from the resolution M_1^infinity of M_1, and hence, to address the initial question of the thesis