A Model Structure on the Category of Small Acyclic Categories
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Arxiv:1403.7027V2 [Math.AG] 21 Oct 2015 Ytnoigit Iebnlson Bundles Line Into Tensoring by N Ytednsyfoundation
ON EQUIVARIANT TRIANGULATED CATEGORIES ALEXEY ELAGIN Abstract. Consider a finite group G acting on a triangulated category T . In this paper we investigate triangulated structure on the category T G of G-equivariant objects in T . We prove (under some technical conditions) that such structure exists. Supposed that an action on T is induced by a DG-action on some DG-enhancement of T , we construct a DG-enhancement of T G. Also, we show that the relation “to be an equivariant category with respect to a finite abelian group action” is symmetric on idempotent complete additive categories. 1. Introduction Triangulated categories became very popular in algebra, geometry and topology in last decades. In algebraic geometry, they arise as derived categories of coherent sheaves on algebraic varieties or stacks. It turned out that some geometry of varieties can be under- stood well through their derived categories and homological algebra of these categories. Therefore it is always interesting and important to understand how different geometrical operations, constructions, relations look like on the derived category side. In this paper we are interested in autoequivalences of derived categories or, more gen- eral, in group actions on triangulated categories. For X an algebraic variety, there are “expected” autoequivalences of Db(coh(X)) which are induced by automorphisms of X or by tensoring into line bundles on X. If X is a smooth Fano or if KX is ample, essentially that is all: Bondal and Orlov have shown in [6] that for smooth irreducible projective b variety X with KX or −KX ample all autoequivalences of D (coh(X)) are generated by automorphisms of X, twists into line bundles on X and translations. -
(Pro-) Étale Cohomology 3. Exercise Sheet
(Pro-) Étale Cohomology 3. Exercise Sheet Department of Mathematics Winter Semester 18/19 Prof. Dr. Torsten Wedhorn 2nd November 2018 Timo Henkel Homework Exercise H9 (Clopen subschemes) (12 points) Let X be a scheme. We define Clopen(X ) := Z X Z open and closed subscheme of X . f ⊆ j g Recall that Clopen(X ) is in bijection to the set of idempotent elements of X (X ). Let X S be a morphism of schemes. We consider the functor FX =S fromO the category of S-schemes to the category of sets, given! by FX =S(T S) = Clopen(X S T). ! × Now assume that X S is a finite locally free morphism of schemes. Show that FX =S is representable by an affine étale S-scheme which is of! finite presentation over S. Exercise H10 (Lifting criteria) (12 points) Let f : X S be a morphism of schemes which is locally of finite presentation. Consider the following diagram of S-schemes:! T0 / X (1) f T / S Let be a class of morphisms of S-schemes. We say that satisfies the 1-lifting property (resp. !-lifting property) ≤ withC respect to f , if for all morphisms T0 T in andC for all diagrams9 of the form (1) there exists9 at most (resp. exactly) one morphism of S-schemes T X!which makesC the diagram commutative. Let ! 1 := f : T0 T closed immersion of S-schemes f is given by a locally nilpotent ideal C f ! j g 2 := f : T0 T closed immersion of S-schemes T is affine and T0 is given by a nilpotent ideal C f ! j g 2 3 := f : T0 T closed immersion of S-schemes T is the spectrum of a local ring and T0 is given by an ideal I with I = 0 C f ! j g Show that the following assertions are equivalent: (i) 1 satisfies the 1-lifting property (resp. -
Classification of Subcategories in Abelian Categories and Triangulated Categories
CLASSIFICATION OF SUBCATEGORIES IN ABELIAN CATEGORIES AND TRIANGULATED CATEGORIES ATHESIS SUBMITTED TO THE FACULTY OF GRADUATE STUDIES AND RESEARCH IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN MATHEMATICS UNIVERSITY OF REGINA By Yong Liu Regina, Saskatchewan September 2016 c Copyright 2016: Yong Liu UNIVERSITY OF REGINA FACULTY OF GRADUATE STUDIES AND RESEARCH SUPERVISORY AND EXAMINING COMMITTEE Yong Liu, candidate for the degree of Doctor of Philosophy in Mathematics, has presented a thesis titled, Classification of Subcategories in Abelian Categories and Triangulated Categories, in an oral examination held on September 8, 2016. The following committee members have found the thesis acceptable in form and content, and that the candidate demonstrated satisfactory knowledge of the subject material. External Examiner: Dr. Henning Krause, University of Bielefeld Supervisor: Dr. Donald Stanley, Department of Mathematics and Statistics Committee Member: Dr. Allen Herman, Department of Mathematics and Statistics Committee Member: *Dr. Fernando Szechtman, Department of Mathematics and Statistics Committee Member: Dr. Yiyu Yao, Department of Computer Science Chair of Defense: Dr. Renata Raina-Fulton, Department of Chemistry and Biochemistry *Not present at defense Abstract Two approaches for classifying subcategories of a category are given. We examine the class of Serre subcategories in an abelian category as our first target, using the concepts of monoform objects and the associated atom spectrum [13]. Then we generalize this idea to give a classification of nullity classes in an abelian category, using premonoform objects instead to form a new spectrum so that there is a bijection between the collection of nullity classes and that of closed and extension closed subsets of the spectrum. -
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. -
On Modeling Homotopy Type Theory in Higher Toposes
Review: model categories for type theory Left exact localizations Injective fibrations On modeling homotopy type theory in higher toposes Mike Shulman1 1(University of San Diego) Midwest homotopy type theory seminar Indiana University Bloomington March 9, 2019 Review: model categories for type theory Left exact localizations Injective fibrations Here we go Theorem Every Grothendieck (1; 1)-topos can be presented by a model category that interprets \Book" Homotopy Type Theory with: • Σ-types, a unit type, Π-types with function extensionality, and identity types. • Strict universes, closed under all the above type formers, and satisfying univalence and the propositional resizing axiom. Review: model categories for type theory Left exact localizations Injective fibrations Here we go Theorem Every Grothendieck (1; 1)-topos can be presented by a model category that interprets \Book" Homotopy Type Theory with: • Σ-types, a unit type, Π-types with function extensionality, and identity types. • Strict universes, closed under all the above type formers, and satisfying univalence and the propositional resizing axiom. Review: model categories for type theory Left exact localizations Injective fibrations Some caveats 1 Classical metatheory: ZFC with inaccessible cardinals. 2 Classical homotopy theory: simplicial sets. (It's not clear which cubical sets can even model the (1; 1)-topos of 1-groupoids.) 3 Will not mention \elementary (1; 1)-toposes" (though we can deduce partial results about them by Yoneda embedding). 4 Not the full \internal language hypothesis" that some \homotopy theory of type theories" is equivalent to the homotopy theory of some kind of (1; 1)-category. Only a unidirectional interpretation | in the useful direction! 5 We assume the initiality hypothesis: a \model of type theory" means a CwF. -
Derived Functors and Homological Dimension (Pdf)
DERIVED FUNCTORS AND HOMOLOGICAL DIMENSION George Torres Math 221 Abstract. This paper overviews the basic notions of abelian categories, exact functors, and chain complexes. It will use these concepts to define derived functors, prove their existence, and demon- strate their relationship to homological dimension. I affirm my awareness of the standards of the Harvard College Honor Code. Date: December 15, 2015. 1 2 DERIVED FUNCTORS AND HOMOLOGICAL DIMENSION 1. Abelian Categories and Homology The concept of an abelian category will be necessary for discussing ideas on homological algebra. Loosely speaking, an abelian cagetory is a type of category that behaves like modules (R-mod) or abelian groups (Ab). We must first define a few types of morphisms that such a category must have. Definition 1.1. A morphism f : X ! Y in a category C is a zero morphism if: • for any A 2 C and any g; h : A ! X, fg = fh • for any B 2 C and any g; h : Y ! B, gf = hf We denote a zero morphism as 0XY (or sometimes just 0 if the context is sufficient). Definition 1.2. A morphism f : X ! Y is a monomorphism if it is left cancellative. That is, for all g; h : Z ! X, we have fg = fh ) g = h. An epimorphism is a morphism if it is right cancellative. The zero morphism is a generalization of the zero map on rings, or the identity homomorphism on groups. Monomorphisms and epimorphisms are generalizations of injective and surjective homomorphisms (though these definitions don't always coincide). It can be shown that a morphism is an isomorphism iff it is epic and monic. -
N-Quasi-Abelian Categories Vs N-Tilting Torsion Pairs 3
N-QUASI-ABELIAN CATEGORIES VS N-TILTING TORSION PAIRS WITH AN APPLICATION TO FLOPS OF HIGHER RELATIVE DIMENSION LUISA FIOROT Abstract. It is a well established fact that the notions of quasi-abelian cate- gories and tilting torsion pairs are equivalent. This equivalence fits in a wider picture including tilting pairs of t-structures. Firstly, we extend this picture into a hierarchy of n-quasi-abelian categories and n-tilting torsion classes. We prove that any n-quasi-abelian category E admits a “derived” category D(E) endowed with a n-tilting pair of t-structures such that the respective hearts are derived equivalent. Secondly, we describe the hearts of these t-structures as quotient categories of coherent functors, generalizing Auslander’s Formula. Thirdly, we apply our results to Bridgeland’s theory of perverse coherent sheaves for flop contractions. In Bridgeland’s work, the relative dimension 1 assumption guaranteed that f∗-acyclic coherent sheaves form a 1-tilting torsion class, whose associated heart is derived equivalent to D(Y ). We generalize this theorem to relative dimension 2. Contents Introduction 1 1. 1-tilting torsion classes 3 2. n-Tilting Theorem 7 3. 2-tilting torsion classes 9 4. Effaceable functors 14 5. n-coherent categories 17 6. n-tilting torsion classes for n> 2 18 7. Perverse coherent sheaves 28 8. Comparison between n-abelian and n + 1-quasi-abelian categories 32 Appendix A. Maximal Quillen exact structure 33 Appendix B. Freyd categories and coherent functors 34 Appendix C. t-structures 37 References 39 arXiv:1602.08253v3 [math.RT] 28 Dec 2019 Introduction In [6, 3.3.1] Beilinson, Bernstein and Deligne introduced the notion of a t- structure obtained by tilting the natural one on D(A) (derived category of an abelian category A) with respect to a torsion pair (X , Y). -
A Godefroy-Kalton Principle for Free Banach Lattices 3
A GODEFROY-KALTON PRINCIPLE FOR FREE BANACH LATTICES ANTONIO AVILES,´ GONZALO MART´INEZ-CERVANTES, JOSE´ RODR´IGUEZ, AND PEDRO TRADACETE Abstract. Motivated by the Lipschitz-lifting property of Banach spaces introduced by Godefroy and Kalton, we consider the lattice-lifting property, which is an analogous notion within the category of Ba- nach lattices and lattice homomorphisms. Namely, a Banach lattice X satisfies the lattice-lifting property if every lattice homomorphism to X having a bounded linear right-inverse must have a lattice homomor- phism right-inverse. In terms of free Banach lattices, this can be rephrased into the following question: which Banach lattices embed into the free Banach lattice which they generate as a lattice-complemented sublattice? We will provide necessary conditions for a Banach lattice to have the lattice-lifting property, and show that this property is shared by Banach spaces with a 1-unconditional basis as well as free Banach lattices. The case of C(K) spaces will also be analyzed. 1. Introduction In a fundamental paper concerning the Lipschitz structure of Banach spaces, G. Godefroy and N.J. Kalton introduced the Lipschitz-lifting property of a Banach space. In order to properly introduce this notion, and as a motivation for our work, let us recall the basic ingredients for this construction (see [9] for details). Given a Banach space E, let Lip0(E) denote the Banach space of all real-valued Lipschitz functions on E which vanish at 0, equipped with the norm |f(x) − f(y)| kfk = sup : x, y ∈ E, x 6= y . Lip0(E) kx − yk E The Lipschitz-free space over E, denoted by F(E), is the canonical predual of Lip0(E), that is the closed ∗ linear span of the evaluation functionals δ(x) ∈ Lip0(E) given by hδ(x),fi = f(x) for all x ∈ E and all f ∈ Lip0(E). -
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 . -
Weak Subobjects and Weak Limits in Categories and Homotopy Categories Cahiers De Topologie Et Géométrie Différentielle Catégoriques, Tome 38, No 4 (1997), P
CAHIERS DE TOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLE CATÉGORIQUES MARCO GRANDIS Weak subobjects and weak limits in categories and homotopy categories Cahiers de topologie et géométrie différentielle catégoriques, tome 38, no 4 (1997), p. 301-326 <http://www.numdam.org/item?id=CTGDC_1997__38_4_301_0> © Andrée C. Ehresmann et les auteurs, 1997, tous droits réservés. L’accès aux archives de la revue « Cahiers de topologie et géométrie différentielle catégoriques » implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ CAHIERS DE TOPOLOGIE ET Volume XXXVIII-4 (1997) GEOMETRIE DIFFERENTIELLE CATEGORIQUES WEAK SUBOBJECTS AND WEAK LIMITS IN CATEGORIES AND HOMOTOPY CATEGORIES by Marco GRANDIS R6sumi. Dans une cat6gorie donn6e, un sousobjet faible, ou variation, d’un objet A est defini comme une classe d’6quivalence de morphismes A valeurs dans A, de faqon a étendre la notion usuelle de sousobjet. Les sousobjets faibles sont lies aux limites faibles, comme les sousobjets aux limites; et ils peuvent 8tre consid6r6s comme remplaqant les sousobjets dans les categories "a limites faibles", notamment la cat6gorie d’homotopie HoTop des espaces topologiques, ou il forment un treillis de types de fibration sur 1’espace donn6. La classification des variations des groupes et des groupes ab£liens est un outil important pour d6terminer ces types de fibration, par les foncteurs d’homotopie et homologie. Introduction We introduce here the notion of weak subobject in a category, as an extension of the notion of subobject. -
Lecture 11 Last Lecture We Defined Connections on Fiber Bundles. This Lecture We Will Attempt to Explain What a Connection Does
Lecture 11 Last lecture we defined connections on fiber bundles. This lecture we will attempt to explain what a connection does for differential ge- ometers. We start with two basic concepts from algebraic topology. The first is that of homotopy lifting property. Let us recall: Definition 1. Let π : E → B be a continuous map and X a space. We say that (X, π) satisfies homotopy lifting property if for every H : X × I → B and map f : X → E with π ◦ f = H( , 0) there exists a homotopy F : X × I → E starting at f such that π ◦ F = H. If ({∗}, π) satisfies this property then we say that π has the homotopy lifting property for a point. Now, as is well known and easily proven, every fiber bundle satisfies the homotopy lifting property. A much stronger property is the notion of a covering space (which we will not recall with precision). Here we have the homotopy lifting property for a point plus uniqueness or the unique path lifting property. I.e. there is exactly one lift of a given path in the base starting at a specified point in the total space. The point of a connection is to take the ambiguity out of the homotopy lifting property for X = {∗} and force a fiber bundle to satisfy the unique path lifting property. We will see many conceptual advantages of this uniqueness, but let us start by making this a bit clearer via parallel transport. Definition 2. Let B be a topological space and PB the category whose objects are points of B and whose morphisms are defined via {γ : → B : ∃ c, d ∈ s.t.