Category Theory from Scratch
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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. -
Sheaves and Homotopy Theory
SHEAVES AND HOMOTOPY THEORY DANIEL DUGGER The purpose of this note is to describe the homotopy-theoretic version of sheaf theory developed in the work of Thomason [14] and Jardine [7, 8, 9]; a few enhancements are provided here and there, but the bulk of the material should be credited to them. Their work is the foundation from which Morel and Voevodsky build their homotopy theory for schemes [12], and it is our hope that this exposition will be useful to those striving to understand that material. Our motivating examples will center on these applications to algebraic geometry. Some history: The machinery in question was invented by Thomason as the main tool in his proof of the Lichtenbaum-Quillen conjecture for Bott-periodic algebraic K-theory. He termed his constructions `hypercohomology spectra', and a detailed examination of their basic properties can be found in the first section of [14]. Jardine later showed how these ideas can be elegantly rephrased in terms of model categories (cf. [8], [9]). In this setting the hypercohomology construction is just a certain fibrant replacement functor. His papers convincingly demonstrate how many questions concerning algebraic K-theory or ´etale homotopy theory can be most naturally understood using the model category language. In this paper we set ourselves the specific task of developing some kind of homotopy theory for schemes. The hope is to demonstrate how Thomason's and Jardine's machinery can be built, step-by-step, so that it is precisely what is needed to solve the problems we encounter. The papers mentioned above all assume a familiarity with Grothendieck topologies and sheaf theory, and proceed to develop the homotopy-theoretic situation as a generalization of the classical case. -
Simplicial Sets, Nerves of Categories, Kan Complexes, Etc
SIMPLICIAL SETS, NERVES OF CATEGORIES, KAN COMPLEXES, ETC FOLING ZOU These notes are taken from Peter May's classes in REU 2018. Some notations may be changed to the note taker's preference and some detailed definitions may be skipped and can be found in other good notes such as [2] or [3]. The note taker is responsible for any mistakes. 1. simplicial approach to defining homology Defnition 1. A simplical set/group/object K is a sequence of sets/groups/objects Kn for each n ≥ 0 with face maps: di : Kn ! Kn−1; 0 ≤ i ≤ n and degeneracy maps: si : Kn ! Kn+1; 0 ≤ i ≤ n satisfying certain commutation equalities. Images of degeneracy maps are said to be degenerate. We can define a functor: ordered abstract simplicial complex ! sSet; K 7! Ks; where s Kn = fv0 ≤ · · · ≤ vnjfv0; ··· ; vng (may have repetition) is a simplex in Kg: s s Face maps: di : Kn ! Kn−1; 0 ≤ i ≤ n is by deleting vi; s s Degeneracy maps: si : Kn ! Kn+1; 0 ≤ i ≤ n is by repeating vi: In this way it is very straightforward to remember the equalities that face maps and degeneracy maps have to satisfy. The simplical viewpoint is helpful in establishing invariants and comparing different categories. For example, we are going to define the integral homology of a simplicial set, which will agree with the simplicial homology on a simplical complex, but have the virtue of avoiding the barycentric subdivision in showing functoriality and homotopy invariance of homology. This is an observation made by Samuel Eilenberg. To start, we construct functors: F C sSet sAb ChZ: The functor F is the free abelian group functor applied levelwise to a simplical set. -
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 . -
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 -
Arxiv:2001.09075V1 [Math.AG] 24 Jan 2020
A topos-theoretic view of difference algebra Ivan Tomašić Ivan Tomašić, School of Mathematical Sciences, Queen Mary Uni- versity of London, London, E1 4NS, United Kingdom E-mail address: [email protected] arXiv:2001.09075v1 [math.AG] 24 Jan 2020 2000 Mathematics Subject Classification. Primary . Secondary . Key words and phrases. difference algebra, topos theory, cohomology, enriched category Contents Introduction iv Part I. E GA 1 1. Category theory essentials 2 2. Topoi 7 3. Enriched category theory 13 4. Internal category theory 25 5. Algebraic structures in enriched categories and topoi 41 6. Topos cohomology 51 7. Enriched homological algebra 56 8. Algebraicgeometryoverabasetopos 64 9. Relative Galois theory 70 10. Cohomologyinrelativealgebraicgeometry 74 11. Group cohomology 76 Part II. σGA 87 12. Difference categories 88 13. The topos of difference sets 96 14. Generalised difference categories 111 15. Enriched difference presheaves 121 16. Difference algebra 126 17. Difference homological algebra 136 18. Difference algebraic geometry 142 19. Difference Galois theory 148 20. Cohomologyofdifferenceschemes 151 21. Cohomologyofdifferencealgebraicgroups 157 22. Comparison to literature 168 Bibliography 171 iii Introduction 0.1. The origins of difference algebra. Difference algebra can be traced back to considerations involving recurrence relations, recursively defined sequences, rudi- mentary dynamical systems, functional equations and the study of associated dif- ference equations. Let k be a commutative ring with identity, and let us write R = kN for the ring (k-algebra) of k-valued sequences, and let σ : R R be the shift endomorphism given by → σ(x0, x1,...) = (x1, x2,...). The first difference operator ∆ : R R is defined as → ∆= σ id, − and, for r N, the r-th difference operator ∆r : R R is the r-th compositional power/iterate∈ of ∆, i.e., → r r ∆r = (σ id)r = ( 1)r−iσi. -
1.Introduction Given a Class Σ of Morphisms of a Category X, We Can Construct a Cat- Egory of Fractions X[Σ−1] Where All Morphisms of Σ Are Invertible
Pre-Publicac¸´ oes˜ do Departamento de Matematica´ Universidade de Coimbra Preprint Number 15–41 A CALCULUS OF LAX FRACTIONS LURDES SOUSA Abstract: We present a notion of category of lax fractions, where lax fraction stands for a formal composition s∗f with s∗s = id and ss∗ ≤ id, and a corresponding calculus of lax fractions which generalizes the Gabriel-Zisman calculus of frac- tions. 1.Introduction Given a class Σ of morphisms of a category X, we can construct a cat- egory of fractions X[Σ−1] where all morphisms of Σ are invertible. More X X Σ−1 precisely, we can define a functor PΣ : → [ ] which takes the mor- Σ phisms of to isomorphisms, and, moreover, PΣ is universal with respect to this property. As shown in [13], if Σ admits a calculus of fractions, then the morphisms of X[Σ−1] can be expressed by equivalence classes of cospans (f,g) of morphisms of X with g ∈ Σ, which correspond to the for- mal compositions g−1f . We recall that categories of fractions are closely related to reflective sub- categories and orthogonality. In particular, if A is a full reflective subcat- egory of X, the class Σ of all morphisms inverted by the corresponding reflector functor – equivalently, the class of all morphisms with respect to which A is orthogonal – admits a left calculus of fractions; and A is, up to equivalence of categories, a category of fractions of X for Σ. In [3] we presented a Finitary Orthogonality Deduction System inspired by the left calculus of fractions, which can be looked as a generalization of the Implicational Logic of [20], see [4]. -
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. -
The Homotopy Category Is a Homotopy Category
Vol. XXIII, 19~2 435 The homotopy category is a homotopy category By A~NE S~o~ In [4] Quillen defines the concept of a category o/models /or a homotopy theory (a model category for short). A model category is a category K together with three distingxtished classes of morphisms in K: F ("fibrations"), C ("cofibrations"), and W ("weak equivalences"). These classes are required to satisfy axioms M0--M5 of [4]. A closed model category is a model category satisfying the extra axiom M6 (see [4] for the statement of the axioms M0--M6). To each model category K one can associate a category Ho K called the homotopy category of K. Essentially, HoK is obtained by turning the morphisms in W into isomorphisms. It is shown in [4] that the category of topological spaces is a closed model category ff one puts F---- (Serre fibrations) and IV = (weak homotopy equivalences}, and takes C to be the class of all maps having a certain lifting property. From an aesthetical point of view, however, it would be nicer to work with ordinary (Hurewicz) fibrations, cofibrations and homotopy equivalences. The corresponding homotopy category would then be the ordinary homotopy category of topological spaces, i.e. the objects would be all topological spaces and the morphisms would be all homotopy classes of continuous maps. In the first section of this paper we prove that this is indeed feasible, and in the last section we consider the case of spaces with base points. 1. The model category structure of Top. Let Top be the category of topolo~cal spaces and continuous maps. -
Faithful Linear-Categorical Mapping Class Group Action 3
A FAITHFUL LINEAR-CATEGORICAL ACTION OF THE MAPPING CLASS GROUP OF A SURFACE WITH BOUNDARY ROBERT LIPSHITZ, PETER S. OZSVÁTH, AND DYLAN P. THURSTON Abstract. We show that the action of the mapping class group on bordered Floer homol- ogy in the second to extremal spinc-structure is faithful. This paper is designed partly as an introduction to the subject, and much of it should be readable without a background in Floer homology. Contents 1. Introduction1 1.1. Acknowledgements.3 2. The algebras4 2.1. Arc diagrams4 2.2. The algebra B(Z) 4 2.3. The algebra C(Z) 7 3. The bimodules 11 3.1. Diagrams for elements of the mapping class group 11 3.2. Type D modules 13 3.3. Type A modules 16 3.4. Practical computations 18 3.5. Equivalence of the two actions 22 4. Faithfulness of the action 22 5. Finite generation 24 6. Further questions 26 References 26 1. Introduction arXiv:1012.1032v2 [math.GT] 11 Feb 2014 Two long-standing, and apparently unrelated, questions in low-dimensional topology are whether the mapping class group of a surface is linear and whether the Jones polynomial detects the unknot. In 2010, Kronheimer-Mrowka gave an affirmative answer to a categorified version of the second question: they showed that Khovanov homology, a categorification of the Jones polynomial, does detect the unknot [KM11]. (Previously, Grigsby and Wehrli had shown that any nontrivially-colored Khovanov homology detects the unknot [GW10].) 2000 Mathematics Subject Classification. Primary: 57M60; Secondary: 57R58. Key words and phrases. Mapping class group, Heegaard Floer homology, categorical group actions. -
Fields Lectures: Simplicial Presheaves
Fields Lectures: Simplicial presheaves J.F. Jardine∗ January 31, 2007 This is a cleaned up and expanded version of the lecture notes for a short course that I gave at the Fields Institute in late January, 2007. I expect the cleanup and expansion processes to continue for a while yet, so the interested reader should check the web site http://www.math.uwo.ca/∼jardine periodi- cally for updates. Contents 1 Simplicial presheaves and sheaves 2 2 Local weak equivalences 4 3 First local model structure 6 4 Other model structures 12 5 Cocycle categories 13 6 Sheaf cohomology 16 7 Descent 22 8 Non-abelian cohomology 25 9 Presheaves of groupoids 28 10 Torsors and stacks 30 11 Simplicial groupoids 35 12 Cubical sets 43 13 Localization 48 ∗This research was supported by NSERC. 1 1 Simplicial presheaves and sheaves In all that follows, C will be a small Grothendieck site. Examples include the site op|X of open subsets and open covers of a topo- logical space X, the site Zar|S of Zariski open subschemes and open covers of a scheme S, or the ´etale site et|S, again of a scheme S. All of these sites have “big” analogues, like the big sites Topop,(Sch|S)Zar and (Sch|S)et of “all” topological spaces with open covers, and “all” S-schemes T → S with Zariski and ´etalecovers, respectivley. Of course, there are many more examples. Warning: All of the “big” sites in question have (infinite) cardinality bounds on the objects which define them so that the sites are small, and we don’t talk about these bounds.