Regular and Exact Completions
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MAT 4162 - Homework Assignment 2
MAT 4162 - Homework Assignment 2 Instructions: Pick at least 5 problems of varying length and difficulty. You do not have to provide full details in all of the problems, but be sure to indicate which details you declare trivial. Due date: Monday June 22nd, 4pm. The category of relations Let Rel be the category whose objects are sets and whose morphisms X ! Y are relations R ⊆ X × Y . Two relations R ⊆ X × Y and S ⊆ Y × Z may be composed via S ◦ R = f(x; z)j9y 2 Y:R(x; y) ^ S(y; z): Exercise 1. Show that this composition is associative, and that Rel is indeed a category. The powerset functor(s) Consider the powerset functor P : Set ! Set. On objects, it sends a set X to its powerset P(X). A function f : X ! Y is sent to P(f): P(X) !P(Y ); U 7! P(f)(U) = f[U] =def ff(x)jx 2 Ug: Exercise 2. Show that P is indeed a functor. For every set X we have a singleton map ηX : X !P(X), defined by 2 S x 7! fxg. Also, we have a union map µX : P (X) !P(X), defined by α 7! α. Exercise 3. Show that η and µ constitute natural transformations 1Set !P and P2 !P, respectively. Objects of the form P(X) are more than mere sets. In class you have seen that they are in fact complete boolean algebras. For now, we regard P(X) as a complete sup-lattice. A partial ordering (P; ≤) is a complete sup-lattice if it is equipped with a supremum map W : P(P ) ! P which sends a subset U ⊆ P to W P , the least upper bound of U in P . -
Relations in Categories
Relations in Categories Stefan Milius A thesis submitted to the Faculty of Graduate Studies in partial fulfilment of the requirements for the degree of Master of Arts Graduate Program in Mathematics and Statistics York University Toronto, Ontario June 15, 2000 Abstract This thesis investigates relations over a category C relative to an (E; M)-factori- zation system of C. In order to establish the 2-category Rel(C) of relations over C in the first part we discuss sufficient conditions for the associativity of horizontal composition of relations, and we investigate special classes of morphisms in Rel(C). Attention is particularly devoted to the notion of mapping as defined by Lawvere. We give a significantly simplified proof for the main result of Pavlovi´c,namely that C Map(Rel(C)) if and only if E RegEpi(C). This part also contains a proof' that the category Map(Rel(C))⊆ is finitely complete, and we present the results obtained by Kelly, some of them generalized, i. e., without the restrictive assumption that M Mono(C). The next part deals with factorization⊆ systems in Rel(C). The fact that each set-relation has a canonical image factorization is generalized and shown to yield an (E¯; M¯ )-factorization system in Rel(C) in case M Mono(C). The setting without this condition is studied, as well. We propose a⊆ weaker notion of factorization system for a 2-category, where the commutativity in the universal property of an (E; M)-factorization system is replaced by coherent 2-cells. In the last part certain limits and colimits in Rel(C) are investigated. -
Knowledge Representation in Bicategories of Relations
Knowledge Representation in Bicategories of Relations Evan Patterson Department of Statistics, Stanford University Abstract We introduce the relational ontology log, or relational olog, a knowledge representation system based on the category of sets and relations. It is inspired by Spivak and Kent’s olog, a recent categorical framework for knowledge representation. Relational ologs interpolate between ologs and description logic, the dominant formalism for knowledge representation today. In this paper, we investigate relational ologs both for their own sake and to gain insight into the relationship between the algebraic and logical approaches to knowledge representation. On a practical level, we show by example that relational ologs have a friendly and intuitive—yet fully precise—graphical syntax, derived from the string diagrams of monoidal categories. We explain several other useful features of relational ologs not possessed by most description logics, such as a type system and a rich, flexible notion of instance data. In a more theoretical vein, we draw on categorical logic to show how relational ologs can be translated to and from logical theories in a fragment of first-order logic. Although we make extensive use of categorical language, this paper is designed to be self-contained and has considerable expository content. The only prerequisites are knowledge of first-order logic and the rudiments of category theory. 1. Introduction arXiv:1706.00526v2 [cs.AI] 1 Nov 2017 The representation of human knowledge in computable form is among the oldest and most fundamental problems of artificial intelligence. Several recent trends are stimulating continued research in the field of knowledge representation (KR). -
Regular Categories and Regular Logic Basic Research in Computer Science
BRICS Basic Research in Computer Science BRICS LS-98-2 C. Butz: Regular Categories and Regular Logic Regular Categories and Regular Logic Carsten Butz BRICS Lecture Series LS-98-2 ISSN 1395-2048 October 1998 Copyright c 1998, BRICS, Department of Computer Science University of Aarhus. All rights reserved. Reproduction of all or part of this work is permitted for educational or research use on condition that this copyright notice is included in any copy. See back inner page for a list of recent BRICS Lecture Series publica- tions. Copies may be obtained by contacting: BRICS Department of Computer Science University of Aarhus Ny Munkegade, building 540 DK–8000 Aarhus C Denmark Telephone: +45 8942 3360 Telefax: +45 8942 3255 Internet: [email protected] BRICS publications are in general accessible through the World Wide Web and anonymous FTP through these URLs: http://www.brics.dk ftp://ftp.brics.dk This document in subdirectory LS/98/2/ Regular Categories and Regular Logic Carsten Butz Carsten Butz [email protected] BRICS1 Department of Computer Science University of Aarhus Ny Munkegade DK-8000 Aarhus C, Denmark October 1998 1Basic Research In Computer Science, Centre of the Danish National Research Foundation. Preface Notes handed out to students attending the course on Category Theory at the Department of Computer Science in Aarhus, Spring 1998. These notes were supposed to give more detailed information about the relationship between regular categories and regular logic than is contained in Jaap van Oosten’s script on category theory (BRICS Lectures Series LS-95-1). Regular logic is there called coherent logic. -
Relations: Categories, Monoidal Categories, and Props
Logical Methods in Computer Science Vol. 14(3:14)2018, pp. 1–25 Submitted Oct. 12, 2017 https://lmcs.episciences.org/ Published Sep. 03, 2018 UNIVERSAL CONSTRUCTIONS FOR (CO)RELATIONS: CATEGORIES, MONOIDAL CATEGORIES, AND PROPS BRENDAN FONG AND FABIO ZANASI Massachusetts Institute of Technology, United States of America e-mail address: [email protected] University College London, United Kingdom e-mail address: [email protected] Abstract. Calculi of string diagrams are increasingly used to present the syntax and algebraic structure of various families of circuits, including signal flow graphs, electrical circuits and quantum processes. In many such approaches, the semantic interpretation for diagrams is given in terms of relations or corelations (generalised equivalence relations) of some kind. In this paper we show how semantic categories of both relations and corelations can be characterised as colimits of simpler categories. This modular perspective is important as it simplifies the task of giving a complete axiomatisation for semantic equivalence of string diagrams. Moreover, our general result unifies various theorems that are independently found in literature and are relevant for program semantics, quantum computation and control theory. 1. Introduction Network-style diagrammatic languages appear in diverse fields as a tool to reason about computational models of various kinds, including signal processing circuits, quantum pro- cesses, Bayesian networks and Petri nets, amongst many others. In the last few years, there have been more and more contributions towards a uniform, formal theory of these languages which borrows from the well-established methods of programming language semantics. A significant insight stemming from many such approaches is that a compositional analysis of network diagrams, enabling their reduction to elementary components, is more effective when system behaviour is thought of as a relation instead of a function. -
Basic Category Theory and Topos Theory
Basic Category Theory and Topos Theory Jaap van Oosten Jaap van Oosten Department of Mathematics Utrecht University The Netherlands Revised, February 2016 Contents 1 Categories and Functors 1 1.1 Definitions and examples . 1 1.2 Some special objects and arrows . 5 2 Natural transformations 8 2.1 The Yoneda lemma . 8 2.2 Examples of natural transformations . 11 2.3 Equivalence of categories; an example . 13 3 (Co)cones and (Co)limits 16 3.1 Limits . 16 3.2 Limits by products and equalizers . 23 3.3 Complete Categories . 24 3.4 Colimits . 25 4 A little piece of categorical logic 28 4.1 Regular categories and subobjects . 28 4.2 The logic of regular categories . 34 4.3 The language L(C) and theory T (C) associated to a regular cat- egory C ................................ 39 4.4 The category C(T ) associated to a theory T : Completeness Theorem 41 4.5 Example of a regular category . 44 5 Adjunctions 47 5.1 Adjoint functors . 47 5.2 Expressing (co)completeness by existence of adjoints; preserva- tion of (co)limits by adjoint functors . 52 6 Monads and Algebras 56 6.1 Algebras for a monad . 57 6.2 T -Algebras at least as complete as D . 61 6.3 The Kleisli category of a monad . 62 7 Cartesian closed categories and the λ-calculus 64 7.1 Cartesian closed categories (ccc's); examples and basic facts . 64 7.2 Typed λ-calculus and cartesian closed categories . 68 7.3 Representation of primitive recursive functions in ccc's with nat- ural numbers object . -
Rewriting Structured Cospans: a Syntax for Open Systems
UNIVERSITY OF CALIFORNIA RIVERSIDE Rewriting Structured Cospans: A Syntax For Open Systems A Dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Mathematics by Daniel Cicala June 2019 Dissertation Committee: Dr. John C. Baez, Chairperson Dr. Wee Liang Gan Dr. Jacob Greenstein Copyright by Daniel Cicala 2019 The Dissertation of Daniel Cicala is approved: Committee Chairperson University of California, Riverside Acknowledgments First and foremost, I would like to thank my advisor John Baez. In these past few years, I have learned more than I could have imagined about mathematics and the job of doing mathematics. I also want to thank the past and current Baez Crew for the many wonderful discussions. I am indebted to Math Department at the University of California, Riverside, which has afforded me numerous opportunities to travel to conferences near and far. Almost certainly, I would never have had a chance to pursue my doctorate had it not been for my parents who were there for me through every twist and turn on this, perhaps, too scenic route that I traveled. Most importantly, this project would have been impossible without the full-hearted support of my love, Elizabeth. I would also like to acknowledge the previously published material in this disser- tation. The interchange law in Section 3.1 was published in [15]. The material in Sections 3.2 and 3.3 appear in [16]. Also, the ZX-calculus example in Section 4.3 appears in [18]. iv Elizabeth. It’s finally over, baby! v ABSTRACT OF THE DISSERTATION Rewriting Structured Cospans: A Syntax For Open Systems by Daniel Cicala Doctor of Philosophy, Graduate Program in Mathematics University of California, Riverside, June 2019 Dr. -
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. -
Categories of Relations As Models of Quantum Theory
Categories of relations as models of quantum theory Chris Heunen∗ and Sean Tull† fheunen,[email protected] University of Oxford, Department of Computer Science Categories of relations over a regular category form a family of models of quantum theory. Using regular logic, many properties of relations over sets lift to these models, including the correspon- dence between Frobenius structures and internal groupoids. Over compact Hausdorff spaces, this lifting gives continuous symmetric encryption. Over a regular Mal’cev category, this correspondence gives a characterization of categories of completely positive maps, enabling the formulation of quan- tum features. These models are closer to Hilbert spaces than relations over sets in several respects: Heisenberg uncertainty, impossibility of broadcasting, and behavedness of rank one morphisms. 1 Introduction Many features of quantum theory can be abstracted to arbitrary compact dagger categories [1]. Thus we can compare models of quantum theory in a unified setting, and look for features that distinguish the category FHilb of finite-dimensional Hilbert spaces that forms the traditional model. The category Rel of sets and relations is an alternative model. It exhibits many features considered typical of quantum theory [2], but also refutes presumed correspondences between them [15, 21]. However, apart from Rel and its subcategory corresponding to Spekkens’ toy model [14], few alternative models have been studied in detail. We consider a new family of models by generalizing to categories Rel(C) of relations over an arbi- trary regular category C, including any algebraic category like that of groups, any abelian category like that of vector spaces, and any topos like that of sets. -
Some Glances at Topos Theory
Some glances at topos theory Francis Borceux Como, 2018 2 Francis Borceux [email protected] Contents 1 Localic toposes 7 1.1 Sheaves on a topological space . 7 1.2 Sheaves on a locale . 10 1.3 Localic toposes . 12 1.4 An application to ring theory . 13 2 Grothendieck toposes 17 2.1 Sheaves on a site . 17 2.2 The associated sheaf functor . 19 2.3 Limits and colimits in Grothendieck toposes . 21 2.4 Closure operator and subobject classifier . 22 3 Elementary toposes 25 3.1 The axioms for a topos . 25 3.2 Some set theoretical notions in a topos . 26 3.3 The slice toposes . 28 3.4 Exactness properties . 29 3.5 Heyting algebras in a topos . 30 4 Internal logic of a topos 33 4.1 The language of a topos . 33 4.2 Interpretation of terms and formulæ . 35 4.3 Propositional calculus in a topos . 37 4.4 Predicate calculus in a topos . 38 4.5 Structure of a topos in its internal language . 40 4.6 Boolean toposes . 41 4.7 The axiom of choice . 42 4.8 The axiom of infinity . 43 5 Morphisms of toposes 45 5.1 Logical morphisms . 45 5.2 Geometric morphisms . 46 5.3 Coherent and geometric formulæ . 48 5.4 Grothendieck topologies revisited . 49 5.5 Internal topologies and sheaves . 50 5.6 Back to Boolean toposes . 52 3 4 CONTENTS 6 Classifying toposes 53 6.1 What is a classifying topos? . 53 6.2 The theory classified by a topos . 54 6.3 Coherent and geometric theories . -
Categories with Fuzzy Sets and Relations
Categories with Fuzzy Sets and Relations John Harding, Carol Walker, Elbert Walker Department of Mathematical Sciences New Mexico State University Las Cruces, NM 88003, USA jhardingfhardy,[email protected] Abstract We define a 2-category whose objects are fuzzy sets and whose maps are relations subject to certain natural conditions. We enrich this category with additional monoidal and involutive structure coming from t-norms and negations on the unit interval. We develop the basic properties of this category and consider its relation to other familiar categories. A discussion is made of extending these results to the setting of type-2 fuzzy sets. 1 Introduction A fuzzy set is a map A : X ! I from a set X to the unit interval I. Several authors [2, 6, 7, 20, 22] have considered fuzzy sets as a category, which we will call FSet, where a morphism from A : X ! I to B : Y ! I is a function f : X ! Y that satisfies A(x) ≤ (B◦f)(x) for each x 2 X. Here we continue this path of investigation. The underlying idea is to lift additional structure from the unit interval, such as t-norms, t-conorms, and negations, to provide additional structure on the category. Our eventual aim is to provide a setting where processes used in fuzzy control can be abstractly studied, much in the spirit of recent categorical approaches to processes used in quantum computation [1]. Order preserving structure on I, such as t-norms and conorms, lifts to provide additional covariant structure on FSet. In fact, each t-norm T lifts to provide a symmetric monoidal tensor ⊗T on FSet. -
DRAFT: Category Theory for Computer Science
DRAFT: Category Theory for Computer Science J.R.B. Cockett 1 Department of Computer Science, University of Calgary, Calgary, T2N 1N4, Alberta, Canada October 12, 2016 1Partially supported by NSERC, Canada. 2 Contents 1 Basic Category Theory 5 1.1 The definition of a category . 5 1.1.1 Categories as graphs with composition . 5 1.1.2 Categories as partial semigroups . 6 1.1.3 Categories as enriched categories . 7 1.1.4 The opposite category and duality . 7 1.1.5 Examples of categories . 8 1.1.6 Exercises . 14 1.2 Basic properties of maps . 16 1.2.1 Epics, monics, retractions, and sections . 16 1.2.2 Idempotents . 17 1.3 Finite set enriched categories and full retractions . 18 1.3.1 Retractive inverses . 18 1.3.2 Fully retracted objects . 21 1.3.3 Exercises . 22 1.4 Orthogonality and Factorization . 24 1.4.1 Orthogonal classes of maps . 24 1.4.2 Introduction to factorization systems . 27 1.4.3 Exercises . 32 1.5 Functors and natural transformations . 35 1.5.1 Functors . 35 1.5.2 Natural transformations . 38 1.5.3 The Yoneda lemma . 41 1.5.4 Exercises . 43 2 Adjoints and Monad 45 2.1 Basic constructions on categories . 45 2.1.1 Slice categories . 45 2.1.2 Comma categories . 46 2.1.3 Inserters . 47 2.1.4 Exercises . 48 2.2 Adjoints . 49 2.2.1 The universal property . 49 2.2.2 Basic properties of adjoints . 52 3 4 CONTENTS 2.2.3 Reflections, coreflections and equivalences .