Michael Paluch Category Theory April 29, 2014 Preface These note are based in part on the the book [2] by Saunders Mac Lane and on the book [3] by Saunders Mac Lane and Ieke Moerdijk. v Contents 1 Foundations ....................................................... 1 1.1 Extensionality and comprehension . .1 1.2 Zermelo Frankel set theory . .3 1.3 Universes.....................................................5 1.4 Classes and Gödel-Bernays . .5 1.5 Categories....................................................6 1.6 Functors .....................................................7 1.7 Natural Transformations. .8 1.8 Basic terminology . 10 2 Constructions on Categories ....................................... 11 2.1 Contravariance and Opposites . 11 2.2 Products of Categories . 13 2.3 Functor Categories . 15 2.4 The category of all categories . 16 2.5 Comma categories . 17 3 Universals and Limits .............................................. 19 3.1 Universal Morphisms. 19 3.2 Products, Coproducts, Limits and Colimits . 20 3.3 YonedaLemma ............................................... 24 3.4 Free cocompletion . 28 4 Adjoints ........................................................... 31 4.1 Adjoint functors and universal morphisms . 31 4.2 Freyd’s adjoint functor theorem . 38 5 Topos Theory ...................................................... 43 5.1 Subobject classifier . 43 5.2 Sieves........................................................ 45 5.3 Exponentials . 47 vii viii Contents Index .................................................................. 53 Acronyms List of categories. Ab The category of small abelian groups and group homomorphisms. AlgA The category of commutative A-algebras. Cb The category Func(Cop,Sets). Cat The category of small categories and functors. CRings The category of commutative ring with an identity and ring homomor- phisms which preserve identities. Grp The category of small groups and group homomorphisms. Sets Category of small set and functions. Sets Category of small pointed set and pointed functions. ¤ Top The category of small topological spaces and continuous maps. List of sets P (a) Set of subsets of a. U Universe. ix Chapter 1 Foundations In the study of categories we shall often find ourselves discussing properties in- volving all sets, all groups, all topological spaces, etc. For example, an abelian group P is called projective when for all pairs (B,C) of abelian groups and each surjective group homomorphism f : B C and each group homomorphism g : P C, there is a group homomorphism h : P B such that g f h. As ! ! Æ ± pointed out by Saunders Mac Lane in [1] “it is the intent of category theory that this all be taken seriously.” Set theory is complicated and interesting, and one needs to be very cautious in order to avoid inconsistencies. Although it is not our objective to describe here all the subtle nuances of set theory, it is important to be aware of the strengths and limitations of set theory. We view sets as building blocks with which one can construct a working theory of categories. Roughly speaking a set is a collection of things, like the set of works of art in a specific museum or a set of stamps in a philatelist’s collection. The items in a specific collection s are called members or elements of s. We write x s and say that “x is a member (or an element) of s,” 2 when x is one of the items in the collection s. We begin with two principles which govern a naive concept of sets. We shall see, however, that these principles lead to an inconsistent theory. 1.1 Extensionality and comprehension A set is determined by its elements; equivalently, if one knows the members of a set one knows the set. We can formalize this with the the following axiom. Axiom 1.1.1. If a and b are sets such that x a if and only if x b, then a b. 2 2 Æ This principle is called the Axiom of Extensionality. The second principle of naive set theory is the so-called unrestricted compre- hension axiom. It asserts that every property specifies a set. 1 2 1 Foundations Axiom 1.1.2. For each formula P(x) of first order logic, there exists a set s such that x s if and only if P(x). 2 If P is a formula and nothing satisfies P, for example P(x) is the formula x x, 6Æ Axiom 1.1.2 provides the existences of the empty set , and if P is a formula with ; the property that there is one and only one object x satisfying P, Axiom 1.1.2 asserts that there exists a set s {x}. Æ Given sets a and b, we say that a is a subset of b and write a b when every el- ⊆ ements of a is an element of b. Note that a b and a b have different meanings. 2 ⊆ We have the following two results. (i) For each set a, a a. ⊆ (ii) For all sets a and b, one has a b if and only if a b and b a. Æ ⊆ ⊆ These axioms also permit one to define the intersection and the union of two sets. Let a and b be two sets. The intersection of a and b is the unique set a b {x : x a x b}, \ Æ 2 ^ 2 and the union of a and b is the unique set a b {x : x a x b}. [ Æ 2 _ 2 Using Axiom 1.1.2 one can form for any two elements x, y of any set a a set containing only x any y. Formally we have Proposition 1.1.3. For any two objects x and y there is a unique set a {x, y}. Æ Proof. Given objects x and y consider the formula {w : w x w y} Æ _ Æ The existence of a set containing only x and y is guaranteed by Axiom 1.1.2, and its uniqueness is guaranteed by Axiom 1.1.1. For objects x and y, the two sets {x, y} and {y,x} are the same. To form an or- dered pair (x, y) we use the set {x,{x, y}}. The cartesian product of A and B is de- termined by the formula a b ©w : x a y b ¡w {x,{x, y}}¢ª. £ Æ 9 2 ^ 9 2 Æ Using the cartesian product one can define functions from a set a to a set b as certain types of subsets of a b. Given r a b, we defined the domain of r by £ ½ £ domr ©x :(x, y) r ª Æ 2 and the range or codomain of r by ranr codr ©y :(x, y) r ª. Æ Æ 2 1.2 Zermelo Frankel set theory 3 Note that r domr ranr . We say that r is a function from domr to ranr when ½ £ ( x)( y)( z) ¡(x, y) r (x,z) r y z¢. 8 8 8 2 ^ 2 ¡! Æ Russell’s Paradox From the point of view of set theory we run into some problems when we wish to analyze statements in which variables are allowed to range over large collections of objects. We can illustrate this with the set of all subsets of a given set. Let a be a set. There is a unique set P (a), called the power set of a , defined by P (a) {x : x a}. Æ ⊆ Proposition 1.1.4. For any set a, P (a) a. 6⊆ Proof. Let a be a set. Consider the set b defined by b {x : x a x x}. (1.1) Æ 2 ^ 62 The set b is a subset of a, and hence b P (a). We shall show that b a from 2 62 which we deduce P (a) a. 6⊆ Suppose that b a. One has either b b or b b. If b b then by the formula 2 2 62 2 defining b in (1.1) one deduces that b b. On the other hand if b b, then b 62 62 satisfies this formula, and hence b b. We deduce that b b if and only if b b 2 62 2 which is absurd. Thus b a. 62 ut We now consider the negation of Proposition 1.1.4 Proposition 1.1.5. There is a set U such that P (U) U. Æ Proof. Using Axiom 1.1.2 there is a universal set U that contains everything. This set U is defined by U {x : x x}. Æ Æ Now every subset of U is a member of U. Thus P (U) is a subset of U. ut Proposition 1.1.5 contradicts Proposition 1.1.4, and hence Axioms1.1.1 and 1.1.2 lead to an inconsistent theory. The problem is that Axiom 1.1.2 is too broad. The inconsistency arises from the supposition that each property determines a set. Specifically, the property of not belonging to itself does not determine a set. To obtain a consistent theory of sets one can restrict Axiom 1.1.2 to rule out so- called “large” collections of sets. 1.2 Zermelo Frankel set theory A consistent theory of sets can be assembled from atomic formulae 4 1 Foundations x y x y 2 Æ by means of connectives ' Ã, ' Ã, ', ' Ã, ' à ^ _ : ¡! Ã! and the universal quantifier x and the existential quantifier x together with 8 9 following axioms. Axiom 1.2.1 (Extensionality). ( a)( b)( x)¡(x a x b) a b)¢ 8 8 8 2 Ã! 2 ¡! Æ Axiom 1.2.2 (Pairing). ( a)( b)( c)( x)¡x c (x a x b)¢ 8 8 9 8 2 Ã! Æ _ Æ Given a and b, we write {a,b} for the c whose existence is alleged by Axiom 1.2.2.