Deposits of clastic carbonate-dominated (calciclastic) sedimentary slope systems in the rock record have been identified mostly as linearly-consistent carbonate apron deposits, even though most ancient clastic carbonate slope deposits fit the submarine fan systems better. Calciclastic submarine fans are consequently rarely described and are poorly understood. Subsequently, very little is known especially in mud-dominated calciclastic submarine fan systems. Presented in this study are a sedimentological core and petrographic characterisation of samples from eleven boreholes from the Lower Carboniferous of Bowland Basin (Northwest England) that reveals a >250 m thick calciturbidite complex deposited in a calciclastic submarine fan setting. Seven facies are recognised from core and thin section characterisation and are grouped into three carbonate turbidite sequences. They include: 1) Calciturbidites, comprising mostly of highto low-density, wavy-laminated bioclast-rich facies; 2) low-density densite mudstones which are characterised by planar laminated and unlaminated muddominated facies; and 3) Calcidebrites which are muddy or hyper-concentrated debrisflow deposits occurring as poorly-sorted, chaotic, mud-supported floatstones. These

Phd by thesis

This book discusses the construction of triangulated categories of mixed motives over a noetherian scheme of finite dimension, extending Voevodsky's definition of motives over a field. In particular, it is shown that motives with rational coefficients satisfy the formalism of the six operations of Grothendieck. This is achieved by studying descent properties of motives, as well as by comparing different presentations of these categories, following and extending insights and constructions of Deligne, Beilinson, Bloch, Thomason, Gabber, Levine, Morel, Voevodsky, Ayoub, Spitzweck, R\"ondigs, {\O}stv{\ae}r, and others. In particular, the relation of motives with $K$-theory is addressed in full, and we prove the absolute purity theorem with rational coefficients, using Quillen's localization theorem in algebraic $K$-theory together with a variation on the Grothendieck-Riemann-Roch theorem. Using resolution of singularities via alterations of de Jong-Gabber, this leads to a version of Grothendieck-Verdier duality for constructible motivic sheaves with rational coefficients over rather general base schemes. We also study versions with integral coefficients, constructed via sheaves with transfers, for which we obtain partial results. Finally, we associate to any mixed Weil cohomology a system of categories of coefficients and well behaved realization functors, establishing a correspondence between mixed Weil cohomologies and suitable systems of coefficients. The results of this book have already served as ground reference in many subsequent works on motivic sheaves and their realizations, and pointers to the most recent developments of the theory are given in the introduction.

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Triangulated categories of mixed motives

Annals of mathematics studies

https://hal.archives-ouvertes.fr/hal-00772995/document

Homotopical algebraic geometry. I. Topos theory.

In this paper we establish a universal characterization of higher algebraic K–theory in the setting of small stable ∞–categories. Specifically, we prove that connective algebraic K–theory is the universal additive invariant, ie the universal functor with values in spectra which inverts Morita equivalences, preserves filtered colimits and satisfies Waldhausen’s additivity theorem. Similarly, we prove that nonconnective algebraic K–theory is the universal localizing invariant, ie the universal functor that moreover satisfies the Thomason–Trobaugh–Neeman Localization Theorem. To prove these results, we construct and study two stable ∞–categories of “noncommutative motives”; one associated to additivity and another to localization. In these stable ∞–categories, Waldhausen’s S∙–construction corresponds to the suspension functor and connective and nonconnective algebraic K–theory spectra become corepresentable by the noncommutative motive of the sphere spectrum. In particular, the algebraic K–theory of every scheme, stack and ring spectrum can be recovered from these categories of noncommutative motives. In the case of a connective ring spectrum R, we prove moreover that its negative K–groups are isomorphic to the negative K–groups of the ordinary ring π0R. In order to work with these categories of noncommutative motives, we establish comparison theorems between the category of spectral categories localized at the Morita equivalences and the category of small idempotent-complete stable ∞–categories. We also explain in detail the comparison between the ∞–categorical version of Waldhausen K–theory and the classical definition. As an application of our theory, we obtain a complete classification of the natural transformations from higher algebraic K–theory to topological Hochschild homology (THH) and topological cyclic homology (TC). Notably, we obtain an elegant conceptual description of the cyclotomic trace map.

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A universal characterization of higher algebraic K-theory

This book provides an introduction to modern homotopy theory through the lens of higher categories after Joyal and Lurie, giving access to methods used at the forefront of research in algebraic topology and algebraic geometry in the twenty-first century. The text starts from scratch - revisiting results from classical homotopy theory such as Serre's long exact sequence, Quillen's theorems A and B, Grothendieck's smooth/proper base change formulas, and the construction of the Kan–Quillen model structure on simplicial sets - and develops an alternative to a significant part of Lurie's definitive reference Higher Topos Theory, with new constructions and proofs, in particular, the Yoneda Lemma and Kan extensions. The strong emphasis on homotopical algebra provides clear insights into classical constructions such as calculus of fractions, homotopy limits and derived functors. For graduate students and researchers from neighbouring fields, this book is a user-friendly guide to advanced tools that the theory provides for application.

Higher Categories and Homotopical Algebra

Grothendieck introduced in Pursuing Stacks the notion of test category  These are by definition small categories on which presheaves of sets are models for homotopy types of CW-complexes A well known example is the category of simplices (the corresponding presheaves are then simplicial sets) Moreover, Grothendieck defined the notion of basic localizer which gives an axiomatic approach to the homotopy theory of small categories, and gives a natural setting to extend the notion of test category with respect some localizations of the homotopy category of CW-complexes This text can be seen as a sequel of Grothendieck's homotopy theory We prove in particular two conjectures made by Grothendieck: any category of presheaves on a test category is canonically endowed with a Quillen closed model category structure, and the smallest basic localizer defines the homotopy theory of CW-complexes Moreover, we show how a local version of the theory allows to consider in a unified setting the equivariant homotopy theory as well The realization of this program goes through the construction and the study of model category structures on any category of presheaves on an abstract small category, as well as the study of the homotopy theory of small categories following and completing the contributions of Quillen, Thomason and Grothendieck

Les préfaisceaux comme modèles des types d'homotopie

R esum e. Ces notes sont consacr ees a la construction des limites homotopiques, et plus g en eralement, des images directes cohomologiques dans une cat egorie de mod eles arbitraire admettant des petites limites projectives. En outre, la th eorie des d erivateurs de Grothendieck est introduite, a la fois en tant que motivation pour l’ etude de telles structures, et en tant qu’outil de d emonstration.

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Images directes cohomologiques dans les catégories de modèles

We introduce the dendroidal analogues of the notions of complete Segal space and of Segal category, and construct two appropriate model categories for which each of these notions corresponds to the property of being fibrant. We prove that these two model categories are Quillen equivalent to each other, and to the monoidal model category for ∞-operads which we constructed in an earlier paper. By slicing over the monoidal unit objects in these model categories, we derive as immediate corollaries the known comparison results between Joyal’s quasi-categories, Rezk’s complete Segal spaces, and Segal categories.

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Denis-Charles Cisinski

Papers

Triangulated categories of mixed motives

Higher Categories and Homotopical Algebra

Les préfaisceaux comme modèles des types d'homotopie

Images directes cohomologiques dans les catégories de modèles

Dendroidal Segal spaces and ∞-operads