Topos Theory
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The Structure of Gunk: Adventures in the Ontology of Space
The Structure of Gunk: Adventures in the Ontology of Space Jeff Sanford Russell February 16, 2009 1 Points and gunk Here are two ways space might be (not the only two): (1) Space is “pointy”. Ev- ery finite region has infinitely many infinitesimal, indivisible parts, called points. Points are zero-dimensional atoms of space. In addition to points, there are other kinds of “thin” boundary regions, like surfaces of spheres. Some regions include their boundaries—the closed regions—others exclude them—the open regions—and others include some bits of boundary and exclude others. Moreover, space includes unextended regions whose size is zero. (2) Space is “gunky”.1 Every region contains still smaller regions—there are no spatial atoms. Every region is “thick”—there are no boundary regions. Every region is extended. Pointy theories of space and space-time—such as Euclidean space or Minkowski space—are the kind that figure in modern physics. A rival tradition, most famously associated in the last century with A. N. Whitehead, instead embraces gunk.2 On the Whiteheadian view, points, curves and surfaces are not parts of space, but rather abstractions from the true regions. Three different motivations push philosophers toward gunky space. The first is that the physical space (or space-time) of our universe might be gunky. We posit spatial reasons to explain what goes on with physical objects; thus the main reason Dean Zimmerman has been a relentless source of suggestions and criticism throughout the writing of this paper. Thanks are also due to Frank Arntzenius, Kenny Easwaran, Peter Forrest, Hilary Greaves, John Hawthorne, and Beatrice Sanford for helpful comments. -
Derived Categories. Winter 2008/09
Derived categories. Winter 2008/09 Igor V. Dolgachev May 5, 2009 ii Contents 1 Derived categories 1 1.1 Abelian categories .......................... 1 1.2 Derived categories .......................... 9 1.3 Derived functors ........................... 24 1.4 Spectral sequences .......................... 38 1.5 Exercises ............................... 44 2 Derived McKay correspondence 47 2.1 Derived category of coherent sheaves ................ 47 2.2 Fourier-Mukai Transform ...................... 59 2.3 Equivariant derived categories .................... 75 2.4 The Bridgeland-King-Reid Theorem ................ 86 2.5 Exercises ............................... 100 3 Reconstruction Theorems 105 3.1 Bondal-Orlov Theorem ........................ 105 3.2 Spherical objects ........................... 113 3.3 Semi-orthogonal decomposition ................... 121 3.4 Tilting objects ............................ 128 3.5 Exercises ............................... 131 iii iv CONTENTS Lecture 1 Derived categories 1.1 Abelian categories We assume that the reader is familiar with the concepts of categories and func- tors. We will assume that all categories are small, i.e. the class of objects Ob(C) in a category C is a set. A small category can be defined by two sets Mor(C) and Ob(C) together with two maps s, t : Mor(C) → Ob(C) defined by the source and the target of a morphism. There is a section e : Ob(C) → Mor(C) for both maps defined by the identity morphism. We identify Ob(C) with its image under e. The composition of morphisms is a map c : Mor(C) ×s,t Mor(C) → Mor(C). There are obvious properties of the maps (s, t, e, c) expressing the axioms of associativity and the identity of a category. For any A, B ∈ Ob(C) we denote −1 −1 by MorC(A, B) the subset s (A) ∩ t (B) and we denote by idA the element e(A) ∈ MorC(A, A). -
Representations of Semisimple Lie Algebras in Prime Characteristic and the Noncommutative Springer Resolution
Annals of Mathematics 178 (2013), 835{919 http://dx.doi.org/10.4007/annals.2013.178.3.2 Representations of semisimple Lie algebras in prime characteristic and the noncommutative Springer resolution By Roman Bezrukavnikov and Ivan Mirkovic´ To Joseph Bernstein with admiration and gratitude Abstract We prove most of Lusztig's conjectures on the canonical basis in homol- ogy of a Springer fiber. The conjectures predict that this basis controls numerics of representations of the Lie algebra of a semisimple algebraic group over an algebraically closed field of positive characteristic. We check this for almost all characteristics. To this end we construct a noncom- mutative resolution of the nilpotent cone which is derived equivalent to the Springer resolution. On the one hand, this noncommutative resolution is closely related to the positive characteristic derived localization equiva- lences obtained earlier by the present authors and Rumynin. On the other hand, it is compatible with the t-structure arising from an equivalence with the derived category of perverse sheaves on the affine flag variety of the Langlands dual group. This equivalence established by Arkhipov and the first author fits the framework of local geometric Langlands duality. The latter compatibility allows one to apply Frobenius purity theorem to deduce the desired properties of the basis. We expect the noncommutative counterpart of the Springer resolution to be of independent interest from the perspectives of algebraic geometry and geometric Langlands duality. Contents 0. Introduction 837 0.1. Notations and conventions 841 1. t-structures on cotangent bundles of flag varieties: statements and preliminaries 842 R.B. -
A Category-Theoretic Approach to Representation and Analysis of Inconsistency in Graph-Based Viewpoints
A Category-Theoretic Approach to Representation and Analysis of Inconsistency in Graph-Based Viewpoints by Mehrdad Sabetzadeh A thesis submitted in conformity with the requirements for the degree of Master of Science Graduate Department of Computer Science University of Toronto Copyright c 2003 by Mehrdad Sabetzadeh Abstract A Category-Theoretic Approach to Representation and Analysis of Inconsistency in Graph-Based Viewpoints Mehrdad Sabetzadeh Master of Science Graduate Department of Computer Science University of Toronto 2003 Eliciting the requirements for a proposed system typically involves different stakeholders with different expertise, responsibilities, and perspectives. This may result in inconsis- tencies between the descriptions provided by stakeholders. Viewpoints-based approaches have been proposed as a way to manage incomplete and inconsistent models gathered from multiple sources. In this thesis, we propose a category-theoretic framework for the analysis of fuzzy viewpoints. Informally, a fuzzy viewpoint is a graph in which the elements of a lattice are used to specify the amount of knowledge available about the details of nodes and edges. By defining an appropriate notion of morphism between fuzzy viewpoints, we construct categories of fuzzy viewpoints and prove that these categories are (finitely) cocomplete. We then show how colimits can be employed to merge the viewpoints and detect the inconsistencies that arise independent of any particular choice of viewpoint semantics. Taking advantage of the same category-theoretic techniques used in defining fuzzy viewpoints, we will also introduce a more general graph-based formalism that may find applications in other contexts. ii To my mother and father with love and gratitude. Acknowledgements First of all, I wish to thank my supervisor Steve Easterbrook for his guidance, support, and patience. -
Generalized Inverse Limits and Topological Entropy
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Repository of Faculty of Science, University of Zagreb FACULTY OF SCIENCE DEPARTMENT OF MATHEMATICS Goran Erceg GENERALIZED INVERSE LIMITS AND TOPOLOGICAL ENTROPY DOCTORAL THESIS Zagreb, 2016 PRIRODOSLOVNO - MATEMATICKIˇ FAKULTET MATEMATICKIˇ ODSJEK Goran Erceg GENERALIZIRANI INVERZNI LIMESI I TOPOLOŠKA ENTROPIJA DOKTORSKI RAD Zagreb, 2016. FACULTY OF SCIENCE DEPARTMENT OF MATHEMATICS Goran Erceg GENERALIZED INVERSE LIMITS AND TOPOLOGICAL ENTROPY DOCTORAL THESIS Supervisors: prof. Judy Kennedy prof. dr. sc. Vlasta Matijevic´ Zagreb, 2016 PRIRODOSLOVNO - MATEMATICKIˇ FAKULTET MATEMATICKIˇ ODSJEK Goran Erceg GENERALIZIRANI INVERZNI LIMESI I TOPOLOŠKA ENTROPIJA DOKTORSKI RAD Mentori: prof. Judy Kennedy prof. dr. sc. Vlasta Matijevic´ Zagreb, 2016. Acknowledgements First of all, i want to thank my supervisor professor Judy Kennedy for accept- ing a big responsibility of guiding a transatlantic student. Her enthusiasm and love for mathematics are contagious. I thank professor Vlasta Matijevi´c, not only my supervisor but also my role model as a professor of mathematics. It was privilege to be guided by her for master's and doctoral thesis. I want to thank all my math teachers, from elementary school onwards, who helped that my love for math rises more and more with each year. Special thanks to Jurica Cudina´ who showed me a glimpse of math theory already in the high school. I thank all members of the Topology seminar in Split who always knew to ask right questions at the right moment and to guide me in the right direction. I also thank Iztok Baniˇcand the rest of the Topology seminar in Maribor who welcomed me as their member and showed me the beauty of a teamwork. -
Topics in Noncommutative Algebraic Geometry, Homological Algebra and K-Theory
Topics in Noncommutative Algebraic Geometry, Homological Algebra and K-Theory Introduction This text is based on my lectures delivered at the School on Algebraic K-Theory and Applications which took place at the International Center for Theoretical Physics (ICTP) in Trieste during the last two weeks of May of 2007. It might be regarded as an introduction to some basic facts of noncommutative algebraic geometry and the related chapters of homological algebra and (as a part of it) a non-conventional version of higher K-theory of noncommutative 'spaces'. Arguments are mostly replaced by sketches of the main steps, or references to complete proofs, which makes the text an easier reading than the ample accounts on different topics discussed here indicated in the bibliography. Lecture 1 is dedicated to the first notions of noncommutative algebraic geometry { preliminaries on 'spaces' represented by categories and morphisms of 'spaces' represented by (isomorphism classes of) functors. We introduce continuous, affine, and locally affine morphisms which lead to definitions of noncommutative schemes and more general locally affine 'spaces'. The notion of a noncommutative scheme is illustrated with important examples related to quantized enveloping algebras: the quantum base affine spaces and flag varieties and the associated quantum D-schemes represented by the categories of (twisted) quantum D-modules introduced in [LR] (see also [T]). Noncommutative projective 'spaces' introduced in [KR1] and more general Grassmannians and flag varieties studied in [KR3] are examples of smooth locally affine noncommutative 'spaces' which are not schemes. In Lecture 2, we recover some fragments of geometry behind the pseudo-geometric picture outlined in the first lecture. -
A Few Points in Topos Theory
A few points in topos theory Sam Zoghaib∗ Abstract This paper deals with two problems in topos theory; the construction of finite pseudo-limits and pseudo-colimits in appropriate sub-2-categories of the 2-category of toposes, and the definition and construction of the fundamental groupoid of a topos, in the context of the Galois theory of coverings; we will take results on the fundamental group of étale coverings in [1] as a starting example for the latter. We work in the more general context of bounded toposes over Set (instead of starting with an effec- tive descent morphism of schemes). Questions regarding the existence of limits and colimits of diagram of toposes arise while studying this prob- lem, but their general relevance makes it worth to study them separately. We expose mainly known constructions, but give some new insight on the assumptions and work out an explicit description of a functor in a coequalizer diagram which was as far as the author is aware unknown, which we believe can be generalised. This is essentially an overview of study and research conducted at dpmms, University of Cambridge, Great Britain, between March and Au- gust 2006, under the supervision of Martin Hyland. Contents 1 Introduction 2 2 General knowledge 3 3 On (co)limits of toposes 6 3.1 The construction of finite limits in BTop/S ............ 7 3.2 The construction of finite colimits in BTop/S ........... 9 4 The fundamental groupoid of a topos 12 4.1 The fundamental group of an atomic topos with a point . 13 4.2 The fundamental groupoid of an unpointed locally connected topos 15 5 Conclusion and future work 17 References 17 ∗e-mail: [email protected] 1 1 Introduction Toposes were first conceived ([2]) as kinds of “generalised spaces” which could serve as frameworks for cohomology theories; that is, mapping topological or geometrical invariants with an algebraic structure to topological spaces. -
Basic Category Theory and Topos Theory
Basic Category Theory and Topos Theory Jaap van Oosten Jaap van Oosten Department of Mathematics Utrecht University The Netherlands Revised, February 2016 Contents 1 Categories and Functors 1 1.1 Definitions and examples . 1 1.2 Some special objects and arrows . 5 2 Natural transformations 8 2.1 The Yoneda lemma . 8 2.2 Examples of natural transformations . 11 2.3 Equivalence of categories; an example . 13 3 (Co)cones and (Co)limits 16 3.1 Limits . 16 3.2 Limits by products and equalizers . 23 3.3 Complete Categories . 24 3.4 Colimits . 25 4 A little piece of categorical logic 28 4.1 Regular categories and subobjects . 28 4.2 The logic of regular categories . 34 4.3 The language L(C) and theory T (C) associated to a regular cat- egory C ................................ 39 4.4 The category C(T ) associated to a theory T : Completeness Theorem 41 4.5 Example of a regular category . 44 5 Adjunctions 47 5.1 Adjoint functors . 47 5.2 Expressing (co)completeness by existence of adjoints; preserva- tion of (co)limits by adjoint functors . 52 6 Monads and Algebras 56 6.1 Algebras for a monad . 57 6.2 T -Algebras at least as complete as D . 61 6.3 The Kleisli category of a monad . 62 7 Cartesian closed categories and the λ-calculus 64 7.1 Cartesian closed categories (ccc's); examples and basic facts . 64 7.2 Typed λ-calculus and cartesian closed categories . 68 7.3 Representation of primitive recursive functions in ccc's with nat- ural numbers object . -
A Small Complete Category
Annals of Pure and Applied Logic 40 (1988) 135-165 135 North-Holland A SMALL COMPLETE CATEGORY J.M.E. HYLAND Department of Pure Mathematics and Mathematical Statktics, 16 Mill Lane, Cambridge CB2 ISB, England Communicated by D. van Dalen Received 14 October 1987 0. Introduction This paper is concerned with a remarkable fact. The effective topos contains a small complete subcategory, essentially the familiar category of partial equiv- alence realtions. This is in contrast to the category of sets (indeed to all Grothendieck toposes) where any small complete category is equivalent to a (complete) poset. Note at once that the phrase ‘a small complete subcategory of a topos’ is misleading. It is not the subcategory but the internal (small) category which matters. Indeed for any ordinary subcategory of a topos there may be a number of internal categories with global sections equivalent to the given subcategory. The appropriate notion of subcategory is an indexed (or better fibred) one, see 0.1. Another point that needs attention is the definition of completeness (see 0.2). In my talk at the Church’s Thesis meeting, and in the first draft of this paper, I claimed too strong a form of completeness for the internal category. (The elementary oversight is described in 2.7.) Fortunately during the writing of [13] my collaborators Edmund Robinson and Giuseppe Rosolini noticed the mistake. Again one needs to pay careful attention to the ideas of indexed (or fibred) categories. The idea that small (sufficiently) complete categories in toposes might exist, and would provide the right setting in which to discuss models for strong polymorphism (quantification over types), was suggested to me by Eugenio Moggi. -
The Hecke Bicategory
Axioms 2012, 1, 291-323; doi:10.3390/axioms1030291 OPEN ACCESS axioms ISSN 2075-1680 www.mdpi.com/journal/axioms Communication The Hecke Bicategory Alexander E. Hoffnung Department of Mathematics, Temple University, 1805 N. Broad Street, Philadelphia, PA 19122, USA; E-Mail: [email protected]; Tel.: +215-204-7841; Fax: +215-204-6433 Received: 19 July 2012; in revised form: 4 September 2012 / Accepted: 5 September 2012 / Published: 9 October 2012 Abstract: We present an application of the program of groupoidification leading up to a sketch of a categorification of the Hecke algebroid—the category of permutation representations of a finite group. As an immediate consequence, we obtain a categorification of the Hecke algebra. We suggest an explicit connection to new higher isomorphisms arising from incidence geometries, which are solutions of the Zamolodchikov tetrahedron equation. This paper is expository in style and is meant as a companion to Higher Dimensional Algebra VII: Groupoidification and an exploration of structures arising in the work in progress, Higher Dimensional Algebra VIII: The Hecke Bicategory, which introduces the Hecke bicategory in detail. Keywords: Hecke algebras; categorification; groupoidification; Yang–Baxter equations; Zamalodchikov tetrahedron equations; spans; enriched bicategories; buildings; incidence geometries 1. Introduction Categorification is, in part, the attempt to shed new light on familiar mathematical notions by replacing a set-theoretic interpretation with a category-theoretic analogue. Loosely speaking, categorification replaces sets, or more generally n-categories, with categories, or more generally (n + 1)-categories, and functions with functors. By replacing interesting equations by isomorphisms, or more generally equivalences, this process often brings to light a new layer of structure previously hidden from view. -
Abelian Categories
Abelian Categories Lemma. In an Ab-enriched category with zero object every finite product is coproduct and conversely. π1 Proof. Suppose A × B //A; B is a product. Define ι1 : A ! A × B and π2 ι2 : B ! A × B by π1ι1 = id; π2ι1 = 0; π1ι2 = 0; π2ι2 = id: It follows that ι1π1+ι2π2 = id (both sides are equal upon applying π1 and π2). To show that ι1; ι2 are a coproduct suppose given ' : A ! C; : B ! C. It φ : A × B ! C has the properties φι1 = ' and φι2 = then we must have φ = φid = φ(ι1π1 + ι2π2) = ϕπ1 + π2: Conversely, the formula ϕπ1 + π2 yields the desired map on A × B. An additive category is an Ab-enriched category with a zero object and finite products (or coproducts). In such a category, a kernel of a morphism f : A ! B is an equalizer k in the diagram k f ker(f) / A / B: 0 Dually, a cokernel of f is a coequalizer c in the diagram f c A / B / coker(f): 0 An Abelian category is an additive category such that 1. every map has a kernel and a cokernel, 2. every mono is a kernel, and every epi is a cokernel. In fact, it then follows immediatly that a mono is the kernel of its cokernel, while an epi is the cokernel of its kernel. 1 Proof of last statement. Suppose f : B ! C is epi and the cokernel of some g : A ! B. Write k : ker(f) ! B for the kernel of f. Since f ◦ g = 0 the map g¯ indicated in the diagram exists. -
Extending Obstructions to Noncommutative Functorial Spectra
EXTENDING OBSTRUCTIONS TO NONCOMMUTATIVE FUNCTORIAL SPECTRA BENNO VAN DEN BERG AND CHRIS HEUNEN Abstract. Any functor from the category of C*-algebras to the category of locales that assigns to each commutative C*-algebra its Gelfand spectrum must be trivial on algebras of n-by-n matrices for n ≥ 3. This obstruction also applies to other spectra such as those named after Zariski, Stone, and Pierce. We extend these no-go results to functors with values in (ringed) topological spaces, (ringed) toposes, schemes, and quantales. The possibility of spectra in other categories is discussed. 1. Introduction The spectrum of a commutative ring is a leading tool of commutative algebra and algebraic geometry. For example, a commutative ring can be reconstructed using (among other ingredients) its Zariski spectrum, a coherent topological space. Spectra are also of central importance to functional analysis and operator algebra. For example, there is a dual equivalence between the category of commutative C*-algebras and compact Hausdorff topological spaces, due to Gelfand.1 A natural question is whether such spectra can be extended to the noncommu- tative setting. Indeed, many candidates have been proposed for noncommutative spectra. In a recent article [23], M. L. Reyes observed that none of the proposed spectra behave functorially, and proved that indeed they cannot, on pain of trivial- izing on the prototypical noncommutative rings Mn(C) of n-by-n matrices with complex entries. To be precise: any functor F : Ringop → Set that satisfies F (C) = Spec(C) for commutative rings C, must also satisfy F (Mn(C)) = ∅ for n ≥ 3.