DOCSLIB.ORG

Weakly Globular Double Categories

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Weakly Globular Double Categories Weakly Globular Double Categories Dorette Pronk (joint work with Simona Paoli) Weak 2-Categories Weakly Globular Double Categories Bicategories Tamsamani weak 2-categories Weakly globular double categories Dorette Pronk (joint work with Simona Paoli) The Correspondence between Bicat and WGDbl Dalhousie University (and the University of Leicester) The fundamental bicategory The double category of marked paths Companions and quasi CMS Summer Meeting 2013 units Precompanions and Progress in Higher Categories equivalences The Weakly Thursday June 6, 2013 Globular Double Category of Fractions The Bicategory of Fractions The Construction The Universal Properties Weakly Globular Outline Double Categories Dorette Pronk (joint work with Simona Paoli) Weak 2-Categories Weak Bicategories 2-Categories Tamsamani weak 2-categories Bicategories Tamsamani weak 2-categories Weakly globular double categories Weakly globular double categories The The Correspondence between Bicat and WGDbl Correspondence between Bicat and The fundamental bicategory WGDbl The fundamental bicategory Companions and quasi units The double category of marked paths Precompanions and equivalences Companions and quasi units Precompanions and equivalences The Weakly Globular Double Category of Fractions The Weakly Globular Double The Bicategory of Fractions Category of Fractions The Construction The Bicategory of Fractions The Construction The Universal Properties The Universal Properties Weakly Globular Bicategories Double Categories Dorette Pronk (joint work with Simona Paoli) Weak 2-Categories Bicategories Tamsamani weak 2-categories I We are considering models for a 2-category of weak Weakly globular double 2-categories. categories The Our first model is the 2-category Bicat of Correspondence I icon between Bicat and WGDbl I Objects: Bicategories The fundamental bicategory I Arrows: Normal Homomorphisms The double category of marked paths I 2-Cells: Icons (identity component pseudo natural Companions and quasi units transformations) Precompanions and equivalences The Weakly Globular Double Category of Fractions The Bicategory of Fractions The Construction The Universal Properties Weakly Globular Bicategories Double Categories Dorette Pronk (joint work with Simona Paoli) Weak 2-Categories Bicategories Tamsamani weak 2-categories I We are considering models for a 2-category of weak Weakly globular double 2-categories. categories The Our first model is the 2-category Bicat of Correspondence I icon between Bicat and WGDbl I Objects: Bicategories The fundamental bicategory I Arrows: Normal Homomorphisms The double category of marked paths I 2-Cells: Icons (identity component pseudo natural Companions and quasi units transformations) Precompanions and equivalences The Weakly Globular Double Category of Fractions The Bicategory of Fractions The Construction The Universal Properties Weakly Globular The 2-Nerve Double Categories Dorette Pronk Definition (Lack-Paoli, 2008) (joint work with The 2-nerve of a bicategory, N(B) : ∆op ! Cat: Simona Paoli) Weak I 0-simplices N(B)0: the discrete category B0 2-Categories Bicategories Tamsamani weak id id 2-categories •X •Y •X •Y Weakly globular double 4 4 categories The I 1-simplices N(B)1: Correspondence between Bicat and WGDbl Objects Arrows The fundamental bicategory The double category of f f marked paths Companions and quasi % units X• •Y • α+ • Precompanions and : > equivalences g g The Weakly h Globular Double S• / •T Category of Fractions The Bicategory of Fractions 2-simplices N(B) has objects The Construction I 2 Ñ@ == The Universal Properties f ÑÑ == g ÑÑ α=∼ == ÑÑ == Ñ / h Weakly Globular The 2-Nerve [Lack-Paoli, 2008] Double Categories Dorette Pronk (joint work with Simona Paoli) Weak I This 2-nerve N is the object part of a fully faithful 2-Categories Bicategories 2-functor Tamsamani weak op 2-categories N : Bicat ! [∆ ; Cat] Weakly globular double icon categories The which has a left biadjoint. Correspondence between Bicat and I N(B) is a Tamsamani weak 2-category. WGDbl The fundamental bicategory The double category of marked paths Theorem (Lack-Paoli, 2008) Companions and quasi units Precompanions and There is a 2-adjoint biequivalence equivalences The Weakly Globular Double Bicaticon ' Ta2: Category of Fractions The Bicategory of Fractions The Construction The Universal Properties Weakly Globular The 2-Nerve [Lack-Paoli, 2008] Double Categories Dorette Pronk (joint work with Simona Paoli) Weak I This 2-nerve N is the object part of a fully faithful 2-Categories Bicategories 2-functor Tamsamani weak op 2-categories N : Bicat ! [∆ ; Cat] Weakly globular double icon categories The which has a left biadjoint. Correspondence between Bicat and I N(B) is a Tamsamani weak 2-category. WGDbl The fundamental bicategory The double category of marked paths Theorem (Lack-Paoli, 2008) Companions and quasi units Precompanions and There is a 2-adjoint biequivalence equivalences The Weakly Globular Double Bicaticon ' Ta2: Category of Fractions The Bicategory of Fractions The Construction The Universal Properties Weakly Globular The 2-Nerve [Lack-Paoli, 2008] Double Categories Dorette Pronk (joint work with Simona Paoli) Weak I This 2-nerve N is the object part of a fully faithful 2-Categories Bicategories 2-functor Tamsamani weak op 2-categories N : Bicat ! [∆ ; Cat] Weakly globular double icon categories The which has a left biadjoint. Correspondence between Bicat and I N(B) is a Tamsamani weak 2-category. WGDbl The fundamental bicategory The double category of marked paths Theorem (Lack-Paoli, 2008) Companions and quasi units Precompanions and There is a 2-adjoint biequivalence equivalences The Weakly Globular Double Bicaticon ' Ta2: Category of Fractions The Bicategory of Fractions The Construction The Universal Properties Weakly Globular Tamsamani Weak 2-Categories Double Categories Dorette Pronk (joint work with Simona Paoli) This leads to our second model: Ta2 Weak 2-Categories I Objects: Bicategories Tamsamani weak op 2-categories X∗ : ∆ ! Cat Weakly globular double categories such that The Correspondence I X0 is discrete between Bicat and WGDbl I the Segal maps The fundamental bicategory The double category of marked paths ηk : Xk ! X1 ×X0 · · · ×X0 X1 Companions and quasi units Precompanions and are equivalences of categories equivalences The Weakly Arrows: pseudo natural transformations Globular Double I Category of Fractions I 2-Cells: modifications The Bicategory of Fractions The Construction The Universal Properties Weakly Globular Remarks Double Categories Dorette Pronk op (joint work with I X∗ : ∆ ! Cat is the horizontal nerve of an internal Simona Paoli) category in Cat, i.e., a double category, if and only if Weak the Segal maps are isomorphisms. 2-Categories Bicategories Tamsamani weak I [Paoli-P] For each Tamsamani 2-category 2-categories Weakly globular double categories op X∗ : ∆ ! Cat; The Correspondence between Bicat and there is a functor WGDbl The fundamental bicategory The double category of op marked paths (RX)∗ : ∆ ! Cat Companions and quasi units Precompanions and such that equivalences The Weakly I (RX)n ' Xn for all n ≥ 0 Globular Double k Category of Fractions I (RX) =∼ (RX) × · · · × (RX) k 1 (RX)0 (RX)0 1 The Bicategory of Fractions The Construction I Note: We have traded a discrete X0 and Segal ' for The Universal Properties ∼ a posetal groupoid (RX)0 and Segal =. Weakly Globular Remarks Double Categories Dorette Pronk op (joint work with I X∗ : ∆ ! Cat is the horizontal nerve of an internal Simona Paoli) category in Cat, i.e., a double category, if and only if Weak the Segal maps are isomorphisms. 2-Categories Bicategories Tamsamani weak I [Paoli-P] For each Tamsamani 2-category 2-categories Weakly globular double categories op X∗ : ∆ ! Cat; The Correspondence between Bicat and there is a functor WGDbl The fundamental bicategory The double category of op marked paths (RX)∗ : ∆ ! Cat Companions and quasi units Precompanions and such that equivalences The Weakly I (RX)n ' Xn for all n ≥ 0 Globular Double k Category of Fractions I (RX) =∼ (RX) × · · · × (RX) k 1 (RX)0 (RX)0 1 The Bicategory of Fractions The Construction I Note: We have traded a discrete X0 and Segal ' for The Universal Properties ∼ a posetal groupoid (RX)0 and Segal =. Weakly Globular Weakly Globular Double Categories Double Categories Dorette Pronk Definition (joint work with Simona Paoli) A (strict) double category is weakly globular if it has the Weak following properties: 2-Categories Bicategories I There is an equivalence of categories Tamsamani weak 2-categories Weakly globular double γ : ! d ; categories X0 X0 The Correspondence d between Bicat and where X0 is the category of vertical arrows and X0 is WGDbl The fundamental bicategory the discrete category of its path components. The double category of marked paths Companions and quasi I γ induces an equivalence of categories, for n ≥ 2, units Precompanions and equivalences n n The Weakly X1× 0 · · · × 0 X1 ' X1× d · · · × d X1 X X X0 X0 Globular Double Category of Fractions The Bicategory of Fractions The Construction Remark The Universal Properties The second condition is satisfied when at least one of d0; d1 : X1 ⇒ X0 is an isofibration. Weakly Globular Weakly Globular Double Categories Double Categories Dorette Pronk (joint work with Simona Paoli) This leads to our third model: WGDblps Weak 2-Categories I Objects: Weakly globular double categories Bicategories Tamsamani weak 2-categories I Arrows: Pseudo functors (which correspond to Weakly globular double categories pseudo
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]
  • 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]
  • 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]
  • 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]
  • A Short Introduction to Category Theory
    A SHORT INTRODUCTION TO CATEGORY THEORY September 4, 2019 Contents 1. Category theory 1 1.1. Definition of a category 1 1.2. Natural transformations 3 1.3. Epimorphisms and monomorphisms 5 1.4. Yoneda Lemma 6 1.5. Limits and colimits 7 1.6. Free objects 9 References 9 1. Category theory 1.1. Definition of a category. Definition 1.1.1. A category C is the following data (1) a class ob(C), whose elements are called objects; (2) for any pair of objects A; B of C a set HomC(A; B), called set of morphisms; (3) for any three objects A; B; C of C a function HomC(A; B) × HomC(B; C) −! HomC(A; C) (f; g) −! g ◦ f; called composition of morphisms; which satisfy the following axioms (i) the sets of morphisms are all disjoints, so any morphism f determines his domain and his target; (ii) the composition is associative; (iii) for any object A of C there exists a morphism idA 2 Hom(A; A), called identity morphism, such that for any object B of C and any morphisms f 2 Hom(A; B) and g 2 Hom(C; A) we have f ◦ idA = f and idA ◦ g = g. Remark 1.1.2. The above definition is the definition of what is called a locally small category. For a general category we should admit that the set of morphisms is not a set. If the class of object is in fact a set then the category is called small. For a general category we should admit that morphisms form a class.
    [Show full text]
  • Universal Homotopy Theories UROP+ Final Paper, Summer 2016
    Universal Homotopy Theories UROP+ Final Paper, Summer 2016 Attilio Castano Mentor: Pablo Boixeda September 2, 2016 Abstract The following paper is aimed to be a self-contained expository paper about Daniel Dugger's construction of a universal homotopy theory assigned to a small category [4]. Contents 0 Introduction 2 1 Motivation 5 1.1 Presheaves...........................................6 1.2 Simplicial Sets........................................ 10 2 A Model Structure for Simplicial Sets 15 2.1 Presentable and Accessible Categories........................... 17 2.2 Combinatorial Model Categories.............................. 19 2.3 Kan Model Structure..................................... 23 3 A Model Structure for Simplicial Presheaves 28 3.1 Diagram Categories..................................... 31 3.2 Directed Categories...................................... 35 3.3 Reedy Categories....................................... 39 4 Homotopy Colimits 43 4.1 Kan Extensions........................................ 47 4.2 Homotopy Kan Extensions.................................. 51 4.3 Computational Techniques.................................. 53 5 Applications 58 5.1 Homotopy Density Theorem................................. 61 5.2 Universal Property of Simplicial Presheaves........................ 66 6 Looking Forward 71 1 0 Introduction The following paper is aimed to be a self-contained expository paper about Daniel Dugger's construc- tion of a universal homotopy theory assigned to a small category [4]. But what does this mean? Throughout the time I have been working on this project, I have felt an intense need to share what I am working on with my friends and family. The problem, as any mathematician would know, is that it is hard to explain one's work to non mathematicians, and even to mathematicians in a different field. As a result of many conversations in which I tried to explain my work, I have decided to write the following fictional short story that I hope serves as a good metaphor for what I have been working on.
    [Show full text]
  • Category Theory and Diagrammatic Reasoning 3 Universal Properties, Limits and Colimits
    Category theory and diagrammatic reasoning 13th February 2019 Last updated: 7th February 2019 3 Universal properties, limits and colimits A division problem is a question of the following form: Given a and b, does there exist x such that a composed with x is equal to b? If it exists, is it unique? Such questions are ubiquitious in mathematics, from the solvability of systems of linear equations, to the existence of sections of fibre bundles. To make them precise, one needs additional information: • What types of objects are a and b? • Where can I look for x? • How do I compose a and x? Since category theory is, largely, a theory of composition, it also offers a unifying frame- work for the statement and classification of division problems. A fundamental notion in category theory is that of a universal property: roughly, a universal property of a states that for all b of a suitable form, certain division problems with a and b as parameters have a (possibly unique) solution. Let us start from universal properties of morphisms in a category. Consider the following division problem. Problem 1. Let F : Y ! X be a functor, x an object of X. Given a pair of morphisms F (y0) f 0 F (y) x , f does there exist a morphism g : y ! y0 in Y such that F (y0) F (g) f 0 F (y) x ? f If it exists, is it unique? 1 This has the form of a division problem where a and b are arbitrary morphisms in X (which need to have the same target), x is constrained to be in the image of a functor F , and composition is composition of morphisms.
    [Show full text]