Brief Introduction to Categories
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Chapter 4. Homomorphisms and Isomorphisms of Groups
Chapter 4. Homomorphisms and Isomorphisms of Groups 4.1 Note: We recall the following terminology. Let X and Y be sets. When we say that f is a function or a map from X to Y , written f : X ! Y , we mean that for every x 2 X there exists a unique corresponding element y = f(x) 2 Y . The set X is called the domain of f and the range or image of f is the set Image(f) = f(X) = f(x) x 2 X . For a set A ⊆ X, the image of A under f is the set f(A) = f(a) a 2 A and for a set −1 B ⊆ Y , the inverse image of B under f is the set f (B) = x 2 X f(x) 2 B . For a function f : X ! Y , we say f is one-to-one (written 1 : 1) or injective when for every y 2 Y there exists at most one x 2 X such that y = f(x), we say f is onto or surjective when for every y 2 Y there exists at least one x 2 X such that y = f(x), and we say f is invertible or bijective when f is 1:1 and onto, that is for every y 2 Y there exists a unique x 2 X such that y = f(x). When f is invertible, the inverse of f is the function f −1 : Y ! X defined by f −1(y) = x () y = f(x). For f : X ! Y and g : Y ! Z, the composite g ◦ f : X ! Z is given by (g ◦ f)(x) = g(f(x)). -
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. -
Product Category Rules
Product Category Rules PRé’s own Vee Subramanian worked with a team of experts to publish a thought provoking article on product category rules (PCRs) in the International Journal of Life Cycle Assessment in April 2012. The full article can be viewed at http://www.springerlink.com/content/qp4g0x8t71432351/. Here we present the main body of the text that explores the importance of global alignment of programs that manage PCRs. Comparing Product Category Rules from Different Programs: Learned Outcomes Towards Global Alignment Vee Subramanian • Wesley Ingwersen • Connie Hensler • Heather Collie Abstract Purpose Product category rules (PCRs) provide category-specific guidance for estimating and reporting product life cycle environmental impacts, typically in the form of environmental product declarations and product carbon footprints. Lack of global harmonization between PCRs or sector guidance documents has led to the development of duplicate PCRs for same products. Differences in the general requirements (e.g., product category definition, reporting format) and LCA methodology (e.g., system boundaries, inventory analysis, allocation rules, etc.) diminish the comparability of product claims. Methods A comparison template was developed to compare PCRs from different global program operators. The goal was to identify the differences between duplicate PCRs from a broad selection of product categories and propose a path towards alignment. We looked at five different product categories: Milk/dairy (2 PCRs), Horticultural products (3 PCRs), Wood-particle board (2 PCRs), and Laundry detergents (4 PCRs). Results & discussion Disparity between PCRs ranged from broad differences in scope, system boundaries and impacts addressed (e.g. multi-impact vs. carbon footprint only) to specific differences of technical elements. -
Covers, Envelopes, and Cotorsion Theories in Locally Presentable
COVERS, ENVELOPES, AND COTORSION THEORIES IN LOCALLY PRESENTABLE ABELIAN CATEGORIES AND CONTRAMODULE CATEGORIES L. POSITSELSKI AND J. ROSICKY´ Abstract. We prove general results about completeness of cotorsion theories and existence of covers and envelopes in locally presentable abelian categories, extend- ing the well-established theory for module categories and Grothendieck categories. These results are then applied to the categories of contramodules over topological rings, which provide examples and counterexamples. 1. Introduction 1.1. An abelian category with enough projective objects is determined by (can be recovered from) its full subcategory of projective objects. An explicit construction providing the recovery procedure is described, in the nonadditive context, in the paper [13] (see also [21]), where it is called “the exact completion of a weakly left exact category”, and in the additive context, in the paper [22, Lemma 1], where it is called “the category of coherent functors”. In fact, the exact completion in the additive context is contained already in [19], as it is explained in [35]. Hence, in particular, any abelian category with arbitrary coproducts and a single small (finitely generated, finitely presented) projective generator P is equivalent to the category of right modules over the ring of endomorphisms of P . More generally, an abelian category with arbitrary coproducts and a set of small projective generators is equivalent to the category of contravariant additive functors from its full subcategory formed by these generators into the category of abelian groups (right modules over arXiv:1512.08119v2 [math.CT] 21 Apr 2017 the big ring of morphisms between the generators). Now let K be an abelian category with a single, not necessarily small projective generator P ; suppose that all coproducts of copies of P exist in K. -
An Introduction to Category Theory and Categorical Logic
An Introduction to Category Theory and Categorical Logic Wolfgang Jeltsch Category theory An Introduction to Category Theory basics Products, coproducts, and and Categorical Logic exponentials Categorical logic Functors and Wolfgang Jeltsch natural transformations Monoidal TTU¨ K¨uberneetika Instituut categories and monoidal functors Monads and Teooriaseminar comonads April 19 and 26, 2012 References An Introduction to Category Theory and Categorical Logic Category theory basics Wolfgang Jeltsch Category theory Products, coproducts, and exponentials basics Products, coproducts, and Categorical logic exponentials Categorical logic Functors and Functors and natural transformations natural transformations Monoidal categories and Monoidal categories and monoidal functors monoidal functors Monads and comonads Monads and comonads References References An Introduction to Category Theory and Categorical Logic Category theory basics Wolfgang Jeltsch Category theory Products, coproducts, and exponentials basics Products, coproducts, and Categorical logic exponentials Categorical logic Functors and Functors and natural transformations natural transformations Monoidal categories and Monoidal categories and monoidal functors monoidal functors Monads and Monads and comonads comonads References References An Introduction to From set theory to universal algebra Category Theory and Categorical Logic Wolfgang Jeltsch I classical set theory (for example, Zermelo{Fraenkel): I sets Category theory basics I functions from sets to sets Products, I composition -
Toy Industry Product Categories
Definitions Document Toy Industry Product Categories Action Figures Action Figures, Playsets and Accessories Includes licensed and theme figures that have an action-based play pattern. Also includes clothing, vehicles, tools, weapons or play sets to be used with the action figure. Role Play (non-costume) Includes role play accessory items that are both action themed and generically themed. This category does not include dress-up or costume items, which have their own category. Arts and Crafts Chalk, Crayons, Markers Paints and Pencils Includes singles and sets of these items. (e.g., box of crayons, bucket of chalk). Reusable Compounds (e.g., Clay, Dough, Sand, etc.) and Kits Includes any reusable compound, or items that can be manipulated into creating an object. Some examples include dough, sand and clay. Also includes kits that are intended for use with reusable compounds. Design Kits and Supplies – Reusable Includes toys used for designing that have a reusable feature or extra accessories (e.g., extra paper). Examples include Etch-A-Sketch, Aquadoodle, Lite Brite, magnetic design boards, and electronic or digital design units. Includes items created on the toy themselves or toys that connect to a computer or tablet for designing / viewing. Design Kits and Supplies – Single Use Includes items used by a child to create art and sculpture projects. These items are all-inclusive kits and may contain supplies that are needed to create the project (e.g., crayons, paint, yarn). This category includes refills that are sold separately to coincide directly with the kits. Also includes children’s easels and paint-by-number sets. -
Introduction to Category Theory (Notes for Course Taught at HUJI, Fall 2020-2021) (UNPOLISHED DRAFT)
Introduction to category theory (notes for course taught at HUJI, Fall 2020-2021) (UNPOLISHED DRAFT) Alexander Yom Din February 10, 2021 It is never true that two substances are entirely alike, differing only in being two rather than one1. G. W. Leibniz, Discourse on metaphysics 1This can be imagined to be related to at least two of our themes: the imperative of considering a contractible groupoid of objects as an one single object, and also the ideology around Yoneda's lemma ("no two different things have all their properties being exactly the same"). 1 Contents 1 The basic language 3 1.1 Categories . .3 1.2 Functors . .7 1.3 Natural transformations . .9 2 Equivalence of categories 11 2.1 Contractible groupoids . 11 2.2 Fibers . 12 2.3 Fibers and fully faithfulness . 12 2.4 A lemma on fully faithfulness in families . 13 2.5 Definition of equivalence of categories . 14 2.6 Simple examples of equivalence of categories . 17 2.7 Theory of the fundamental groupoid and covering spaces . 18 2.8 Affine algebraic varieties . 23 2.9 The Gelfand transform . 26 2.10 Galois theory . 27 3 Yoneda's lemma, representing objects, limits 27 3.1 Yoneda's lemma . 27 3.2 Representing objects . 29 3.3 The definition of a limit . 33 3.4 Examples of limits . 34 3.5 Dualizing everything . 39 3.6 Examples of colimits . 39 3.7 General colimits in terms of special ones . 41 4 Adjoint functors 42 4.1 Bifunctors . 42 4.2 The definition of adjoint functors . 43 4.3 Some examples of adjoint functors . -
CS E6204 Lecture 5 Automorphisms
CS E6204 Lecture 5 Automorphisms Abstract An automorphism of a graph is a permutation of its vertex set that preserves incidences of vertices and edges. Under composition, the set of automorphisms of a graph forms what algbraists call a group. In this section, graphs are assumed to be simple. 1. The Automorphism Group 2. Graphs with Given Group 3. Groups of Graph Products 4. Transitivity 5. s-Regularity and s-Transitivity 6. Graphical Regular Representations 7. Primitivity 8. More Automorphisms of Infinite Graphs * This lecture is based on chapter [?] contributed by Mark E. Watkins of Syracuse University to the Handbook of Graph The- ory. 1 1 The Automorphism Group Given a graph X, a permutation α of V (X) is an automor- phism of X if for all u; v 2 V (X) fu; vg 2 E(X) , fα(u); α(v)g 2 E(X) The set of all automorphisms of a graph X, under the operation of composition of functions, forms a subgroup of the symmetric group on V (X) called the automorphism group of X, and it is denoted Aut(X). Figure 1: Example 2.2.3 from GTAIA. Notation The identity of any permutation group is denoted by ι. (JG) Thus, Watkins would write ι instead of λ0 in Figure 1. 2 Rigidity and Orbits A graph is asymmetric if the identity ι is its only automor- phism. Synonym: rigid. Figure 2: The smallest rigid graph. (JG) The orbit of a vertex v in a graph G is the set of all vertices α(v) such that α is an automorphism of G. -
1 the Spin Homomorphism SL2(C) → SO1,3(R) a Summary from Multiple Sources Written by Y
1 The spin homomorphism SL2(C) ! SO1;3(R) A summary from multiple sources written by Y. Feng and Katherine E. Stange Abstract We will discuss the spin homomorphism SL2(C) ! SO1;3(R) in three manners. Firstly we interpret SL2(C) as acting on the Minkowski 1;3 spacetime R ; secondly by viewing the quadratic form as a twisted 1 1 P × P ; and finally using Clifford groups. 1.1 Introduction The spin homomorphism SL2(C) ! SO1;3(R) is a homomorphism of classical matrix Lie groups. The lefthand group con- sists of 2 × 2 complex matrices with determinant 1. The righthand group consists of 4 × 4 real matrices with determinant 1 which preserve some fixed real quadratic form Q of signature (1; 3). This map is alternately called the spinor map and variations. The image of this map is the identity component + of SO1;3(R), denoted SO1;3(R). The kernel is {±Ig. Therefore, we obtain an isomorphism + PSL2(C) = SL2(C)= ± I ' SO1;3(R): This is one of a family of isomorphisms of Lie groups called exceptional iso- morphisms. In Section 1.3, we give the spin homomorphism explicitly, al- though these formulae are unenlightening by themselves. In Section 1.4 we describe O1;3(R) in greater detail as the group of Lorentz transformations. This document describes this homomorphism from three distinct per- spectives. The first is very concrete, and constructs, using the language of Minkowski space, Lorentz transformations and Hermitian matrices, an ex- 4 plicit action of SL2(C) on R preserving Q (Section 1.5). -
A STUDY on the ALGEBRAIC STRUCTURE of SL 2(Zpz)
A STUDY ON THE ALGEBRAIC STRUCTURE OF SL2 Z pZ ( ~ ) A Thesis Presented to The Honors Tutorial College Ohio University In Partial Fulfillment of the Requirements for Graduation from the Honors Tutorial College with the degree of Bachelor of Science in Mathematics by Evan North April 2015 Contents 1 Introduction 1 2 Background 5 2.1 Group Theory . 5 2.2 Linear Algebra . 14 2.3 Matrix Group SL2 R Over a Ring . 22 ( ) 3 Conjugacy Classes of Matrix Groups 26 3.1 Order of the Matrix Groups . 26 3.2 Conjugacy Classes of GL2 Fp ....................... 28 3.2.1 Linear Case . .( . .) . 29 3.2.2 First Quadratic Case . 29 3.2.3 Second Quadratic Case . 30 3.2.4 Third Quadratic Case . 31 3.2.5 Classes in SL2 Fp ......................... 33 3.3 Splitting of Classes of(SL)2 Fp ....................... 35 3.4 Results of SL2 Fp ..............................( ) 40 ( ) 2 4 Toward Lifting to SL2 Z p Z 41 4.1 Reduction mod p ...............................( ~ ) 42 4.2 Exploring the Kernel . 43 i 4.3 Generalizing to SL2 Z p Z ........................ 46 ( ~ ) 5 Closing Remarks 48 5.1 Future Work . 48 5.2 Conclusion . 48 1 Introduction Symmetries are one of the most widely-known examples of pure mathematics. Symmetry is when an object can be rotated, flipped, or otherwise transformed in such a way that its appearance remains the same. Basic geometric figures should create familiar examples, take for instance the triangle. Figure 1: The symmetries of a triangle: 3 reflections, 2 rotations. The red lines represent the reflection symmetries, where the trianlge is flipped over, while the arrows represent the rotational symmetry of the triangle. -
Group Properties and Group Isomorphism
GROUP PROPERTIES AND GROUP ISOMORPHISM Evelyn. M. Manalo Mathematics Honors Thesis University of California, San Diego May 25, 2001 Faculty Mentor: Professor John Wavrik Department of Mathematics GROUP PROPERTIES AND GROUP ISOMORPHISM I n t r o d u c t i o n T H E I M P O R T A N C E O F G R O U P T H E O R Y is relevant to every branch of Mathematics where symmetry is studied. Every symmetrical object is associated with a group. It is in this association why groups arise in many different areas like in Quantum Mechanics, in Crystallography, in Biology, and even in Computer Science. There is no such easy definition of symmetry among mathematical objects without leading its way to the theory of groups. In this paper we present the first stages of constructing a systematic method for classifying groups of small orders. Classifying groups usually arise when trying to distinguish the number of non-isomorphic groups of order n. This paper arose from an attempt to find a formula or an algorithm for classifying groups given invariants that can be readily determined without any other known assumptions about the group. This formula is very useful if we want to know if two groups are isomorphic. Mathematical objects are considered to be essentially the same, from the point of view of their algebraic properties, when they are isomorphic. When two groups Γ and Γ’ have exactly the same group-theoretic structure then we say that Γ is isomorphic to Γ’ or vice versa. -
Isomorphisms, Automorphisms, Homomorphisms, Kernels
Isomorphisms, Automorphisms, Homomorphisms, Kernels September 10, 2009 Definition 1 Let G; G0 be groups and let f : G ! G0 be a mapping (of the underlying sets). We say that f is a (group) homomorphism if f(x · y) = f(x) · f(y) for all x; y 2 G. The composition f G−!G0−!h G" of two homomorphisms (as displayed) is again a homomorphism. For any two groups G, G0 by the trivial homomorphism from G to G0 we mean the mapping G ! G0 sending all elements to the identity element in G0. Recall: Definition 2 A group homomorphism f : G ! G0 is called a group isomorphism|or, for short, an isomorphism|if f is bijective. An isomorphism from a group G to itself is called an automorphism. Exercise 1 Let f : G ! G0 be a homomorphism of groups. The image under the homomorphism 0 f of any subgroup of G is a subgroup of G ; if H ⊂ G is generated by elements fx1; x2; : : : ; xng then its image f(H) is generated by the images ff(x1); f(x2); : : : ; f(xn)g; the image of any cyclic subgroup is cyclic; If f is an isomorphism from G to G0 and x 2 G then the order of x (which|by definition—is the order of the cyclic subgroup generated by x) is equal to the order of f(x). For G a group, let Aut(G) be the set of all automorphisms of G, with `composition of automorphisms" as its `composition law' (denoted by a center-dot (·). As noted in class, Proposition 1 If G is a group then Aut(G) is also a group.