Introduction to Category Theory
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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)). -
The General Linear Group
18.704 Gabe Cunningham 2/18/05 [email protected] The General Linear Group Definition: Let F be a field. Then the general linear group GLn(F ) is the group of invert- ible n × n matrices with entries in F under matrix multiplication. It is easy to see that GLn(F ) is, in fact, a group: matrix multiplication is associative; the identity element is In, the n × n matrix with 1’s along the main diagonal and 0’s everywhere else; and the matrices are invertible by choice. It’s not immediately clear whether GLn(F ) has infinitely many elements when F does. However, such is the case. Let a ∈ F , a 6= 0. −1 Then a · In is an invertible n × n matrix with inverse a · In. In fact, the set of all such × matrices forms a subgroup of GLn(F ) that is isomorphic to F = F \{0}. It is clear that if F is a finite field, then GLn(F ) has only finitely many elements. An interesting question to ask is how many elements it has. Before addressing that question fully, let’s look at some examples. ∼ × Example 1: Let n = 1. Then GLn(Fq) = Fq , which has q − 1 elements. a b Example 2: Let n = 2; let M = ( c d ). Then for M to be invertible, it is necessary and sufficient that ad 6= bc. If a, b, c, and d are all nonzero, then we can fix a, b, and c arbitrarily, and d can be anything but a−1bc. This gives us (q − 1)3(q − 2) matrices. -
Nearly Locally Presentable Categories Are Locally Presentable Is Equivalent to Vopˇenka’S Principle
NEARLY LOCALLY PRESENTABLE CATEGORIES L. POSITSELSKI AND J. ROSICKY´ Abstract. We introduce a new class of categories generalizing locally presentable ones. The distinction does not manifest in the abelian case and, assuming Vopˇenka’s principle, the same happens in the regular case. The category of complete partial orders is the natural example of a nearly locally finitely presentable category which is not locally presentable. 1. Introduction Locally presentable categories were introduced by P. Gabriel and F. Ulmer in [6]. A category K is locally λ-presentable if it is cocomplete and has a strong generator consisting of λ-presentable objects. Here, λ is a regular cardinal and an object A is λ-presentable if its hom-functor K(A, −): K → Set preserves λ-directed colimits. A category is locally presentable if it is locally λ-presentable for some λ. This con- cept of presentability formalizes the usual practice – for instance, finitely presentable groups are precisely groups given by finitely many generators and finitely many re- lations. Locally presentable categories have many nice properties, in particular they are complete and co-wellpowered. Gabriel and Ulmer [6] also showed that one can define locally presentable categories by using just monomorphisms instead all morphisms. They defined λ-generated ob- jects as those whose hom-functor K(A, −) preserves λ-directed colimits of monomor- phisms. Again, this concept formalizes the usual practice – finitely generated groups are precisely groups admitting a finite set of generators. This leads to locally gener- ated categories, where a cocomplete category K is locally λ-generated if it has a strong arXiv:1710.10476v2 [math.CT] 2 Apr 2018 generator consisting of λ-generated objects and every object of K has only a set of strong quotients. -
Categories, Functors, and Natural Transformations I∗
Lecture 2: Categories, functors, and natural transformations I∗ Nilay Kumar June 4, 2014 (Meta)categories We begin, for the moment, with rather loose definitions, free from the technicalities of set theory. Definition 1. A metagraph consists of objects a; b; c; : : :, arrows f; g; h; : : :, and two operations, as follows. The first is the domain, which assigns to each arrow f an object a = dom f, and the second is the codomain, which assigns to each arrow f an object b = cod f. This is visually indicated by f : a ! b. Definition 2. A metacategory is a metagraph with two additional operations. The first is the identity, which assigns to each object a an arrow Ida = 1a : a ! a. The second is the composition, which assigns to each pair g; f of arrows with dom g = cod f an arrow g ◦ f called their composition, with g ◦ f : dom f ! cod g. This operation may be pictured as b f g a c g◦f We require further that: composition is associative, k ◦ (g ◦ f) = (k ◦ g) ◦ f; (whenever this composition makese sense) or diagrammatically that the diagram k◦(g◦f)=(k◦g)◦f a d k◦g f k g◦f b g c commutes, and that for all arrows f : a ! b and g : b ! c, we have 1b ◦ f = f and g ◦ 1b = g; or diagrammatically that the diagram f a b f g 1b g b c commutes. ∗This talk follows [1] I.1-4 very closely. 1 Recall that a diagram is commutative when, for each pair of vertices c and c0, any two paths formed from direct edges leading from c to c0 yield, by composition of labels, equal arrows from c to c0. -
Irreducible Representations of Finite Monoids
U.U.D.M. Project Report 2019:11 Irreducible representations of finite monoids Christoffer Hindlycke Examensarbete i matematik, 30 hp Handledare: Volodymyr Mazorchuk Examinator: Denis Gaidashev Mars 2019 Department of Mathematics Uppsala University Irreducible representations of finite monoids Christoffer Hindlycke Contents Introduction 2 Theory 3 Finite monoids and their structure . .3 Introductory notions . .3 Cyclic semigroups . .6 Green’s relations . .7 von Neumann regularity . 10 The theory of an idempotent . 11 The five functors Inde, Coinde, Rese,Te and Ne ..................... 11 Idempotents and simple modules . 14 Irreducible representations of a finite monoid . 17 Monoid algebras . 17 Clifford-Munn-Ponizovski˘ıtheory . 20 Application 24 The symmetric inverse monoid . 24 Calculating the irreducible representations of I3 ........................ 25 Appendix: Prerequisite theory 37 Basic definitions . 37 Finite dimensional algebras . 41 Semisimple modules and algebras . 41 Indecomposable modules . 42 An introduction to idempotents . 42 1 Irreducible representations of finite monoids Christoffer Hindlycke Introduction This paper is a literature study of the 2016 book Representation Theory of Finite Monoids by Benjamin Steinberg [3]. As this book contains too much interesting material for a simple master thesis, we have narrowed our attention to chapters 1, 4 and 5. This thesis is divided into three main parts: Theory, Application and Appendix. Within the Theory chapter, we (as the name might suggest) develop the necessary theory to assist with finding irreducible representations of finite monoids. Finite monoids and their structure gives elementary definitions as regards to finite monoids, and expands on the basic theory of their structure. This part corresponds to chapter 1 in [3]. The theory of an idempotent develops just enough theory regarding idempotents to enable us to state a key result, from which the principal result later follows almost immediately. -
Coreflective Subcategories
transactions of the american mathematical society Volume 157, June 1971 COREFLECTIVE SUBCATEGORIES BY HORST HERRLICH AND GEORGE E. STRECKER Abstract. General morphism factorization criteria are used to investigate categorical reflections and coreflections, and in particular epi-reflections and mono- coreflections. It is shown that for most categories with "reasonable" smallness and completeness conditions, each coreflection can be "split" into the composition of two mono-coreflections and that under these conditions mono-coreflective subcategories can be characterized as those which are closed under the formation of coproducts and extremal quotient objects. The relationship of reflectivity to closure under limits is investigated as well as coreflections in categories which have "enough" constant morphisms. 1. Introduction. The concept of reflections in categories (and likewise the dual notion—coreflections) serves the purpose of unifying various fundamental con- structions in mathematics, via "universal" properties that each possesses. His- torically, the concept seems to have its roots in the fundamental construction of E. Cech [4] whereby (using the fact that the class of compact spaces is productive and closed-hereditary) each completely regular F2 space is densely embedded in a compact F2 space with a universal extension property. In [3, Appendice III; Sur les applications universelles] Bourbaki has shown the essential underlying similarity that the Cech-Stone compactification has with other mathematical extensions, such as the completion of uniform spaces and the embedding of integral domains in their fields of fractions. In doing so, he essentially defined the notion of reflections in categories. It was not until 1964, when Freyd [5] published the first book dealing exclusively with the theory of categories, that sufficient categorical machinery and insight were developed to allow for a very simple formulation of the concept of reflections and for a basic investigation of reflections as entities themselvesi1). -
Categories of Modules for Idempotent Rings and Morita Equivalences
Master's Thesis Categories of Modules for Idempotent Rings and Morita Equivalences I Leandro Marm 1997 t Department of Mathematics University of Glasgow University Gardens Glasgow, G12 8QW ProQuest Number: 13834261 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a com plete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. uest ProQuest 13834261 Published by ProQuest LLC(2019). Copyright of the Dissertation is held by the Author. All rights reserved. This work is protected against unauthorized copying under Title 17, United States C ode Microform Edition © ProQuest LLC. ProQuest LLC. 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 48106- 1346 10$ af 7 En resolucion, el se enfrasco tanto en su lectura, que se le pasaban las noches leyendo de claro en claro, y los dias de turbio en turbio; y asi, del poco dormir y del mucho leer se le seco el celebro 1 de manera, que vino a perder el juicio. M iguel de C ervantes Sa a v e d r a : El Ingenioso Hidalgo don Quijote de la Mancha. 1In modern Spanish this word is written ”cerebro” C ontents Chapter 1, Introduction 4 Chapter 2. Categories of Modules for Rings I 7 1. Noncommutative Localization 7 2. The Construction of the Categories 11 3. The Equivalence of the Categories 24 4. The Independence of the Base Ring 28 Chapter 3. -
The Category of Sheaves Is a Topos Part 2
The category of sheaves is a topos part 2 Patrick Elliott Recall from the last talk that for a small category C, the category PSh(C) of presheaves on C is an elementary topos. Explicitly, PSh(C) has the following structure: • Given two presheaves F and G on C, the exponential GF is the presheaf defined on objects C 2 obC by F G (C) = Hom(hC × F; G); where hC = Hom(−;C) is the representable functor associated to C, and the product × is defined object-wise. • Writing 1 for the constant presheaf of the one object set, the subobject classifier true : 1 ! Ω in PSh(C) is defined on objects by Ω(C) := fS j S is a sieve on C in Cg; and trueC : ∗ ! Ω(C) sends ∗ to the maximal sieve t(C). The goal of this talk is to refine this structure to show that the category Shτ (C) of sheaves on a site (C; τ) is also an elementary topos. To do this we must make use of the sheafification functor defined at the end of the first talk: Theorem 0.1. The inclusion functor i : Shτ (C) ! PSh(C) has a left adjoint a : PSh(C) ! Shτ (C); called sheafification, or the associated sheaf functor. Moreover, this functor commutes with finite limits. Explicitly, a(F) = (F +)+, where + F (C) := colimS2τ(C)Match(S; F); where Match(S; F) is the set of matching families for the cover S of C, and the colimit is taken over all covering sieves of C, ordered by reverse inclusion. -
Universality of Multiplicative Infinite Loop Space Machines
UNIVERSALITY OF MULTIPLICATIVE INFINITE LOOP SPACE MACHINES DAVID GEPNER, MORITZ GROTH AND THOMAS NIKOLAUS Abstract. We establish a canonical and unique tensor product for commutative monoids and groups in an ∞-category C which generalizes the ordinary tensor product of abelian groups. Using this tensor product we show that En-(semi)ring objects in C give rise to En-ring spectrum objects in C. In the case that C is the ∞-category of spaces this produces a multiplicative infinite loop space machine which can be applied to the algebraic K-theory of rings and ring spectra. The main tool we use to establish these results is the theory of smashing localizations of presentable ∞-categories. In particular, we identify preadditive and additive ∞-categories as the local objects for certain smashing localizations. A central theme is the stability of algebraic structures under basechange; for example, we show Ring(D ⊗ C) ≃ Ring(D) ⊗ C. Lastly, we also consider these algebraic structures from the perspective of Lawvere algebraic theories in ∞-categories. Contents 0. Introduction 1 1. ∞-categories of commutative monoids and groups 4 2. Preadditive and additive ∞-categories 6 3. Smashing localizations 8 4. Commutative monoids and groups as smashing localizations 11 5. Canonical symmetric monoidal structures 13 6. More functoriality 15 7. ∞-categories of semirings and rings 17 8. Multiplicative infinite loop space theory 19 Appendix A. Comonoids 23 Appendix B. Algebraic theories and monadic functors 23 References 26 0. Introduction The Grothendieck group K0(M) of a commutative monoid M, also known as the group completion, is the universal abelian group which receives a monoid map from M. -
1. Introduction
Pré-Publicações do Departamento de Matemática Universidade de Coimbra Preprint Number 14–18 A CRITERION FOR REFLECTIVENESS OF NORMAL EXTENSIONS WITH AN APPLICATION TO MONOIDS ANDREA MONTOLI, DIANA RODELO AND TIM VAN DER LINDEN Dedicated to Manuela Sobral on the occasion of her seventieth birthday Abstract: We prove that the so-called special homogeneous surjections are reflec- tive amongst surjective homomorphisms of monoids. To do so, we use the recent result that these special homogeneous surjections are the normal (= central) extensi- ons with respect to the admissible Galois structure ΓMon determined by the Grothen- dieck group adjunction together with the classes of surjective homomorphisms. It is well known that such a reflection exists when the left adjoint functor of an admissible Galois structure preserves all pullbacks of fibrations along split epimorphic fibrati- ons, a property which we show to fail for ΓMon. We give a new sufficient condition for the normal extensions in an admissible Galois structure to be reflective, and we then show that this condition is indeed fulfilled by ΓMon. Keywords: categorical Galois theory; admissible Galois structure; central, nor- mal, trivial extension; Grothendieck group; group completion; homogeneous split epimorphism, special homogeneous surjection of monoids. AMS Subject Classification (2010): 20M32, 20M50, 11R32, 19C09, 18F30. 1. Introduction The original aim of our present work was to answer the following question: Is the category of special homogeneous surjections of monoids [3, 4] a reflective subcategory of the category of surjective monoid homomorphisms? Since we recently showed [17] that these special homogeneous surjections are the normal extensions in an admissible Galois structure [10, 11], we were at first convinced that this would be an immediate consequence of some known abstract Galois- theoretical result such as the ones in [13, 12]. -
Monomorphism - Wikipedia, the Free Encyclopedia
Monomorphism - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Monomorphism Monomorphism From Wikipedia, the free encyclopedia In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from X to Y is often denoted with the notation . In the more general setting of category theory, a monomorphism (also called a monic morphism or a mono) is a left-cancellative morphism, that is, an arrow f : X → Y such that, for all morphisms g1, g2 : Z → X, Monomorphisms are a categorical generalization of injective functions (also called "one-to-one functions"); in some categories the notions coincide, but monomorphisms are more general, as in the examples below. The categorical dual of a monomorphism is an epimorphism, i.e. a monomorphism in a category C is an epimorphism in the dual category Cop. Every section is a monomorphism, and every retraction is an epimorphism. Contents 1 Relation to invertibility 2 Examples 3 Properties 4 Related concepts 5 Terminology 6 See also 7 References Relation to invertibility Left invertible morphisms are necessarily monic: if l is a left inverse for f (meaning l is a morphism and ), then f is monic, as A left invertible morphism is called a split mono. However, a monomorphism need not be left-invertible. For example, in the category Group of all groups and group morphisms among them, if H is a subgroup of G then the inclusion f : H → G is always a monomorphism; but f has a left inverse in the category if and only if H has a normal complement in G. -
Arxiv:Math/0206124V2 [Math.CT] 13 Aug 2002
On Regular Closure Operators and Cowellpowered Subcategories Vishvajit V S Gautam The Institute of Mathematical Sciences, Chennai -600113 India. [email protected] Abstract Many properties of a category X , as for instance the existence of an adjoint or of a factorization system, are a consequence of the cow- ellpoweredness of X . In the absence of cowellpoweredness, for general results, fairly strong assumption on the category are needed. This paper provides a number of novel and useful observations to tackle the cowellpoweredness problem of subcategories by means of regular closure operators. Our exposition focusses on the question when two subcategories A and B induce the same regular closure operators (up to isomorphism), then information about (non)-cowellpoweredness of A may be gained from the corresponding property of B, and vice versa. Keywords : A-regular morphism, (strongly) epireflective subcategory, cowellpowered category, regular closure operator, (weakly) reflective subcategory, wellpowered category. AMS subject classification 2000 : 18A20, 18B30, 18A32. Introduction arXiv:math/0206124v2 [math.CT] 13 Aug 2002 Subcategories are always assumed to be full and isomorphism closed. A morphism f : X −→ Y in a category X is an epimorphism if for each pair of morphisms g, h : Y −→ Z in a category X such that g · f = h · f implies g = h. Regular closure operators were originally introduced by Salbany [20]. These operators provide a key instrument for attacking the cowellpowerednes problem in a category X . A category X is said to be cowellpowered if each object X in X admits only a (small) set of non- equivalent X -epimorphisms with domain in X.