Groups and Categories
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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. -
A General Theory of Localizations
GENERAL THEORY OF LOCALIZATION DAVID WHITE • Localization in Algebra • Localization in Category Theory • Bousfield localization Thank them for the invitation. Last section contains some of my PhD research, under Mark Hovey at Wesleyan University. For more, please see my website: dwhite03.web.wesleyan.edu 1. The right way to think about localization in algebra Localization is a systematic way of adding multiplicative inverses to a ring, i.e. given a commutative ring R with unity and a multiplicative subset S ⊂ R (i.e. contains 1, closed under product), localization constructs a ring S−1R and a ring homomorphism j : R ! S−1R that takes elements in S to units in S−1R. We want to do this in the best way possible, and we formalize that via a universal property, i.e. for any f : R ! T taking S to units we have a unique g: j R / S−1R f g T | Recall that S−1R is just R × S= ∼ where (r; s) is really r=s and r=s ∼ r0=s0 iff t(rs0 − sr0) = 0 for some t (i.e. fractions are reduced to lowest terms). The ring structure can be verified just as −1 for Q. The map j takes r 7! r=1, and given f you can set g(r=s) = f(r)f(s) . Demonstrate commutativity of the triangle here. The universal property is saying that S−1R is the closest ring to R with the property that all s 2 S are units. A category theorist uses the universal property to define the object, then uses R × S= ∼ as a construction to prove it exists. -
Group Homomorphisms
1-17-2018 Group Homomorphisms Here are the operation tables for two groups of order 4: · 1 a a2 + 0 1 2 1 1 a a2 0 0 1 2 a a a2 1 1 1 2 0 a2 a2 1 a 2 2 0 1 There is an obvious sense in which these two groups are “the same”: You can get the second table from the first by replacing 0 with 1, 1 with a, and 2 with a2. When are two groups the same? You might think of saying that two groups are the same if you can get one group’s table from the other by substitution, as above. However, there are problems with this. In the first place, it might be very difficult to check — imagine having to write down a multiplication table for a group of order 256! In the second place, it’s not clear what a “multiplication table” is if a group is infinite. One way to implement a substitution is to use a function. In a sense, a function is a thing which “substitutes” its output for its input. I’ll define what it means for two groups to be “the same” by using certain kinds of functions between groups. These functions are called group homomorphisms; a special kind of homomorphism, called an isomorphism, will be used to define “sameness” for groups. Definition. Let G and H be groups. A homomorphism from G to H is a function f : G → H such that f(x · y)= f(x) · f(y) forall x,y ∈ G. -
Complete Objects in Categories
Complete objects in categories James Richard Andrew Gray February 22, 2021 Abstract We introduce the notions of proto-complete, complete, complete˚ and strong-complete objects in pointed categories. We show under mild condi- tions on a pointed exact protomodular category that every proto-complete (respectively complete) object is the product of an abelian proto-complete (respectively complete) object and a strong-complete object. This to- gether with the observation that the trivial group is the only abelian complete group recovers a theorem of Baer classifying complete groups. In addition we generalize several theorems about groups (subgroups) with trivial center (respectively, centralizer), and provide a categorical explana- tion behind why the derivation algebra of a perfect Lie algebra with trivial center and the automorphism group of a non-abelian (characteristically) simple group are strong-complete. 1 Introduction Recall that Carmichael [19] called a group G complete if it has trivial cen- ter and each automorphism is inner. For each group G there is a canonical homomorphism cG from G to AutpGq, the automorphism group of G. This ho- momorphism assigns to each g in G the inner automorphism which sends each x in G to gxg´1. It can be readily seen that a group G is complete if and only if cG is an isomorphism. Baer [1] showed that a group G is complete if and only if every normal monomorphism with domain G is a split monomorphism. We call an object in a pointed category complete if it satisfies this latter condi- arXiv:2102.09834v1 [math.CT] 19 Feb 2021 tion. -
Category of G-Groups and Its Spectral Category
Communications in Algebra ISSN: 0092-7872 (Print) 1532-4125 (Online) Journal homepage: http://www.tandfonline.com/loi/lagb20 Category of G-Groups and its Spectral Category María José Arroyo Paniagua & Alberto Facchini To cite this article: María José Arroyo Paniagua & Alberto Facchini (2017) Category of G-Groups and its Spectral Category, Communications in Algebra, 45:4, 1696-1710, DOI: 10.1080/00927872.2016.1222409 To link to this article: http://dx.doi.org/10.1080/00927872.2016.1222409 Accepted author version posted online: 07 Oct 2016. Published online: 07 Oct 2016. Submit your article to this journal Article views: 12 View related articles View Crossmark data Full Terms & Conditions of access and use can be found at http://www.tandfonline.com/action/journalInformation?journalCode=lagb20 Download by: [UNAM Ciudad Universitaria] Date: 29 November 2016, At: 17:29 COMMUNICATIONS IN ALGEBRA® 2017, VOL. 45, NO. 4, 1696–1710 http://dx.doi.org/10.1080/00927872.2016.1222409 Category of G-Groups and its Spectral Category María José Arroyo Paniaguaa and Alberto Facchinib aDepartamento de Matemáticas, División de Ciencias Básicas e Ingeniería, Universidad Autónoma Metropolitana, Unidad Iztapalapa, Mexico, D. F., México; bDipartimento di Matematica, Università di Padova, Padova, Italy ABSTRACT ARTICLE HISTORY Let G be a group. We analyse some aspects of the category G-Grp of G-groups. Received 15 April 2016 In particular, we show that a construction similar to the construction of the Revised 22 July 2016 spectral category, due to Gabriel and Oberst, and its dual, due to the second Communicated by T. Albu. author, is possible for the category G-Grp. -
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. -
Categories of Sets with a Group Action
Categories of sets with a group action Bachelor Thesis of Joris Weimar under supervision of Professor S.J. Edixhoven Mathematisch Instituut, Universiteit Leiden Leiden, 13 June 2008 Contents 1 Introduction 1 1.1 Abstract . .1 1.2 Working method . .1 1.2.1 Notation . .1 2 Categories 3 2.1 Basics . .3 2.1.1 Functors . .4 2.1.2 Natural transformations . .5 2.2 Categorical constructions . .6 2.2.1 Products and coproducts . .6 2.2.2 Fibered products and fibered coproducts . .9 3 An equivalence of categories 13 3.1 G-sets . 13 3.2 Covering spaces . 15 3.2.1 The fundamental group . 15 3.2.2 Covering spaces and the homotopy lifting property . 16 3.2.3 Induced homomorphisms . 18 3.2.4 Classifying covering spaces through the fundamental group . 19 3.3 The equivalence . 24 3.3.1 The functors . 25 4 Applications and examples 31 4.1 Automorphisms and recovering the fundamental group . 31 4.2 The Seifert-van Kampen theorem . 32 4.2.1 The categories C1, C2, and πP -Set ................... 33 4.2.2 The functors . 34 4.2.3 Example . 36 Bibliography 38 Index 40 iii 1 Introduction 1.1 Abstract In the 40s, Mac Lane and Eilenberg introduced categories. Although by some referred to as abstract nonsense, the idea of categories allows one to talk about mathematical objects and their relationions in a general setting. Its origins lie in the field of algebraic topology, one of the topics that will be explored in this thesis. First, a concise introduction to categories will be given. -
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. -
Notes and Solutions to Exercises for Mac Lane's Categories for The
Stefan Dawydiak Version 0.3 July 2, 2020 Notes and Exercises from Categories for the Working Mathematician Contents 0 Preface 2 1 Categories, Functors, and Natural Transformations 2 1.1 Functors . .2 1.2 Natural Transformations . .4 1.3 Monics, Epis, and Zeros . .5 2 Constructions on Categories 6 2.1 Products of Categories . .6 2.2 Functor categories . .6 2.2.1 The Interchange Law . .8 2.3 The Category of All Categories . .8 2.4 Comma Categories . 11 2.5 Graphs and Free Categories . 12 2.6 Quotient Categories . 13 3 Universals and Limits 13 3.1 Universal Arrows . 13 3.2 The Yoneda Lemma . 14 3.2.1 Proof of the Yoneda Lemma . 14 3.3 Coproducts and Colimits . 16 3.4 Products and Limits . 18 3.4.1 The p-adic integers . 20 3.5 Categories with Finite Products . 21 3.6 Groups in Categories . 22 4 Adjoints 23 4.1 Adjunctions . 23 4.2 Examples of Adjoints . 24 4.3 Reflective Subcategories . 28 4.4 Equivalence of Categories . 30 4.5 Adjoints for Preorders . 32 4.5.1 Examples of Galois Connections . 32 4.6 Cartesian Closed Categories . 33 5 Limits 33 5.1 Creation of Limits . 33 5.2 Limits by Products and Equalizers . 34 5.3 Preservation of Limits . 35 5.4 Adjoints on Limits . 35 5.5 Freyd's adjoint functor theorem . 36 1 6 Chapter 6 38 7 Chapter 7 38 8 Abelian Categories 38 8.1 Additive Categories . 38 8.2 Abelian Categories . 38 8.3 Diagram Lemmas . 39 9 Special Limits 41 9.1 Interchange of Limits . -
Arithmetical Foundations Recursion.Evaluation.Consistency Ωi 1
••• Arithmetical Foundations Recursion.Evaluation.Consistency Ωi 1 f¨ur AN GELA & FRAN CISCUS Michael Pfender version 3.1 December 2, 2018 2 Priv. - Doz. M. Pfender, Technische Universit¨atBerlin With the cooperation of Jan Sablatnig Preprint Institut f¨urMathematik TU Berlin submitted to W. DeGRUYTER Berlin book on demand [email protected] The following students have contributed seminar talks: Sandra An- drasek, Florian Blatt, Nick Bremer, Alistair Cloete, Joseph Helfer, Frank Herrmann, Julia Jonczyk, Sophia Lee, Dariusz Lesniowski, Mr. Matysiak, Gregor Myrach, Chi-Thanh Christopher Nguyen, Thomas Richter, Olivia R¨ohrig,Paul Vater, and J¨orgWleczyk. Keywords: primitive recursion, categorical free-variables Arith- metic, µ-recursion, complexity controlled iteration, map code evaluation, soundness, decidability of p. r. predicates, com- plexity controlled iterative self-consistency, Ackermann dou- ble recursion, inconsistency of quantified arithmetical theo- ries, history. Preface Johannes Zawacki, my high school teacher, told us about G¨odel'ssec- ond theorem, on non-provability of consistency of mathematics within mathematics. Bonmot of Andr´eWeil: Dieu existe parceque la Math´e- matique est consistente, et le diable existe parceque nous ne pouvons pas prouver cela { God exists since Mathematics is consistent, and the devil exists since we cannot prove that. The problem with 19th/20th century mathematical foundations, clearly stated in Skolem 1919, is unbound infinitistic (non-constructive) formal existential quantification. In his 1973 -
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. -
Självständiga Arbeten I Matematik
SJÄLVSTÄNDIGA ARBETEN I MATEMATIK MATEMATISKA INSTITUTIONEN, STOCKHOLMS UNIVERSITET Geometric interpretation of non-associative composition algebras av Fredrik Cumlin 2020 - No K13 MATEMATISKA INSTITUTIONEN, STOCKHOLMS UNIVERSITET, 106 91 STOCKHOLM Geometric interpretation of non-associative composition algebras Fredrik Cumlin Självständigt arbete i matematik 15 högskolepoäng, grundnivå Handledare: Wushi Goldring 2020 Abstract This paper aims to discuss the connection between non-associative composition algebra and geometry. It will first recall the notion of an algebra, and investigate the properties of an algebra together with a com- position norm. The composition norm will induce a law on the algebra, which is stated as the composition law. This law is then used to derive the multiplication and conjugation laws, where the last is also known as convolution. These laws are then used to prove Hurwitz’s celebrated the- orem concerning the different finite composition algebras. More properties of composition algebras will be covered, in order to look at the structure of the quaternions H and octonions O. The famous Fano plane will be the finishing touch of the relationship between the standard orthogonal vectors which construct the octonions. Lastly, the notion of invertible maps in relation to invertible loops will be covered, to later show the connection between 8 dimensional rotations − and multiplication of unit octonions. 2 Contents 1 Algebra 4 1.1 The multiplication laws . .6 1.2 The conjugation laws . .7 1.3 Dickson double . .8 1.4 Hurwitz’s theorem . 11 2 Properties of composition algebras 14 2.1 The left-, right- and bi-multiplication maps . 16 2.2 Basic properties of quaternions and octonions .