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Autour De La Géométrie Algébrique Dérivée Autour de la Géométrie Algébrique Dérivée groupe de travail—spring 2013 Institut de Mathématiques de Jussieu Université Paris Diderot Organised by: Gabriele VEZZOSI Speakers: Pieter BELMANS, Brice LE GRIGNOU, Valerio MELANI, Mauro PORTA, Marco ROBALO, Pietro VERTECHI, Yan ZHAO Contents Exposé 1 Model categories 1 by Mauro Porta 1.1 Motivations and main ideas . 1 1.2 Model categories . 4 1.3 The homotopy category . 8 1.4 Examples . 16 1.5 Quillen adjunctions and total derived functors . 23 1.6 Homotopy limits and colimits . 29 1.7 Mapping spaces . 33 1.8 Bousfield localization . 40 1.9 Complements to Chapter 1 . 45 Exposé 2 Simplicial localization 51 by Brice Le Grignou 2.1 Introduction . 51 2.2 Motivation of the simplicial localization . 52 2.3 Two definitions of the simplicial localization . 55 2.4 Homotopy simplicial category and conclusion . 62 2.5 Complements to Chapter 2 . 63 Exposé 3 Quasi-categories 65 by Valerio Melani 3.1 Intuition and first models . 65 3.2 Quasi-categories . 66 Exposé 4 Segal spaces and Segal categories 71 by Yan Zhao 4.1 Preliminaries . 71 4.2 Segal spaces . 74 4.3 Segal Categories . 82 4.4 Comparison theorems . 85 Exposé 5 Simplicial presheaves 91 by Mauro Porta 5.1 Review of sheaf theory . 91 ii 5.2 Fibered categories and stacks . 92 5.3 Simplicial presheaves . 103 5.4 Complements to Chapter 5 . 112 Exposé 6 Higher derived stacks 113 by Pietro Vertechi 6.1 Higher stacks . 113 6.2 Derived algebraic geometry . 116 6.3 Affinisation . 117 6.4 Cohomology . 118 6.5 Groups and group action . 119 Exposé 7 Differential graded categories 121 by Pieter Belmans 7.1 Introduction . 121 7.2 Differential graded categories . 122 7.3 The category of differential graded categories . 124 7.4 Differential graded modules . 125 7.5 Model category structures on dg Catk . 127 7.6 Mapping spaces . 128 7.7 Monoidal structure . 129 7.8 Applications . 130 7.9 ( , 1)- and dg categories . 131 1 Exposé 8 Affine derived schemes 135 by Mauro Porta and Yohann Segalat 8.1 Lifting criteria . 135 8.2 Affine derived schemes . 135 Exposé 9 Derived moduli stacks 145 by Pieter Belmans 9.1 Introduction . 145 9.2 Preliminaries . 147 9.3 Properties of derived moduli stacks . 151 9.4 The tangent complex of a derived moduli stack . 154 9.5 Example: derived moduli stack of perfect complexes on a scheme . 155 Exposé 10 Algebraic structures in higher categories 157 by Pietro Vertechi 10.1 Infinity operads . 157 10.2 Weak algebraic structures . 158 10.3 Looping and delooping . 159 Exposé 11 Derived loop spaces and de Rham cohomology 161 by Pietro Vertechi 11.1 Topological spaces and groups in derived algebraic geometry . 161 11.2 Loop space and the algebra of differential forms . 164 Exposé 12 Derived formal deformation theory 167 by Brice Le Grignou and Mauro Porta 12.1 Introduction . 167 iii 12.2 An overview of classical deformation theory . 168 12.3 Derived deformation theory . 170 12.4 The tangent spectrum of a formal moduli problem . 173 12.5 The relationship between spectra and complexes . 174 12.6 Review of cdga and dgla . 174 12.7 The formal moduli problem associated to a dgla . 175 Exposé 13 Noncommutative motives and non-connective K-theory of dg-categories 177 by Marco Robalo 13.1 Noncommutative Algebraic Geometry . 177 13.2 Motives . 180 13.3 Noncommutative Motives . 183 13.4 K-Theory and Noncommutative Motives . 185 13.5 Relation with the work of Cisinski-Tabuada . 188 13.6 Future Works . 189 A Simplicial sets 193 by Mauro Porta A.1 The category ∆ and (co)simplicial objects . 193 A.2 Simplicial sets . 195 A.3 Geometric realization . 197 A.4 Kan fibrations . 198 A.5 Anodyne extensions . 198 B Algebras and modules in category theory 201 by Mauro Porta B.1 The theory of monads . 201 B.2 Monoidal categories . 204 C Enriched category theory 211 by Mauro Porta C.1 Enriched categories . 211 C.2 Tensor and cotensor . 212 C.3 Induced structures . 212 Introduction The purpose of these notes is to provide a written background for the seminars taught by the working group organized by Gabriele Vezzosi in the Spring of 2013. The main goal is to give a wide introduction to Derived Algebraic Geometry, starting from the very foundations. In the present notes, we added more details with respect to the expositions, to make easier for the not (yet) specialised reader to go over them. The work is organized as follows: 1. Chapter 1 contains a survey of the theory of model categories: the homotopy category, function complexes, left and right (Bousfield) localizations, standard simplicial localization and hammock localization. The relative seminar has been taught the 3/15/2013 by Mauro Porta and Brice Legrignou; 2. Chapter 2 contains a survey of all the known models for ( , 1)-categories, and the details of three of those constructions (quasicategories, Segal categories1 and Segal spaces). The relative seminar will be taught the 3/22/2013 by Yan Zhao and Valerio Melani; 3. Chapter 3 contains a survey of of the theory of simplicial presheaves and their application to the theory of (higher) stacks. The relative seminar has been taught the 3/29/2013 by Mauro Porta and Pietro Vertechi; 4. Finally, the Appendices contain material which is somehow more foundational: we collected some basic results from the theory of simplicial sets, enriched category theory... Essentially because the authors wanted to learn all these theories in a better way. This is a work in progress v EXPOSÉ 1 Model categories In this chapter we will introduce one of the main tools of this cycle of seminars, namely model cate- gories. After an informal section concerning motivations coming from others areas of Mathematics, we will introduce the basic definitions and we will state the main theorems, sketching some proof. We will try, whenever possible and within our capacities, to explain the intuitions motivating the definitions, why we should expect a certain theorem to be true and so on; moreover, we selected four examples where we can test the result we will obtain: 1. simplicial sets sSet: this is the main example in homotopy theory; in a sense, we could say that simplicial sets stands to homotopy theory as sets stand to the whole mathematics (which can be seen as constructions on the topos Set, in a very broad sense); 2. topological spaces: this is the “continuous” version of simplicial sets; for many purpose there is no distinction from the homotopy theory for topological spaces and the one for simplicial sets; 3. chain complexes: this example relates homotopy theory to homological algebra. It gives the “correct motivation” for a bunch of facts; for example, that a homotopy of topological spaces gives rise to a chain homotopy between singular complexes; 4. groupoids: this will be needed in future chapters; it also provides an interesting example where the model structure is uniquely determined by the weak equivalences. The main topics include the homotopy category, Quillen functors, Reedy categories and (co)simplicial resolutions, function complexes. The last part.
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