Notes and Solutions to Exercises for Mac Lane's Categories for The
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UNIVERSAL PROPERTY & CASUALTY INSURANCE COMPANY MASSACHUSETTS PERSONAL PROPERTY MANUAL HOMEOWNERS PROGRAM EDITION 02/15 TABLE OF CONTENTS RULE CONTENT PAGE NO. 1 Application of Program Rules 1 2 Reserved for Future Use 1 3 Extent of Coverage 1 4 Reserved for Future Use 1 5 Policy Fee 1 6 Reserved for Future Use 1 7 Policy Period, Minimum Premium and Waiver of Premium 1 8 Rounding of Premiums 1 9 Changes and Mid-Term Premium Adjustments 2 10 Effective Date and Important Notices 2 11 Protective Device Credits 2 12 Townhouses or Rowhouses 2 13 Underwriting Surcharges 3 14 Mandatory Additional Charges 3 15 Payment and Payment Plan Options 4 16 Building Code Compliance 5 100 Reserved for Future Use 5 101 Limits of Liability and Coverage Relationship 6 102 Description of Coverages 6 103 Mandatory Coverages 7 104 Eligibility 7 105 Secondary Residence Premises 9 106 Reserved for Future Use 9 107 Construction Definitions 9 108 Seasonal Dwelling Definition 9 109 Single Building Definition 9 201 Policy Period 9 202 Changes or Cancellations 9 203 Manual Premium Revision 9 204 Reserved for Future Use 10 205 Reserved for Future Use 10 206 Transfer or Assignment 10 207 Reserved for Future Use 10 208 Premium Rounding Rule 10 209 Reserved for Future Use 10 300 Key Factor Interpolation Computation 10 301 Territorial Base Rates 11 401 Reserved for Future Use 20 402 Personal Property (Coverage C ) Replacement Cost Coverage 20 403 Age of Home 20 404 Ordinance or Law Coverage - Increased Limits 21 405 Debris Removal – HO 00 03 21 406 Deductibles 22 407 Additional -
Arxiv:1705.02246V2 [Math.RT] 20 Nov 2019 Esyta Ulsubcategory Full a That Say [15]
WIDE SUBCATEGORIES OF d-CLUSTER TILTING SUBCATEGORIES MARTIN HERSCHEND, PETER JØRGENSEN, AND LAERTIS VASO Abstract. A subcategory of an abelian category is wide if it is closed under sums, summands, kernels, cokernels, and extensions. Wide subcategories provide a significant interface between representation theory and combinatorics. If Φ is a finite dimensional algebra, then each functorially finite wide subcategory of mod(Φ) is of the φ form φ∗ mod(Γ) in an essentially unique way, where Γ is a finite dimensional algebra and Φ −→ Γ is Φ an algebra epimorphism satisfying Tor (Γ, Γ) = 0. 1 Let F ⊆ mod(Φ) be a d-cluster tilting subcategory as defined by Iyama. Then F is a d-abelian category as defined by Jasso, and we call a subcategory of F wide if it is closed under sums, summands, d- kernels, d-cokernels, and d-extensions. We generalise the above description of wide subcategories to this setting: Each functorially finite wide subcategory of F is of the form φ∗(G ) in an essentially φ Φ unique way, where Φ −→ Γ is an algebra epimorphism satisfying Tord (Γ, Γ) = 0, and G ⊆ mod(Γ) is a d-cluster tilting subcategory. We illustrate the theory by computing the wide subcategories of some d-cluster tilting subcategories ℓ F ⊆ mod(Φ) over algebras of the form Φ = kAm/(rad kAm) . Dedicated to Idun Reiten on the occasion of her 75th birthday 1. Introduction Let d > 1 be an integer. This paper introduces and studies wide subcategories of d-abelian categories as defined by Jasso. The main examples of d-abelian categories are d-cluster tilting subcategories as defined by Iyama. -
Universal Properties, Free Algebras, Presentations
Universal Properties, Free Algebras, Presentations Modern Algebra 1 Fall 2016 Modern Algebra 1 (Fall 2016) Universal Properties, Free Algebras, Presentations 1 / 14 (1) f ◦ g exists if and only if dom(f ) = cod(g). (2) Composition is associative when it is defined. (3) dom(f ◦ g) = dom(g), cod(f ◦ g) = cod(f ). (4) If A = dom(f ) and B = cod(f ), then f ◦ idA = f and idB ◦ f = f . (5) dom(idA) = cod(idA) = A. (1) Ob(C) = O is a class whose members are called objects, (2) Mor(C) = M is a class whose members are called morphisms, (3) ◦ : M × M ! M is a binary partial operation called composition, (4) id : O ! M is a unary function assigning to each object A 2 O a morphism idA called the identity of A, (5) dom; cod : M ! O are unary functions assigning to each morphism f objects called the domain and codomain of f respectively. The laws defining categories are: Defn (Category) A category is a 2-sorted partial algebra C = hO; M; ◦; id; dom; codi where Modern Algebra 1 (Fall 2016) Universal Properties, Free Algebras, Presentations 2 / 14 (1) f ◦ g exists if and only if dom(f ) = cod(g). (2) Composition is associative when it is defined. (3) dom(f ◦ g) = dom(g), cod(f ◦ g) = cod(f ). (4) If A = dom(f ) and B = cod(f ), then f ◦ idA = f and idB ◦ f = f . (5) dom(idA) = cod(idA) = A. (2) Mor(C) = M is a class whose members are called morphisms, (3) ◦ : M × M ! M is a binary partial operation called composition, (4) id : O ! M is a unary function assigning to each object A 2 O a morphism idA called the identity of A, (5) dom; cod : M ! O are unary functions assigning to each morphism f objects called the domain and codomain of f respectively. -
Balanced Category Theory II
BALANCED CATEGORY THEORY II CLAUDIO PISANI ABSTRACT. In the first part, we further advance the study of category theory in a strong balanced factorization category C [Pisani, 2008], a finitely complete category en- dowed with two reciprocally stable factorization systems such that M/1 = M′/1. In particular some aspects related to “internal” (co)limits and to Cauchy completeness are considered. In the second part, we maintain that also some aspects of topology can be effectively synthesized in a (weak) balanced factorization category T , whose objects should be considered as possibly “infinitesimal” and suitably “regular” topological spaces. While in C the classes M and M′ play the role of discrete fibrations and opfibrations, in T they play the role of local homeomorphisms and perfect maps, so that M/1 and M′/1 are the subcategories of discrete and compact spaces respectively. One so gets a direct abstract link between the subjects, with mutual benefits. For example, the slice projection X/x → X and the coslice projection x\X → X, obtained as the second factors of x :1 → X according to (E, M) and (E′, M′) in C, correspond in T to the “infinitesimal” neighborhood of x ∈ X and to the closure of x. Furthermore, the open-closed complementation (generalized to reciprocal stability) becomes the key tool to internally treat, in a coherent way, some categorical concepts (such as (co)limits of presheaves) which are classically related by duality. Contents 1 Introduction 2 2 Bicartesian arrows of bimodules 4 3 Factorization systems 6 4 Balanced factorization categories 10 5 Cat asastrongbalancedfactorizationcategory 12 arXiv:0904.1790v3 [math.CT] 27 Apr 2009 6 Slices and colimits in a bfc 14 7 Internalaspectsofbalancedcategorytheory 16 8 The tensor functor and the internal hom 20 9 Retracts of slices 24 2000 Mathematics Subject Classification: 18A05, 18A22, 18A30, 18A32, 18B30, 18D99, 54B30, 54C10. -
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 -
A Subtle Introduction to Category Theory
A Subtle Introduction to Category Theory W. J. Zeng Department of Computer Science, Oxford University \Static concepts proved to be very effective intellectual tranquilizers." L. L. Whyte \Our study has revealed Mathematics as an array of forms, codify- ing ideas extracted from human activates and scientific problems and deployed in a network of formal rules, formal definitions, formal axiom systems, explicit theorems with their careful proof and the manifold in- terconnections of these forms...[This view] might be called formal func- tionalism." Saunders Mac Lane Let's dive right in then, shall we? What can we say about the array of forms MacLane speaks of? Claim (Tentative). Category theory is about the formal aspects of this array of forms. Commentary: It might be tempting to put the brakes on right away and first grab hold of what we mean by formal aspects. Instead, rather than trying to present \the formal" as the object of study and category theory as our instrument, we would nudge the reader to consider another perspective. Category theory itself defines formal aspects in much the same way that physics defines physical concepts or laws define legal (and correspondingly illegal) aspects: it embodies them. When we speak of what is formal/physical/legal we inescapably speak of category/physical/legal theory, and vice versa. Thus we pass to the somewhat grammatically awkward revision of our initial claim: Claim (Tentative). Category theory is the formal aspects of this array of forms. Commentary: Let's unpack this a little bit. While individual forms are themselves tautologically formal, arrays of forms and everything else networked in a system lose this tautological formality. -
Category Theory
CMPT 481/731 Functional Programming Fall 2008 Category Theory Category theory is a generalization of set theory. By having only a limited number of carefully chosen requirements in the definition of a category, we are able to produce a general algebraic structure with a large number of useful properties, yet permit the theory to apply to many different specific mathematical systems. Using concepts from category theory in the design of a programming language leads to some powerful and general programming constructs. While it is possible to use the resulting constructs without knowing anything about category theory, understanding the underlying theory helps. Category Theory F. Warren Burton 1 CMPT 481/731 Functional Programming Fall 2008 Definition of a Category A category is: 1. a collection of ; 2. a collection of ; 3. operations assigning to each arrow (a) an object called the domain of , and (b) an object called the codomain of often expressed by ; 4. an associative composition operator assigning to each pair of arrows, and , such that ,a composite arrow ; and 5. for each object , an identity arrow, satisfying the law that for any arrow , . Category Theory F. Warren Burton 2 CMPT 481/731 Functional Programming Fall 2008 In diagrams, we will usually express and (that is ) by The associative requirement for the composition operators means that when and are both defined, then . This allow us to think of arrows defined by paths throught diagrams. Category Theory F. Warren Burton 3 CMPT 481/731 Functional Programming Fall 2008 I will sometimes write for flip , since is less confusing than Category Theory F. -
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. -
Free Globularily Generated Double Categories
Free Globularily Generated Double Categories I Juan Orendain Abstract: This is the first part of a two paper series studying free globular- ily generated double categories. In this first installment we introduce the free globularily generated double category construction. The free globularily generated double category construction canonically associates to every bi- category together with a possible category of vertical morphisms, a double category fixing this set of initial data in a free and minimal way. We use the free globularily generated double category to study length, free prod- ucts, and problems of internalization. We use the free globularily generated double category construction to provide formal functorial extensions of the Haagerup standard form construction and the Connes fusion operation to inclusions of factors of not-necessarily finite Jones index. Contents 1 Introduction 1 2 The free globularily generated double category 11 3 Free globularily generated internalizations 29 4 Length 34 5 Group decorations 38 6 von Neumann algebras 42 1 Introduction arXiv:1610.05145v3 [math.CT] 21 Nov 2019 Double categories were introduced by Ehresmann in [5]. Bicategories were later introduced by Bénabou in [13]. Both double categories and bicategories express the notion of a higher categorical structure of second order, each with its advantages and disadvantages. Double categories and bicategories relate in different ways. Every double category admits an underlying bicategory, its horizontal bicategory. The horizontal bicategory HC of a double category C ’flattens’ C by discarding vertical morphisms and only considering globular squares. 1 There are several structures transferring vertical information on a double category to its horizontal bicategory, e.g. -
Lecture 2: Spaces of Maps, Loop Spaces and Reduced Suspension
LECTURE 2: SPACES OF MAPS, LOOP SPACES AND REDUCED SUSPENSION In this section we will give the important constructions of loop spaces and reduced suspensions associated to pointed spaces. For this purpose there will be a short digression on spaces of maps between (pointed) spaces and the relevant topologies. To be a bit more specific, one aim is to see that given a pointed space (X; x0), then there is an entire pointed space of loops in X. In order to obtain such a loop space Ω(X; x0) 2 Top∗; we have to specify an underlying set, choose a base point, and construct a topology on it. The underlying set of Ω(X; x0) is just given by the set of maps 1 Top∗((S ; ∗); (X; x0)): A base point is also easily found by considering the constant loop κx0 at x0 defined by: 1 κx0 :(S ; ∗) ! (X; x0): t 7! x0 The topology which we will consider on this set is a special case of the so-called compact-open topology. We begin by introducing this topology in a more general context. 1. Function spaces Let K be a compact Hausdorff space, and let X be an arbitrary space. The set Top(K; X) of continuous maps K ! X carries a natural topology, called the compact-open topology. It has a subbasis formed by the sets of the form B(T;U) = ff : K ! X j f(T ) ⊆ Ug where T ⊆ K is compact and U ⊆ X is open. Thus, for a map f : K ! X, one can form a typical basis open neighborhood by choosing compact subsets T1;:::;Tn ⊆ K and small open sets Ui ⊆ X with f(Ti) ⊆ Ui to get a neighborhood Of of f, Of = B(T1;U1) \ ::: \ B(Tn;Un): One can even choose the Ti to cover K, so as to `control' the behavior of functions g 2 Of on all of K. -
Arxiv:1005.0156V3
FOUR PROBLEMS REGARDING REPRESENTABLE FUNCTORS G. MILITARU C Abstract. Let R, S be two rings, C an R-coring and RM the category of left C- C C comodules. The category Rep (RM, SM) of all representable functors RM→ S M is C shown to be equivalent to the opposite of the category RMS . For U an (S, R)-bimodule C we give necessary and sufficient conditions for the induction functor U ⊗R − : RM→ SM to be: a representable functor, an equivalence of categories, a separable or a Frobenius functor. The latter results generalize and unify the classical theorems of Morita for categories of modules over rings and the more recent theorems obtained by Brezinski, Caenepeel et al. for categories of comodules over corings. Introduction Let C be a category and V a variety of algebras in the sense of universal algebras. A functor F : C →V is called representable [1] if γ ◦ F : C → Set is representable in the classical sense, where γ : V → Set is the forgetful functor. Four general problems concerning representable functors have been identified: Problem A: Describe the category Rep (C, V) of all representable functors F : C→V. Problem B: Give a necessary and sufficient condition for a given functor F : C→V to be representable (possibly predefining the object of representability). Problem C: When is a composition of two representable functors a representable functor? Problem D: Give a necessary and sufficient condition for a representable functor F : C → V and for its left adjoint to be separable or Frobenius. The pioneer of studying problem A was Kan [10] who described all representable functors from semigroups to semigroups. -
On Iterated Semidirect Products of Finite Semilattices
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF ALGEBRA 142, 2399254 (1991) On Iterated Semidirect Products of Finite Semilattices JORGE ALMEIDA INIC-Centro de Mawmdtica, Faculdade de Ci&cias, Universidade do Porro, 4000 Porte, Portugal Communicated by T. E. Hull Received December 15, 1988 Let SI denote the class of all finite semilattices and let SI” be the pseudovariety of semigroups generated by all semidirect products of n finite semilattices. The main result of this paper shows that, for every n 2 3, Sl” is not finitely based. Nevertheless, a simple basis of identities is described for each SI”. We also show that SI" is generated by a semigroup with 2n generators but, except for n < 2, we do not know whether the bound 2n is optimal. ‘(” 1991 Academic PKSS, IW 1. INTRODUCTION Among the various operations which allow us to construct semigroups from simpler ones, the semidirect product has thus far received the most attention in the theory of finite semigroups. It plays a special role in the connections with language theory (Eilenberg [S], Pin [9]) and in the Krohn-Rodes decomposition theorem [S]. Questions dealing with semidirect products of finite semigroups are often properly dealt with in the context of pseudovarieties. For pseudovarieties V and W, their semidirect product V * W is defined to be the pseudovariety generated by all semidirect products S * T with SE V and T E W. It is well known that this operation on pseudovarieties is associative and that V * W consists of all homomorphic images of subsemigroups of S * T with SE V and TEW [S].