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 purpose of this book is twofold: to provide a general introduction to higher category theory (using the formalism of "quasicategories" or "weak Kan complexes"), and to apply this theory to the study of higher versions of Grothendieck topoi. A few applications to classical topology are included.

Higher Topos Theory

This is the second part of a series of papers devoted to develop Homotopical Algebraic Geometry. We start by defining and studying generalizations of standard notions of linear and commutative algebra in an abstract monoidal model category, such as derivations, etale and smooth maps, flat and projective modules, etc. We then use the theory of stacks over model categories introduced in \cite{hagI} in order to define a general notion of geometric stack over a base symmetric monoidal model category C, and prove that this notion satisfies the expected properties. The rest of the paper consists in specializing C to several different contexts. First of all, when C=k-Mod is the category of modules over a ring k, with the trivial model structure, we show that our notion gives back the algebraic n-stacks of C. Simpson. Then we set C=sk-Mod, the model category of simplicial k-modules, and obtain this way a notion of geometric derived stacks which are the main geometric objects of Derived Algebraic Geometry. We give several examples of derived version of classical moduli stacks, as for example the derived stack of local systems on a space, of algebra structures over an operad, of flat bundles on a projective complex manifold, etc. Finally, we present the cases where C=(k) is the model category of unbounded complexes of modules over a char 0 ring k, and C=Sp^{\Sigma} the model category of symmetric spectra. In these two contexts, called respectively Complicial and Brave New Algebraic Geometry, we give some examples of geometric stacks such as the stack of associative dg-algebras, the stack of dg-categories, and a geometric stack constructed using topological modular forms.

/pdf/homotopical-algebraic-geometry-ii-geometric-stacks-and-dyi1d00ajz.pdf

Homotopical Algebraic Geometry II: Geometric Stacks and Applications

The main purpose of this work is the study of the homotopy theory of dg-categories up to quasi-equivalences. Our main result provides a natural description of the mapping spaces between two dg-categories $C$ and $D$ in terms of the nerve of a certain category of $(C,D)$-bimodules. We also prove that the homotopy category $Ho(dg-Cat)$ is cartesian closed (i.e. possesses internal Hom's relative to the tensor product). We use these two results in order to prove a derived version of Morita theory, describing the morphisms between dg-categories of modules over two dg-categories $C$ and $D$ as the dg-category of $(C,D)$-bi-modules. Finally, we give three applications of our results. The first one expresses Hochschild cohomology as endomorphisms of the identity functor, as well as higher homotopy groups of the \emph{classifying space of dg-categories} (i.e. the nerve of the category of dg-categories and quasi-equivalences between them). The second application is the existence of a good theory of localization for dg-categories, defined in terms of a natural universal property. Our last application states that the dg-category of (continuous) morphisms between the dg-categories of quasi-coherent (resp. perfect) complexes on two schemes (resp. smooth and proper schemes) is quasi-equivalent to the dg-category of quasi-coherent complexes (resp. perfect) on their product.

/pdf/the-homotopy-theory-of-dg-categories-and-derived-morita-1tg9nxz1cq.pdf

The homotopy theory of dg-categories and derived Morita theory

Higher Algebraic K-Theory

Combinatorial Model Categories Have Presentations

We use hypercovers to study the homotopy theory of simplicial presheaves. The main result says that model structures for simplicial presheaves involving local weak equivalences can be constructed by localizing at the hypercovers. One consequence is that the fibrant objects can be explicitly described in terms of a hypercover descent condition, and the fibrations can be described by a relative descent condition. We give a few applications for this new description of the homotopy theory of simplicial presheaves.

Hypercovers and simplicial presheaves

https://ncatlab.org/nlab/files/DuggerUniv.pdf

Universal homotopy theories

We show that if U
 * is a hypercover of a topological space X then the natural map hocolim U
 * → X is a weak equivalence. This fact is used to construct topological realization functors for the 1-homotopy theory of schemes over real and complex fields. In an appendix, we also prove a theorem about computing homotopy colimits of spaces that are not cofibrant.

Topological hypercovers and 1-realizations

Given a small category C, we show that there is a universal way of expanding C into a model category, essentially by formally adjoining homotopy colimits. The technique of localization becomes a method for imposing `relations' into these universal gadgets. The paper develops this formalism and discusses various applications, for instance to the study of homotopy colimits, the Dwyer-Kan theory of framings, and to the homotopy theory of schemes.

Daniel Dugger

Papers

Combinatorial Model Categories Have Presentations

Hypercovers and simplicial presheaves

Universal homotopy theories

Topological hypercovers and 1-realizations

Universal homotopy theories