DOCSLIB.ORG

Universal Categories

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Universal Categories Universal categories Alistair Savage University of Ottawa Slides available online: alistairsavage.ca/talks Alistair Savage (Ottawa) Universal categories January 11, 2019 1 / 37 Outline Goal: Give overview of universal categories and their importance. Overview: 1 Universal constructions in algebra 2 Warm-up 3 Monoidal categories and string diagrams 4 More warm-up 5 (Oriented) Temperley{Lieb category 6 (Oriented) Brauer category 7 Current work (very brief) Alistair Savage (Ottawa) Universal categories January 11, 2019 2 / 37 Free monoid Let S be a set. The free monoid MS on S has elements given by words in S, multiplication given by concatenation: (ab2c) · (c4ba) = ab2c5ba Universal property: Given any function f : S ! M, with M a monoid, there exists a unique monoid homomorphism ': MS ! M such that s7!s S / MS ' f M commutes. Alistair Savage (Ottawa) Universal categories January 11, 2019 3 / 37 Free group Let S be a set. The free group GS on S has elements given by words in S (+ inverses), multiplication given by concatenation with reduction: (ab2c) · (c−1ba) = ab2cc−1ba = ab3a Universal property: Given any function f : S ! G, with G a group, there exists a unique group homomorphism ': GS ! G such that s7!s S / GS ' f G commutes. Alistair Savage (Ottawa) Universal categories January 11, 2019 4 / 37 Free algebras Let | be a commutative ring. Free commutative algebra The polynomial ring |[X1;:::;Xn] is the free commutative |-algebra on n generators. Universal property: Given any commutative |-algebra R and elements r1; : : : ; rn 2 R, there exists a unique ring homomorphism |[X1;:::;Xn] ! R; Xi 7! ri: Free associative algebra One can also define free associative (not necessarily commutative) algebras. Alistair Savage (Ottawa) Universal categories January 11, 2019 5 / 37 Warm-up: Some simple universal categories Free category F(X) on one object X Objects: X Morphisms: Hom(X; X) = f1X g Universal property: If C is any category and Y is an object of C, then there exists a unique functor F(X) !C;X 7! Y: Alistair Savage (Ottawa) Universal categories January 11, 2019 6 / 37 Warm-up: Some simple universal categories Free category F(X; f) on one object X and one generating morphism f : X ! X Objects: X Morphisms: Hom(X; X) is the free monoid on the generator f. Universal property: If C is any category, Y is an object of C, and g : Y ! Y is a morphism in C, then there exists a unique functor F(X; F ) !C;X 7! Y; f 7! g: Alistair Savage (Ottawa) Universal categories January 11, 2019 7 / 37 Strict monoidal categories A strict monoidal category is a category C equipped with a bifunctor (the tensor product) ⊗: C × C ! C, and a unit object 1, such that (A ⊗ B) ⊗ C = A ⊗ (B ⊗ C) for all objects A, B, C, 1 ⊗ A = A = A ⊗ 1 for all objects A, (f ⊗ g) ⊗ h = f ⊗ (g ⊗ h) for all morphisms f, g, h, 11 ⊗ f = f = f ⊗ 11 for all morphisms f. Remark: Non-strict monoidal categories In a (not necessarily strict) monoidal category, the equalities above are replaced by isomorphism, and we impose some coherence conditions. Every monoidal category is monoidally equivalent to a strict one. Alistair Savage (Ottawa) Universal categories January 11, 2019 8 / 37 |-linear monoidal categories Fix a commutative ground ring |. A |-linear category is a category such that each morphism space is a |-module, composition of morphisms is |-bilinear. In a strict |-linear monoidal category, we also require that tensor product of morphisms is |-bilinear. The interchange law The axioms of a strict monoidal category imply the interchange law: For f g A1 −! A2 and B1 −! B2, the following diagram commutes: 1⊗g A1 ⊗ B1 / A1 ⊗ B2 f⊗g f⊗1 f⊗1 & A ⊗ B / A ⊗ B 2 1 1⊗g 2 2 Alistair Savage (Ottawa) Universal categories January 11, 2019 9 / 37 Warm-up: Some simple universal categories Free |-linear category L(X; f) on one object X and one generating morphism f : X ! X Objects: X Morphisms: Hom(X; X) = |[f]. Universal property: If C is any |-linear category, Y is an object of C, and g : Y ! Y is a morphism in C, then there exists a unique |-linear functor F(X; F ) !C;X 7! Y; f 7! g: Alistair Savage (Ottawa) Universal categories January 11, 2019 10 / 37 Warm-up: Some simple universal categories Free monoidal category M(X) on one generating object X Objects: X⊗n for n ≥ 0 (where X⊗0 = 1). Morphisms: ( 1 ⊗n if n = m; Hom(X⊗n;X⊗m) = X ? if n 6= m: Universal property: If C is any monoidal category and Y is an object of C, then there exists a unique monoidal functor M(X) !C;X 7! Y: Alistair Savage (Ottawa) Universal categories January 11, 2019 11 / 37 Warm-up: Some simple universal categories Free |-linear monoidal category M(X; f) on one generating object X and one generating morphism f : X ! X Objects: X⊗n for n ≥ 0. Morphisms: ( [f ; : : : ; f ] if n = m; Hom(X⊗n;X⊗m) = | 1 n 0 if n 6= m; ⊗(i−1) ⊗(n−i) where fi = 1X ⊗ f ⊗ 1X . (The fi commute because of the interchange law!) Universal property: If C is any |-linear monoidal category, Y is an object of C, and g : Y ! Y is a morphism in C, then there exists a unique |-linear monoidal functor M(X; f) !C;X 7! Y; f 7! g: Alistair Savage (Ottawa) Universal categories January 11, 2019 12 / 37 String diagrams Fix a strict monoidal category C. We will denote a morphism f : A ! B by: B f A The identity map 1A : A ! A is a string with no label: A A We sometimes omit the object labels when they are clear or unimportant. Alistair Savage (Ottawa) Universal categories January 11, 2019 13 / 37 String diagrams Composition is vertical stacking and tensor product is horizontal juxtaposition: f = fg f ⊗ g = f g g The interchange law then becomes: f g = f g = g f A morphism f : A1 ⊗ A2 ! B1 ⊗ B2 can be depicted: B1 B2 f A1 A2 Alistair Savage (Ottawa) Universal categories January 11, 2019 14 / 37 Presentations of strict monoidal categories One can give presentations of some strict |-linear monoidal categories, just as for monoids, groups, algebras, etc. Objects: If the objects are generated by some collection Ai, i 2 I, then we have all possible tensor products of these objects: 1;Ai;Ai ⊗ Aj ⊗ Ak ⊗ A`; etc. Morphisms: If the morphisms are generated by some collection fj, j 2 J, then we have all possible compositions and tensor products of these morphisms (whenever these make sense): 1Ai ; fj ⊗ (fifk) ⊗ (f`); etc. We then often impose some relations on these morphism spaces. String diagrams: We can build complex diagrams out of our simple generating diagrams. Alistair Savage (Ottawa) Universal categories January 11, 2019 15 / 37 Symmetric monoidal categories A symmetric monoidal category is a monoidal category C such that, for each pair of objects X; Y in C, there is an isomorphism =∼ sXY : X ⊗ Y −! Y ⊗ X that is natural in both X and Y , sYX sXY = 1X⊗Y for all objects X, Y , the sXY satisfy certain (associativity and unit) coherence conditions. Examples Sets (cartesian product) Groups (cartesian product) Vector spaces over a given field (tensor product) Alistair Savage (Ottawa) Universal categories January 11, 2019 16 / 37 A universal symmetric monoidal category Define a strict monoidal category S with one generating object " and denote 1" = We have one generating morphism : " ⊗ "!" ⊗ " : We impose the relations: = ; = : Then ⊗n EndS (" ) = Sn is the symmetric group on n letters. Alistair Savage (Ottawa) Universal categories January 11, 2019 17 / 37 A universal symmetric monoidal category Universal property: If C is any symmetric monoidal category and X is an object of C, then there is a unique monoidal functor S!C; " 7! X; 7! sXX : ⊗n Immediate consequence: We have an Sn-action on EndC(X ) coming from the induced map ⊗n ⊗n Sn = EndS (" ) ! EndC(X ): Linear version: We can repeat the above with |-linear categories. Then we get an action of the group algebra ⊗n ⊗n |Sn = EndS (" ) ! EndC(X ): Alistair Savage (Ottawa) Universal categories January 11, 2019 18 / 37 Duals Suppose a strict monoidal category C has two objects " and #, with 1" = ; 1# = : A morphism 1 ! # ⊗ " would have string diagram ; where = 11: We typically omit the dotted line and draw: : 1 !# ⊗ " : Similarly, we can have : " ⊗ #! 1: Alistair Savage (Ottawa) Universal categories January 11, 2019 19 / 37 Duals We say that # is right dual to " (and " is left dual to # ) if there exist morphisms : 1 !# ⊗ " : and : " ⊗ #! 1: such that = and = : Objects " and # are both left and right dual to each other if we also have : 1 !" ⊗ # and : # ⊗ "! 1 such that = and = : Alistair Savage (Ottawa) Universal categories January 11, 2019 20 / 37 Duals: example Let | be a field and consider the category Vect| of f.d. |-vector spaces. Unit object: | Fix a f.d. |-vector space V . Claim: The dual vector space V ∗ is both left and right dual to V , in the sense mentioned above. ∗ Proof: Fix a basis B of V . Let fδv : v 2 Bg be the dual basis of V . Viewing V and " and V ∗ as #, we define ∗ P : | ! V ⊗ V; 1 7! v2B δv ⊗ v; ∗ : V ⊗ V ! |; v ⊗ f 7! f(v); ∗ P : | ! V ⊗ V ; 1 7! v2B v ⊗ δv; ∗ : V ⊗ V ! |; f ⊗ v 7! f(v): Alistair Savage (Ottawa) Universal categories January 11, 2019 21 / 37 Duals: example Let's check the relation = : The left-hand side is the composition ∼ 1V ⊗ ∗ ⊗1V ∼ V = V ⊗ | −−−−−! V ⊗ V ⊗ V −−−−−! | ⊗ V = V; X X X w 7! w ⊗ 1 7! w ⊗ δv ⊗ v 7! δv(w) ⊗ v 7! δv(w)v = w: v2B v2B v2V The verification of the other relations is analogous.
Load more

Recommended publications
  • Table of Contents
    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
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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 .
    [Show full text]
  • 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
    [Show full text]
  • Universal Property & Casualty Insurance Company
    UNIVERSAL PROPERTY & CASUALTY INSURANCE COMPANY FLORIDA PERSONAL PROPERTY MANUAL DWELLING SECTION TABLE OF CONTENTS RULE CONTENT PAGE NO. 1 Introduction 1 2 Applications for Insurance 1 3 Extent of Coverage 6 4 Cancellations 6 5 Policy Fee 7 6 Commissions 8 7 Policy Period, Minimum Premium and Waiver of Premium 8 8 Rounding of Premiums 8 9 Changes and Mid-Term Premium Adjustments 8 10 Effective Date and Important Notices 8 11 Protective Device Discounts 9 12 Townhouse or Rowhouse 9 13 Underwriting Surcharges 10 14 Mandatory Additional Charges 10 15 Payment and Payment Plan Options 11 16 Building Code Compliance 12 100 Additional Underwriting Requirements – Dwelling Program 15 101 Forms, Coverage, Minimum Limits of Liability 16 102 Perils Insured Against 17 103 Eligibility 18 104 Reserved for Future Use 19 105 Seasonal Dwelling Definition 19 106 Construction Definitions 19 107 Single Building Definition 19 201 Policy Period 20 202 Changes or Cancellations 20 203 Manual Premium Revision 20 204 Multiple Locations 20 205 Reserved for Future Use 20 206 Transfer or Assignment 20 207 Reserved for Future Use 20 208 Reserved for Future Use 20 209 Whole Dollar Premium Rule 20 301 Base Premium Computation 21 302 Vandalism and Malicious Mischief – DP 00 01 Only 32 303 Reserved for Future Use 32 304 Permitted Incidental Occupancies 32 306 Hurricane Rates 33 402 Superior Construction 36 403 Reserved for Future Use 36 404 Dwelling Under Construction 36 405 Reserved for Future Use 36 406 Water Back-up and Sump Discharge or Overflow - Florida 36 407 Deductibles
    [Show full text]
  • Limits Commutative Algebra May 11 2020 1. Direct Limits Definition 1
    Limits Commutative Algebra May 11 2020 1. Direct Limits Definition 1: A directed set I is a set with a partial order ≤ such that for every i; j 2 I there is k 2 I such that i ≤ k and j ≤ k. Let R be a ring. A directed system of R-modules indexed by I is a collection of R modules fMi j i 2 Ig with a R module homomorphisms µi;j : Mi ! Mj for each pair i; j 2 I where i ≤ j, such that (i) for any i 2 I, µi;i = IdMi and (ii) for any i ≤ j ≤ k in I, µi;j ◦ µj;k = µi;k. We shall denote a directed system by a tuple (Mi; µi;j). The direct limit of a directed system is defined using a universal property. It exists and is unique up to a unique isomorphism. Theorem 2 (Direct limits). Let fMi j i 2 Ig be a directed system of R modules then there exists an R module M with the following properties: (i) There are R module homomorphisms µi : Mi ! M for each i 2 I, satisfying µi = µj ◦ µi;j whenever i < j. (ii) If there is an R module N such that there are R module homomorphisms νi : Mi ! N for each i and νi = νj ◦µi;j whenever i < j; then there exists a unique R module homomorphism ν : M ! N, such that νi = ν ◦ µi. The module M is unique in the sense that if there is any other R module M 0 satisfying properties (i) and (ii) then there is a unique R module isomorphism µ0 : M ! M 0.
    [Show full text]
  • 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.
    [Show full text]
  • Homeowners Endorsements
    HOMEOWNERS ENDORSEMENTS POLICY FORMS AND ENDORSEMENTS FOR FLORIDA Name Description Credit Disclosure Credit Disclosure Notice HO 00 03 04 91 Homeowners 3 Special Form HO 00 04 04 91 Homeowners 4 Contents Broad Form HO 00 06 04 91 Homeowners 6 Unit Owners Form HO 00 08 04 91 Homeowners 8 Modified Coverage Form HO 04 10 04 91 Additional Interests - Residence Premises HO 04 16 04 91 Premises Alarm or Fire Protection System HO 04 30 04 91 Theft Coverage Increase HO 04 40 04 91 Structures Rented To Others - Residence Premises HO 04 41 04 91 Additional Insured - Residence Premises HO 04 42 04 91 Permitted Incidental Occupancies HO 04 48 04 91 Other Structures HO 04 81 12 13 Actual Cash Value Loss Settlement HO 04 94 06 97 (06-07) Exclusion for Windstorm Coverage HO 04 96 04 91 No Coverage for Home Day Care Business HO 17 32 04 91R (06-07) Unit Owners Coverage A - Special Coverage HO 17 33 04 91 Unit Owners Rental to Others HO 23 70 06 97 Windstorm Exterior Paint or Waterproofing Endorsement HO 23 74 12 13 Replacement Cost Loss Settlement Endorsement UPCIC 00 07 (02-12) Sinkhole Loss Coverage - Florida UPCIC 01 03 06 07 Law and Ordinance Increase to 50% UPCIC 03 33 07 08 Limited Fungi, Wet or Dry Rot, or Bacteria UPCIC 04 33 07 08 Limited Fungi, Wet or Dry Rot, or Bacteria UPCIC 04 90 04 91 (06-07) Personal Property Replacement Cost UPCIC 06 03 32 08 Limited Fungi, Wet or Dry Rot, or Bacteria UPCIC 06 33 07 08 (1) Limited Fungi, Wet or Dry Rot, or Bacteria UPCIC 08 33 07 08 Limited Fungi, Wet or Dry Rot, or Bacteria UPCIC 10 01 98 (06-07) Existing Damage Exclusion UPCIC 14 01 98 Amendment of Loss Settlement Condition - Florida UPCIC 16 01 98 Loss Assessment Coverage UPCIC 19 01 98 Windstorm Protective Devices UPCIC 23 12 13 Special Provisions - Florida UPCIC 23 01 16 Special Provisions - Florida UPCIC 25 01 98 (06-07) Hurricane Deductible UPCIC 3 01 98 Outline of Your Homeowner Policy UPCIC Privacy UPCIC Privacy Statement UPCIC SPL (05-08) Swimming Pool Liability Exclusion Universal Property & Casualty Insurance Company 1110 W.
    [Show full text]
  • Categorical Models of Type Theory
    Categorical models of type theory Michael Shulman February 28, 2012 1 / 43 Theories and models Example The theory of a group asserts an identity e, products x · y and inverses x−1 for any x; y, and equalities x · (y · z) = (x · y) · z and x · e = x = e · x and x · x−1 = e. I A model of this theory (in sets) is a particularparticular group, like Z or S3. I A model in spaces is a topological group. I A model in manifolds is a Lie group. I ... 3 / 43 Group objects in categories Definition A group object in a category with finite products is an object G with morphisms e : 1 ! G, m : G × G ! G, and i : G ! G, such that the following diagrams commute. m×1 (e;1) (1;e) G × G × G / G × G / G × G o G F G FF xx 1×m m FF xx FF m xx 1 F x 1 / F# x{ x G × G m G G ! / e / G 1 GO ∆ m G × G / G × G 1×i 4 / 43 Categorical semantics Categorical semantics is a general procedure to go from 1. the theory of a group to 2. the notion of group object in a category. A group object in a category is a model of the theory of a group. Then, anything we can prove formally in the theory of a group will be valid for group objects in any category. 5 / 43 Doctrines For each kind of type theory there is a corresponding kind of structured category in which we consider models.
    [Show full text]
  • Groups and Categories
    \chap04" 2009/2/27 i i page 65 i i 4 GROUPS AND CATEGORIES This chapter is devoted to some of the various connections between groups and categories. If you already know the basic group theory covered here, then this will give you some insight into the categorical constructions we have learned so far; and if you do not know it yet, then you will learn it now as an application of category theory. We will focus on three different aspects of the relationship between categories and groups: 1. groups in a category, 2. the category of groups, 3. groups as categories. 4.1 Groups in a category As we have already seen, the notion of a group arises as an abstraction of the automorphisms of an object. In a specific, concrete case, a group G may thus consist of certain arrows g : X ! X for some object X in a category C, G ⊆ HomC(X; X) But the abstract group concept can also be described directly as an object in a category, equipped with a certain structure. This more subtle notion of a \group in a category" also proves to be quite useful. Let C be a category with finite products. The notion of a group in C essentially generalizes the usual notion of a group in Sets. Definition 4.1. A group in C consists of objects and arrows as so: m i G × G - G G 6 u 1 i i i i \chap04" 2009/2/27 i i page 66 66 GROUPSANDCATEGORIES i i satisfying the following conditions: 1.
    [Show full text]
  • Math 395: Category Theory Northwestern University, Lecture Notes
    Math 395: Category Theory Northwestern University, Lecture Notes Written by Santiago Can˜ez These are lecture notes for an undergraduate seminar covering Category Theory, taught by the author at Northwestern University. The book we roughly follow is “Category Theory in Context” by Emily Riehl. These notes outline the specific approach we’re taking in terms the order in which topics are presented and what from the book we actually emphasize. We also include things we look at in class which aren’t in the book, but otherwise various standard definitions and examples are left to the book. Watch out for typos! Comments and suggestions are welcome. Contents Introduction to Categories 1 Special Morphisms, Products 3 Coproducts, Opposite Categories 7 Functors, Fullness and Faithfulness 9 Coproduct Examples, Concreteness 12 Natural Isomorphisms, Representability 14 More Representable Examples 17 Equivalences between Categories 19 Yoneda Lemma, Functors as Objects 21 Equalizers and Coequalizers 25 Some Functor Properties, An Equivalence Example 28 Segal’s Category, Coequalizer Examples 29 Limits and Colimits 29 More on Limits/Colimits 29 More Limit/Colimit Examples 30 Continuous Functors, Adjoints 30 Limits as Equalizers, Sheaves 30 Fun with Squares, Pullback Examples 30 More Adjoint Examples 30 Stone-Cech 30 Group and Monoid Objects 30 Monads 30 Algebras 30 Ultrafilters 30 Introduction to Categories Category theory provides a framework through which we can relate a construction/fact in one area of mathematics to a construction/fact in another. The goal is an ultimate form of abstraction, where we can truly single out what about a given problem is specific to that problem, and what is a reflection of a more general phenomenom which appears elsewhere.
    [Show full text]