Category Theory by Example Ivan Murashko October 28, 2018 2 Contents 1 Base denitions9 1.1 Denitions.............................9 1.1.1 Object...........................9 1.1.2 Morphism......................... 10 1.1.3 Category.......................... 12 1.2 Set category example....................... 14 1.3 Programming languages examples................ 18 1.3.1 Hask toy category.................... 19 1.3.2 C++ toy category.................... 20 1.3.3 Scala toy category.................... 21 1.4 Quantum mechanics examples.................. 22 2 Objects and morphisms 27 2.1 Equality.............................. 27 2.2 Initial and terminal objects................... 28 2.3 Product and sum......................... 30 2.4 Category as a monoid....................... 31 2.5 Exponential............................ 32 2.6 Type algebra and Curry-Howard-Lambek correspondence... 34 2.6.1 Proof category...................... 34 2.6.2 Hask category...................... 36 2.6.3 C++ category...................... 38 2.6.4 Scala category...................... 39 2.7 Quantum mechanics....................... 40 3 Functors 41 3.1 Denitions............................. 41 3.2 Cat category........................... 43 3.3 Bifunctors............................. 44 3 4 CONTENTS 4 Natural transformation 47 4.1 Denitions............................. 47 4.2 Operations with natural transformations............ 49 4.3 Polymorphism and natural transformation........... 52 4.3.1 Hask category...................... 52 5 Monads 55 5.1 Monoid in Set category..................... 55 5.2 Monoidal category........................ 59 5.3 Category of endofunctors..................... 60 5.4 Monads in programming languages............... 62 5.4.1 Haskell........................... 62 5.4.2 C++............................ 63 5.4.3 Scala............................ 63 5.5 Kleisli category.......................... 63 5.5.1 Partiality and Exception................. 65 5.5.2 Non-Determinism..................... 67 5.5.3 Side eects and interactive input/output........ 68 5.5.4 Continuation....................... 68 5.6 Examples............................. 68 5.6.1 Quantum mechanics................... 68 6 Yoneda's lemma 69 6.1 Examples............................. 69 6.1.1 Quantum mechanics................... 69 Index 71 Notations [C; D] Fun category(Example 4.2) α ◦ β Vertical composition of natural transformations (circle dot) α ? β Horizontal composition of natural transformations (star dot) αH Left whiskering α; β Natural transformation (Greek small letters) α : F −!: G Natural transformation (arrow with dot) CM Kleisli category C Category (bold capital Latin letter) Cop Opposite category cod f Codomain dom f Domain hom(a; b) set of Morphisms between a and b homC(a; b) Set of morphisms 1C)C Identity functor 1a!a Identity morphism : Identity natural transformation 1F −!F hM; µ, ηi Monad Hn nite dimensional Hilbert space ∼ a =f b there is an Isomorphism f between a and b 5 6 CONTENTS a; b Objects (Latin small letters) ab Exponential F ◦ G Functor composition (circle dot) f ◦ g Morphism composition (circle dot) F; G Functor (capital Latin letter) f; g; h Morphism (Latin small letter) F : C ) D Functor (double arrow) f : a ! b Morphism (simple arrow) Hα Right whiskering TBD To Be Dened (later) Introduction You just looked at yet another introduction to Category Theory. The subject mostly consists of a lot of denitions that are related each others and I wrote the book to collect all of them in one place to be easy checked and updated in future when I decide to refresh my knowledge about the eld of math. Therefore the book was written mostly for my category theory studying purposes but I will appreciate if somebody else nd it useful. The topics(chapters) cover the base denitions (Object, Morphism and Category), Functor, Natural transformation, Monad and also include impor- tant results from the category theory such as Yoneda's lemma (see chapter6) and Curry-Howard-Lambek correspondence (see section 2.6). There are a lot of examples in each chapter. The examples cover dierent category theory application areas. I assume that the reader is familiar with the corresponding area and the example(s) can be passed if not. I.e. anyone can choose the suitable example(s) for (s)he. The most important examples are related to the set theory. The set theory and category theory are very close related. Each one can be considered as an alternative view to another one. There are a lot of examples from programming languages which include Haskell, Scala, C++. The source les for programming languages examples (Haskell, C++, Scala) can be found on github repository [8]. The examples from physics are related to quantum mechanics that is the most known for me. For the examples I am inspired by the Bob Coecke article [1]. The text is distributed under Creative Common Public License (see the text of the license at the end of the book) i.e. any reader has right to copy, store, modify, distribute or build upon the book until the original author is pointed in the derivative products. 7 8 CONTENTS Chapter 1 Base denitions 1.1 Denitions 1.1.1 Object Denition 1.1 (Class). A class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously dened by a property that all its members share. Denition 1.2 (Object). In category theory object is considered as some- thing that does not have internal structure (aka point) but has a property that makes dierent objects belong to the same Class Remark 1.3 (Class of Objects). The Class of Objects will be marked as ob(C) (see g. 1.1). C c a d b Figure 1.1: Class of objects ob(C) = fa; b; c; dg 9 10 CHAPTER 1. BASE DEFINITIONS 1.1.2 Morphism Morphism is a kind of relation between 2 Objects. Denition 1.4 (Morphism). A relation between two Objects a and b fab : a ! b is called morphism. Morphism assumes a direction i.e. one Object( a) is called source and another one (b) target. The Set of all morphisms between objects a and b is denoted as hom (a; b). The important remark about morphisms is below Remark 1.5 (Morphism). The morphism has to be considered as a relation between objects. We will avoid standard (from set theory) notation for mor- phisms: f(a) = b. The reason for this is the following. Let f1 : a ! b and f2 : a ! b are 2 dierent morphisms. The notation f1(a) = b; f2(a) = b leads to incorrect conclusion that f1 = f2. For instance if then 2 functions set 2 a = b = R f1(x) = x; f2(x) = −x dierent ordering on R and as result have not to be considered as the same functions. Denition 1.6 (Domain). Given a Morphism f : a ! b, the Object a is called domain and denoted as dom f. Denition 1.7 (Codomain). Given a Morphism f : a ! b, the Object b is called codomain and denoted as cod f. Morphisms have several properties. 1 Axiom 1.8 (Composition). If we have 3 Objects a; b and c and 2 Morphisms fab : a ! b and fbc : b ! c then there exists Morphism fac : a ! c such that fac = fbc ◦ fab 1The properties don't have any proof and postulated as axioms 1.1. DEFINITIONS 11 Remark 1.9 (Composition). The equation fac = fbc ◦ fab means that we apply fab rst and then we apply fbc to the result of the application i.e. if our objects are sets and x 2 a then fac(x) = fbc(fab(x)); where fab(x) 2 b. Axiom 1.10 (Associativity). The Morphisms Composition(Axiom 1.8) s should follow associativity property: fce ◦ (fbc ◦ fab) = (fce ◦ fbc) ◦ fab = fce ◦ fbc ◦ fab: Denition 1.11 (Identity morphism). For every Object a we dene a special Morphism 1a!a : a ! a with the following properties: 8fab : a ! b 1a!a ◦ fab = fab (1.1) and 8fba : b ! a fba ◦ 1a!a = fba: (1.2) This morphism is called as identity morphism. Note that Identity morphism is unique, see Identity is unique(Theo- rem 2.3) below. Denition 1.12 (Commutative diagram). A commutative diagram is a di- agram of Objects (also known as vertices) and Morphisms (also known as arrows or edges) such that all directed paths in the diagram with the same start and endpoint lead to the same result by composition The following diagram commutes if fab = fcb ◦ fac. a fab b fac fcb c Remark 1.13 (Class of Morphisms). The Class of Morphisms will be marked as hom(C) (see g. 1.2) 12 CHAPTER 1. BASE DEFINITIONS C f a c g h b d Figure 1.2: Class of morphisms hom(C) = ff; g; hg, where h = f ◦ g Denition 1.14 (Monomorphism). If 8g1; g2 the equation f ◦ g1 = f ◦ g2 leads to g1 = g2 then f is called monomorphism. Denition 1.15 (Epimorphism). If 8g1; g2 the equation g1 ◦ f = g2 ◦ f leads to g1 = g2 then f is called epimorphism. Denition 1.16 (Isomorphism). A Morphism f : a ! b is called isomor- phism if 9g : b ! a such that f ◦ g = 1a!a and g ◦ f = 1b!b. If there is an ∼ isomorphism f between objects a and b then it is denoted as a =f b. Remark 1.17 (Isomorphism). There are can be many dierent Isomor- phisms between 2 Objects. If there is an unique isomorphism between 2 objects then the objects can be treated as the same object. 1.1.3 Category Denition 1.18 (Category). A category C consists of Class of Objects ob(C) 1.1. DEFINITIONS 13 C 1a!a 1c!c f a c g h b d 1b!b 1d!d Figure 1.3: Category C. It consists of 4 objects ob(C) = fa; b; c; dg and 7 morphisms ob(C) = ff; g; h = f ◦ g; 1a!a; 1b!b; 1c!c; 1d!dg Class of Morphisms hom(C) dened for ob(C), i.e.