DOCSLIB.ORG

MAT 4162 - Homework Assignment 2

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
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 . Exercise 4. Show that P(X) is a complete sup-lattice. Also show that if (P; ≤; W) is a complete sup-lattice, then the ordering may be recovered from the map W. A morphism of complete sup-lattices (P; W) ! (Q; W) is a function f : P ! Q such that _ _ f( U) = ff(x)jx 2 Ug: 1 Exercise 5. Show that a morphism of complete sup-lattices automatically pre- serves the ordering. Show that complete sup-lattices and their morphisms form a category, called Sup. We shall now set up an adjunction between the category of sup-lattices and the category of sets. In one direction there is a forgetful functor U : Sup ! Set, sending a sup-lattice (P; W) to the underlying set P . In the other direction we have the powerset functor, which sends a set X to the complete sup-lattice P(X). Exercise 6. Prove that the powerset functor is left adjoint to the forgetful functor. The above result explains what the universal property of the powerset oper- ation is: it transforms a set into the free complete sup-lattice on that set! We now connect this with the category of relations Rel. Exercise 7. Show that a relation R ⊆ X × Y is the same thing as a function X !P(Y ) by setting up a bijection between the homsets Rel(X; Y ) and Set(X; P(Y )): Show that this bijection is natural in X and Y . Exercise 8. Show that any relation R ⊆ X × Y gives rise to a function P(R): P(X) !P(Y ), defined by [ P(R)(U) = r(U); x2U where r : X !P(Y ) is the function which corresponds to R under the bijection from the previous exercise. Show also that this makes P into a functor Rel ! Set. The previous exercises prove that there is an adjunction between the category of sets and the category of relations, with left adjoint the inclusion I : Set ! Rel, which is the identity on objects and which sends a function to its graph, and with right adjoint the powerset functor. Thus both Rel and mathsfSup have an adjunction with Set, and one won- ders how these two categories relate. Exercise 9. Show that the functor from Rel to Sup which takes X to P(X) (what does this do on arrows?) makes the following diagram commute: Rel / Sup D DD yy DD yy P DD yy U D! |yy Set Show that the functor Rel ! Sup is full and faithful, and that its image consists of the free sup-lattices. We wrap up by saying that the category of relations is equivalent to the category of free complete sup-lattices. 2 Other problems Exercise 10. Show that there is a natural bijection between the collection of subsets of X and the homset Rel(1;X) (where 1 is a singleton set). Conclude that the functor Rel ! Sets from above is representable. Exercise 11. Consider Grph, the category of directed graphs and graph ho- momorphisms. By forgetting identities and composition we have a forgetful functor U : Cat ! Grph: This functor has a left adjoint which takes a graph G to the free category on G. The objects of this category are the vertices of G, and an arrow f : X ! Y is a path in G which starts at X and ends at Y . Prove that this indeed gives a category, and that the construction gives a left adjoint to the forgetful functor. Exercise 12. Let X and Y be sets, and consider the powersets P(X); P(Y ). If f : X ! Y is a function we may define a function f ∗ : P(Y ) !P(X); f ∗(U) = f −1(U) = fx 2 Xjf(x) 2 Ug: Note that f ∗ goes in the other direction! Show that this makes P into a functor Set ! Setop. The functor from the previous exercise is called the contravariant powerset functor. Exercise 13. Let again X; Y be sets, and regard now P(X) and P(Y ) as posets, hence as categories. For f : X ! Y a function show that f ∗ (which is now regarded as functor) has a left adjoint. (Hint: consider the function P(f).) 3.
Load more

Recommended publications
  • 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.
    [Show full text]
  • 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).
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [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]
  • 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.
    [Show full text]
  • 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 .
    [Show full text]
  • A Concrete Introduction to Category Theory
    A CONCRETE INTRODUCTION TO CATEGORIES WILLIAM R. SCHMITT DEPARTMENT OF MATHEMATICS THE GEORGE WASHINGTON UNIVERSITY WASHINGTON, D.C. 20052 Contents 1. Categories 2 1.1. First Definition and Examples 2 1.2. An Alternative Definition: The Arrows-Only Perspective 7 1.3. Some Constructions 8 1.4. The Category of Relations 9 1.5. Special Objects and Arrows 10 1.6. Exercises 14 2. Functors and Natural Transformations 16 2.1. Functors 16 2.2. Full and Faithful Functors 20 2.3. Contravariant Functors 21 2.4. Products of Categories 23 3. Natural Transformations 26 3.1. Definition and Some Examples 26 3.2. Some Natural Transformations Involving the Cartesian Product Functor 31 3.3. Equivalence of Categories 32 3.4. Categories of Functors 32 3.5. The 2-Category of all Categories 33 3.6. The Yoneda Embeddings 37 3.7. Representable Functors 41 3.8. Exercises 44 4. Adjoint Functors and Limits 45 4.1. Adjoint Functors 45 4.2. The Unit and Counit of an Adjunction 50 4.3. Examples of adjunctions 57 1 1. Categories 1.1. First Definition and Examples. Definition 1.1. A category C consists of the following data: (i) A set Ob(C) of objects. (ii) For every pair of objects a, b ∈ Ob(C), a set C(a, b) of arrows, or mor- phisms, from a to b. (iii) For all triples a, b, c ∈ Ob(C), a composition map C(a, b) ×C(b, c) → C(a, c) (f, g) 7→ gf = g · f. (iv) For each object a ∈ Ob(C), an arrow 1a ∈ C(a, a), called the identity of a.
    [Show full text]
  • A Study of Categories of Algebras and Coalgebras
    A Study of Categories of Algebras and Coalgebras Jesse Hughes May, 2001 Department of Philosophy Carnegie Mellon University Pittsburgh PA 15213 Thesis Committee Steve Awodey, Co-Chair Dana Scott, Co-Chair Jeremy Avigad Lawrence Moss, Indiana University Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Abstract This thesis is intended to help develop the theory of coalgebras by, first, taking classic theorems in the theory of universal algebras and dualizing them and, second, developing an internal logic for categories of coalgebras. We begin with an introduction to the categorical approach to algebras and the dual notion of coalgebras. Following this, we discuss (co)algebras for a (co)monad and develop a theory of regular subcoalgebras which will be used in the internal logic. We also prove that categories of coalgebras are complete, under reasonably weak conditions, and simultaneously prove the well-known dual result for categories of algebras. We close the second chapter with a discussion of bisimulations in which we introduce a weaker notion of bisimulation than is current in the literature, but which is well-behaved and reduces to the standard definition under the assumption of choice. The third chapter is a detailed look at three theorem's of G. Birkhoff [Bir35, Bir44], presenting categorical proofs of the theorems which generalize the classical results and which can be easily dualized to apply to categories of coalgebras. The theorems of interest are the variety theorem, the equational completeness theorem and the subdirect product representation theorem. The duals of each of these theorems is discussed in detail, and the dual notion of \coequation" is introduced and several examples given.
    [Show full text]
  • On the Characterization of Monadic Categories Over Set Cahiers De Topologie Et Géométrie Différentielle Catégoriques, Tome 35, No 4 (1994), P
    CAHIERS DE TOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLE CATÉGORIQUES ENRICO M. VITALE On the characterization of monadic categories over set Cahiers de topologie et géométrie différentielle catégoriques, tome 35, no 4 (1994), p. 351-358 <http://www.numdam.org/item?id=CTGDC_1994__35_4_351_0> © Andrée C. Ehresmann et les auteurs, 1994, tous droits réservés. L’accès aux archives de la revue « Cahiers de topologie et géométrie différentielle catégoriques » implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ CAHIERS DE TOPOLOGIE ET Volume XXXV-4 (1994) GEOMETRIE DIFFERENTIELLE CATEGORIQUES ON THE CHARACTERIZATION OF MONADIC CATEGORIES OVER SET by Enrico M. VITALE R6sum6. Dans la premibre partie de 1’article on examine des con- ditions pour 1’exactitude des categories des algèbres pour une monade. Dans la deuxibme partie on d6montre une condition n6cessaire et suffi- sante pour 1’6quivalence entre categories exactes avec assez de projec- tifs ; on utilise ensuite cette condition pour obtenir une d6monstration élémentaire de la caract6risation des categories alg6briques sur les en- sembles et des categories des pr6faisceaux. Introduction In this work we look for a new proof of the theorem characterizing monadic
    [Show full text]
  • A General Framework for Relational Parametricity
    A General Framework for Relational Parametricity Kristina Sojakova and Patricia Johann January 2018 Abstract Reynolds’ original theory of relational parametricity intended to capture the idea that polymorphically typed System F programs preserve all relations between inputs. But as Reynolds himself later showed, his theory can only be formalized in a meta-theory with an impredicative universe, such as Martin-Lof¨ Type The- ory. A number of more abstract treatments of relational parametricity have since appeared; however, as we show, none of these truly generalize Reynolds’ original theory, in the sense of having it as a direct instance. Indeed, they all require cer- tain strictness conditions that Reynolds’ theory does not satisfy. To correct this, we develop an abstract framework for relational parametricity that does deliver Reynolds’ theory as a direct instance in a natural way. This framework is para- metric and uniform with respect to a choice of meta-theory, which allows us to obtain the well-known PER model of Longo and Moggi as a direct instance in a natural way as well. Moreover, we demonstrate on a concrete example that our no- tion of parametricity also encompasses proof-relevant parametric models, which does not seem to be the case for the well-known definitions. Our framework is thus both descriptive, in that it accounts for well-known models, and prescriptive, in that it identifies properties that good models of relational parametricity should satisfy. It is constructed using the new notion of a split λ2-fibration with isomor- phisms, introduced in this paper, which relaxes certain strictness requirements on split λ2-fibrations.
    [Show full text]
  • Props in Network Theory
    Props in Network Theory John C. Baez∗y Brandon Coya∗ Franciscus Rebro∗ [email protected] [email protected] [email protected] January 23, 2018 Abstract Long before the invention of Feynman diagrams, engineers were using similar diagrams to reason about electrical circuits and more general networks containing mechanical, hydraulic, thermo- dynamic and chemical components. We can formalize this reasoning using props: that is, strict symmetric monoidal categories where the objects are natural numbers, with the tensor product of objects given by addition. In this approach, each kind of network corresponds to a prop, and each network of this kind is a morphism in that prop. A network with m inputs and n outputs is a morphism from m to n, putting networks together in series is composition, and setting them side by side is tensoring. Here we work out the details of this approach for various kinds of elec- trical circuits, starting with circuits made solely of ideal perfectly conductive wires, then circuits with passive linear components, and then circuits that also have voltage and current sources. Each kind of circuit corresponds to a mathematically natural prop. We describe the `behavior' of these circuits using morphisms between props. In particular, we give a new construction of the black-boxing functor of Fong and the first author; unlike the original construction, this new one easily generalizes to circuits with nonlinear components. We also use a morphism of props to clarify the relation between circuit diagrams and the signal-flow diagrams in control theory. Technically, the key tools are the Rosebrugh{Sabadini{Walters result relating circuits to special commutative Frobenius monoids, the monadic adjunction between props and signatures, and a result saying which symmetric monoidal categories are equivalent to props.
    [Show full text]