CATEGORIES. 1.1 Introduction

Transcription

1 1 CATEGORIES 1.1 Introduction What is category theory? As a irst approximation, one could say that category theory is the mathematical study o (abstract) algebras o unctions. Just as group theory is the abstraction o the idea o a system o permutations o a set or symmetries o a geometric object, category theory arises rom the idea o a system o unctions among some objects. A B g g C We think o the composition g as a sort o product o the unctions and g, and consider abstract algebras o the sort arising rom collections o unctions. A category is just such an algebra, consisting o objects A, B, C,... and arrows : A B, g : B C,..., that are closed under composition and satisy certain conditions typical o the composition o unctions. A precise deinition is given later in this chapter. A branch o abstract algebra, category theory was invented in the tradition o Felix Klein s Erlanger Programm, as a way o studying and characterizing dierent kinds o mathematical structures in terms o their admissible transormations. The general notion o a category provides a characterization o the notion o a structure-preserving transormation, and thereby o a species o structures admitting such transormations. The historical development o the subject has been, very roughly, as ollows: 1945 Eilenberg and Mac Lane s General theory o natural equivalences was the original paper, in which the theory was irst ormulated. late 1940s The main applications were originally in the ields o algebraic topology, particularly homology theory, and abstract algebra.

2 2 CATEGORIES 1950s A. Grothendieck et al. began using category theory with great success in algebraic geometry. 1960s F.W. Lawvere and others began applying categories to logic, revealing some deep and surprising connections. 1970s Applications were already appearing in computer science, linguistics, cognitive science, philosophy, and many other areas. One very striking thing about the ield is that it has such wide-ranging applications. In act, it turns out to be a kind o universal mathematical language like set theory. As a result o these various applications, category theory also tends to reveal certain connections between dierent ields like logic and geometry. For example, the important notion o an adjoint unctor occurs in logic as the existential quantiier and in topology as the image operation along a continuous unction. From a categorical point o view these turn out to be essentially the same operation. The concept o adjoint unctor is in act one o the main things that the reader should take away rom the study o this book. It is a strictly category-theoretical notion that has turned out to be a conceptual tool o the irst magnitude on par with the idea o a continuous unction. In act, just as the idea o a topological space arose in connection with continuous unctions, so also the notion o a category arose in order to deine that o a unctor, at least according to one o the inventors. The notion o a unctor arose so the story goes on in order to deine natural transormations. One might as well continue that natural transormations serve to deine adjoints: Category Functor Natural transormation Adjunction Indeed, that gives a pretty good outline o this book. Beore getting down to business, let us ask why it should be that category theory has such ar-reaching applications. Well, we said that it is the abstract theory o unctions, so the answer is simply this: Functions are everywhere! And everywhere that unctions are, there are categories. Indeed, the subject might better have been called abstract unction theory, or, perhaps even better: archery.

3 FUNCTIONS OF SETS Functions o sets We begin by considering unctions between sets. I am not going to say here what a unction is, anymore than what a set is. Instead, we will assume a working knowledge o these terms. They can in act be deined using category theory, but that is not our purpose here. Let be a unction rom a set A to another set B, we write : A B. To be explicit, this means that is deined on all o A and all the values o are in B. In set theoretic terms, range() B. Now suppose we also have a unction g : B C, A B... g g C then there is a composite unction g : A C, given by (g )(a) = g((a)) a A. (1.1) Now this operation o composition o unctions is associative, as ollows. I we have a urther unction h : C D A g B g C h g h D and orm h g and g then we can compare (h g) and h (g ) as indicated in the above diagram. It turns out that these two unctions are always identical, since or any a A, we have (h g) = h (g ) ((h g) )(a) = h(g((a))) = (h (g ))(a) using (1.1). By the way, this is o course what it means or two unctions to be equal: or every argument, they have the same value.

4 4 CATEGORIES Finally, note that every set A has an identity unction given by 1 A : A A 1 A (a) = a. These identity unctions act as units or the operation o composition, in the sense o abstract algebra. That is to say, or any : A B. 1 A = = 1 B A 1 A A 1 A B 1 B B 1 B These are all the properties o set unctions that we want to consider or the abstract notion o unction: composition and identities. Thus, we now want to abstract away everything else, so to speak. That is what is accomplished by the ollowing deinition. 1.3 Deinition o a category Deinition 1.1. A category consists o the ollowing data: Objects: A, B, C,... Arrows:, g, h,... For each arrow there are given objects: dom(), cod() called the domain and codomain o. We write: : A B to indicate that A = dom() and B = cod(). Given arrows : A B and g : B C, that is, with: cod() = dom(g) there is given an arrow: g : A C

5 EXAMPLES OF CATEGORIES 5 called the composite o and g. For each object A there is given an arrow: called the identity arrow o A. 1 A : A A These data are required to satisy the ollowing laws: Associativity: h (g ) = (h g) or all : A B, g : B C, h : C D. Unit: or all : A B. 1 A = = 1 B A category is anything that satisies this deinition and we will have plenty o examples very soon. For now I want to emphasize that, unlike in the previous section, the objects do not have to be sets and the arrows need not be unctions. In this sense, a category is an abstract algebra o unctions, or arrows (sometimes also called morphisms ), with the composition operation as primitive. I you are amiliar with groups, you may think o a category as a sort o generalized group. 1.4 Examples o categories 1. We have already encountered the category Sets o sets and unctions. There is also the category Sets in o all inite sets and unctions between them. Indeed, there are many categories like this, given by restricting the sets that are to be the objects and the unctions that are to be the arrows. For example, take inite sets as objects and injective (i.e., 1 to 1 ) unctions as arrows. Since injective unctions compose to give an injective unction, and since the identity unctions are injective, this also gives a category. What i we take sets as objects and as arrows, those : A B such that or all b B, the subset 1 (b) A has at most two elements (rather than one)? Is this still a category? What i we take the unctions such that 1 (b) is inite? ininite? There are lots o such restricted categories o sets and unctions.

6 6 CATEGORIES 2. Another kind o example one oten sees in mathematics is categories o structured sets, that is, sets with some urther structure and unctions which preserve it, where these notions are determined in some independent way. Examples o this kind you may be amiliar with are: groups and group homomorphisms, vector spaces and linear mappings, graphs and graph homomorphisms, the real numbers R and continuous unctions R R, open subsets U R and continuous unctions : U V R deined on them, topological spaces and continuous mappings, dierentiable maniolds and smooth mappings, the natural numbers N and all recursive unctions N N, or as in the example o continuous unctions, one can take partial recursive unctions deined on subsets U N. posets and monotone unctions. Do not worry i some o these examples are unamiliar to you. Later on, we will take a closer look at some o them. For now, let us just consider the last o the above examples in more detail. 3. A partially ordered set or poset is a set A equipped with a binary relation a A b such that the ollowing conditions hold or all a, b, c A: relexivity: a A a, transitivity: i a A b and b A c, then a A c, antisymmetry: i a A b and b A a, then a = b. For example, the real numbers R with their usual ordering x y orm a poset that is also linearly ordered: either x y or y x or any x, y. An arrow rom a poset A to a poset B is a unction m : A B that is monotone, in the sense that, or all a, a A, a A a implies m(a) B m(a ). What does it take or this to be a category? We need to know that 1 A : A A is monotone, but that is clear since a A a implies a A a. We also need to know that i : A B and g : B C are monotone, then g : A C is monotone. This also holds, since a a implies (a) (a ) implies g((a)) g((a )) implies (g )(a) (g )(a ). So we have the category Pos o posets and monotone unctions.

7 EXAMPLES OF CATEGORIES 7 4. The categories that we have been considering so ar are examples o what are sometimes called concrete categories. Inormally, these are categories in which the objects are sets, possibly equipped with some structure, and the arrows are certain, possibly structure-preserving, unctions (we shall see later on that this notion is not entirely coherent; see Remark 1.7). But in act, one way o understanding what category theory is all about is doing without elements, and replacing them by arrows instead. Let us now take a look at some examples where this point o view is not just optional, but essential. Let Rel be the ollowing category: take sets as objects and take binary relations as arrows. That is, an arrow : A B is an arbitrary subset A B. The identity arrow on a set A is the identity relation. 1 A = {(a, a) A A a A} A A. Given R A B and S B C, deine composition S R by (a, c) S R i b. (a, b) R & (b, c) S that is, the relative product o S and R. We leave it as an exercise to show that Rel is in act a category. (What needs to be done?) For another example o a category in which the arrows are not unctions, let the objects be inite sets A, B, C and an arrow F : A B is a rectangular matrix F = (n ij ) i<a,j<b o natural numbers with a = A and b = B, where C is the number o elements in a set C. The composition o arrows is by the usual matrix multiplication, and the identity arrows are the usual unit matrices. The objects here are serving simply to ensure that the matrix multiplication is deined, but the matrices are not unctions between them. 5. Finite categories O course, the objects o a category do not have to be sets, either. Here are some very simple examples: The category 1 looks like this: It has one object and its identity arrow, which we do not draw. The category 2 looks like this: It has two objects, their required identity arrows, and exactly one arrow between the objects.

8 8 CATEGORIES The category 3 looks like this: It has three objects, their required identity arrows, exactly one arrow rom the irst to the second object, exactly one arrow rom the second to the third object, and exactly one arrow rom the irst to the third object (which is thereore the composite o the other two). The category 0 looks like this: It has no objects or arrows. As above, we will omit the identity arrows in drawing categories rom now on. It is easy to speciy inite categories just take some objects and start putting arrows between them, but make sure to put in the necessary identities and composites, as required by the axioms or a category. Also, i there are any loops, then they need to be cut o by equations to keep the category inite. For example, consider the ollowing speciication: A B g Unless we stipulate an equation like g = 1 A, we will end up with ininitely many arrows g, gg, ggg,.... This is still a category, o course, but it is not a inite category. We will come back to this situation when we discuss ree categories later in this chapter. 6. One important slogan o category theory is, It s the arrows that really matter! So we should also look at the arrows or mappings between categories. A homomorphism o categories is called a unctor. Deinition 1.2. A unctor F : C D between categories C and D is a mapping o objects to objects and arrows to arrows, in such a way that: (a) F ( : A B) = F () : F (A) F (B),

9 EXAMPLES OF CATEGORIES 9 (b) F (1 A ) = 1 F (A), (c) F (g ) = F (g) F (). That is, F preserves domains and codomains, identity arrows, and compostion. A unctor F : C D thus gives a sort o picture perhaps distorted o C in D. A B C g g C F F (B) D F () F (A) F (g ) F (g) F (C) Now, one can easily see that unctors compose in the expected way, and that every category C has an identity unctor 1 C : C C. So we have another example o a category, namely Cat, the category o all categories and unctors. 7. A preorder is a set P equipped with a binary relation p q that is both relexive and transitive: a a, and i a b and b c, then a c. Any preorder P can be regarded as a category by taking the objects to be the elements o P and taking a unique arrow, a b i and only i a b. (1.2) The relexive and transitive conditions on ensure that this is indeed a category. Going in the other direction, any category with at most one arrow between any two objects determines a preorder, simply by deining a binary relation on the objects by (1.2). 8. A poset is evidently a preorder satisying the additional condition o antisymmetry: i a b and b a, then a = b. So, in particular, a poset is also a category. Such poset categories are very common; or example, or

10 10 CATEGORIES any set X, the powerset P(X) is a poset under the usual inclusion relation U V between the subsets U, V o X. What is a unctor F : P Q between poset categories P and Q? It must satisy the identity and composition laws.... Clearly, these are just the monotone unctions already considered above. It is oten useul to think o a category as a kind o generalized poset, one with more structure than just p q. Thus, one can also think o a unctor as a generalized monotone map. 9. An example rom topology: Let X be a topological space with collection o open sets O(X). Ordered by inclusion, O(X) is a poset category. Moreover, the points o X can be preordered by specialization by setting x y i x U implies y U or every open set U, i.e. y is contained in every open set that contains x. I X is suiciently separated ( T 1 ), then this ordering becomes trivial, but it can be quite interesting otherwise, as happens in the spaces o algebraic geometry and denotational semantics. It is an exercise to show that T 0 spaces are actually posets under the specialization ordering. 10. An example rom logic: Given a deductive system o logic, there s an associated category category o proos,in which the objects are ormulas: ϕ, ψ,... An arrow rom ϕ to ψ is a deduction o ψ rom the (uncanceled) assumption ϕ. ϕ. ψ Composition o arrows is given by putting together such deductions in the obvious way, which is clearly associative. (What should the identity arrows 1 ϕ be?) Observe that there can be many dierent arrows p : ϕ ψ, since there may be many dierent such proos. This category turns out to have a very rich structure, which we will consider later in connection with the lambda-calculus. 11. An example rom computer science: Given a unctional programming language L, there is an associated category, where the objects are the data types o L, and the arrows are the computable unctions o L ( processes, procedures, programs ). The composition o two such programs X Y g Z is given by applying g to the output o, sometimes

11 EXAMPLES OF CATEGORIES 11 also written g = ; g. The identity is the do nothing program. Categories such as this are basic to the idea o denotational semantics o programming languages. For example, i C(L) is the category just deined, then the denotational semantics o the language L in a category D o, say, Scott domains is simply a unctor S : C(L) D since S assigns domains to the types o L and continuous unctions to the programs. Both this example and the previous one are related to the notion o cartesian closed category that is considered later. 12. Let X be a set. We can regard X as a category Dis(X) by taking the objects to be the elements o X and taking the arrows to be just the required identity arrows, one or each x X. Such categories, in which the only arrows are identities, are called discrete. Note that discrete categories are just very special posets. 13. A monoid (sometimes called a semigroup with unit) is a set M equipped with a binary operation : M M M and a distinguished unit element u M such that or all x, y, z M, and x (y z) = (x y) z u x = x = x u. Equivalently, a monoid is a category with just one object. The arrows o the category are the elements o the monoid. In particular, the identity arrow is the unit element u. Composition o arrows is the binary operation m n o the monoid. Monoids are very common: there are the monoids o numbers like N, Q or R with addition and 0, or multiplication and 1. But also or any set X, the set o unctions rom X to X, written Hom Sets (X, X) is a monoid under the operation o composition. More generally, or any object C in any category C, the set o arrows rom C to C, written as Hom C (C, C), is a monoid under the composition operation o C. Since monoids are structured sets, there is a category Mon whose objects are monoids and whose arrows are unctions that preserve the monoid structure. In detail, a homomorphism rom a monoid M to a monoid N is a unction h : M N such that or all m, n M, h(m M n) = h(m) N h(n)

12 12 CATEGORIES and h(u M ) = u N. Observe that a monoid homomorphism rom M to N is the same thing as a unctor rom M regarded as a category to N regarded as a category. In this sense, categories are also generalized monoids, and unctors are generalized homomorphisms. 1.5 Isomorphisms Deinition 1.3. In any category C, an arrow : A B is called an isomorphism i there is an arrow g : B A in C such that g = 1 A and g = 1 B. Since inverses are unique (proo!), we write g = 1. We say that A is isomorphic to B, written A = B, i there exists an isomorphism between them. The deinition o isomorphism is our irst example o an abstract, category theoretic deinition o an important notion. It is abstract in the sense that it makes use only o the category theoretic notions, rather than some additional inormation about the objects and arrows. It has the advantage over other possible deinitions that it applies in any category. For example, one sometimes deines an isomorphism o sets (monoids, etc.) as a bijective unction (resp. homomorphism), i.e., one that is 1-1 and onto making use o the elements o the objects. This is equivalent to our deinition in some cases, such as sets and monoids. But note that, or example in Pos, the category theoretic deinition gives the right notion, while there are bijective homomorphisms between nonisomorphic posets. Moreover, in many cases only the abstract deinition makes sense, as or example, in the case o a monoid regarded as a category. Deinition 1.4. A group G is a monoid with an inverse g 1 or every element g. Thus G is a category with one object, in which every arrow is an isomorphism. The natural numbers N do not orm a group under either addition or multiplication, but the integers Z and the positive rationals Q +, respectively, do. For any set X, we have the group Aut(X) o automorphisms (or permutations ) o X, that is, isomorphisms : X X. (Why is this closed under?) A group o permutations is a subgroup G Aut(X) or some set X, that is, a group o (some) automorphisms o X. Thus the set G must satisy the ollowing: 1. The identity unction 1 X on X is in G. 2. I g, g G, then g g G. 3. I g G, then g 1 G.

13 ISOMORPHISMS 13 A homomorphism o groups h : G H is just a homomorphism o monoids, which then necessarily also preserves the inverses (proo!). Now consider the ollowing basic, classical result about abstract groups: Theorem (Cayley). Every group G is isomorphic to a group o permutations. Proo. (sketch) 1. First, deine the Cayley representation Ḡ o G to be the ollowing group o permutations o a set: the set is just G itsel, and or each element g G, we have the permutation ḡ : G G, deined or all h G by acting on the let : ḡ(h) = g h. This is indeed a permutation, since it has the action o g 1 as an inverse. 2. Next deine homomorphisms i : G Ḡ by i(g) = ḡ, and j : Ḡ G by j(ḡ) = ḡ(u). 3. Finally show that i j = 1 Ḡ and j i = 1 G. Warning 1.5. Note the two dierent levels o isomorphisms that occur in the proo o Cayley s theorem. There are permutations o the set o elements o G, which are isomorphisms in Sets, and there is the isomorphism between G and Ḡ, which is in the category Groups o groups and group homomorphisms. Cayley s theorem says that any abstract group can be represented as a concrete one, that is, a group o permutations o a set. The theorem can in act be generalized to show that any category that is not too big can be represented as one that is concrete, that is, a category o sets and unctions. (There is a technical sense o not being too big that will be introduced in Section 1.8.) Theorem 1.6. Every category C with a set o arrows is isomorphic to one in which the objects are sets and the arrows are unctions. Proo. (sketch) Deine the Cayley representation C o C to be the ollowing concrete category: objects are sets o the orm or all C C, arrows are unctions C = { C cod() = C} ḡ : C D

14 14 CATEGORIES or g : C D in C, deined or any : X C in C by ḡ() = g. X C g g D Remark 1.7. This shows us what is wrong with the naive notion o a concrete category o sets and unctions: while not every category has special sets and unctions as its objects and arrows, every category is isomorphic to such a one. Thus, the only special properties such categories can possess are ones that are categorically irrelevant, such as eatures o the objects that do not aect the arrows in any way (like the dierence between the real numbers constructed as Dedekind cuts or as Cauchy sequences). A better attempt to capture what is intended by the rather vague idea o a concrete category is that arbitrary arrows : C D are completely determined by their composites with arrows x : T C rom some test object T, in the sense that x = gx or all such x implies = g. As we shall see later, this amounts to considering a particular representation o the category, determined by T. A category is then said to be concrete when this condition holds or T a terminal object, in the sense o Section 2.2; but there are also good reasons or considering other objects T, as we see in the next chapter. Note that the condition that C have a set o arrows is needed to ensure that the collections { C cod() = C} really are sets we return to this point in Section Constructions on categories Now that we have a stock o categories to work with, we can consider some constructions that produce new categories rom old. 1. The product o two categories C and D, written C D has objects o the orm (C, D), or C C and D D, and arrows o the orm (, g) : (C, D) (C, D )

15 CONSTRUCTIONS ON CATEGORIES 15 or : C C C and g : D D D. Composition and units are deined componentwise; that is, (, g ) (, g) = (, g g) 1 (C,D) = (1 C, 1 D ). There are two obvious projection unctors C π 1 C D π2 D deined by π 1 (C, D) = C and π 1 (, g) =, and similarly or π 2. The reader amiliar with groups will recognize that or groups G and H, the product category G H is the usual (direct) product o groups. 2. The opposite (or dual ) category C op o a category C has the same objects as C, and an arrow : C D in C op is an arrow : D C in C. That is C op is just C with all o the arrows ormally turned around. It is convenient to have a notation to distinguish an object (resp. arrow) in C rom the same one in C op. Thus, let us write : D C in C op or : C D in C. With this notation we can deine composition and units in C op in terms o the corresponding operations in C, namely, Thus, a diagram in C A 1 C = (1 C ) g = (g ). B looks like this in C op g A g g C B g Many duality theorems o mathematics express the act that one category is (a subcategory o) the opposite o another. An example o this sort which C

16 16 CATEGORIES we will prove later is that Sets is dual to the category o complete, atomic Boolean algebras. 3. The arrow category C o a category C has the arrows o C as objects, and an arrow g rom : A B to : A B in C is a commutative square A g 1 A B B g 2 where g 1 and g 2 are arrows in C. That is, such an arrow is a pair o arrows g = (g 1, g 2 ) in C such that g 2 = g 1. The identity arrow 1 on an object : A B is the pair (1 A, 1 B ). Composition o arrows is done componentwise: (h 1, h 2 ) (g 1, g 2 ) = (h 1 g 1, h 2 g 2 ) The reader should veriy that this works out by drawing the appropriate commutative diagram. Observe that there are two unctors: C dom C cod C 4. The slice category C/C o a category C over an object C C has: objects: all arrows C such that cod() = C, arrows: an arrow a rom : X C to : X C is an arrow a : X X in C such that a =, as indicated in X a X C The identity arrows and composites are inherited rom those o C, just as in the arrow category. Note that there is a unctor U : C/C C that orgets about the base object C. I g : C D is any arrow, then there is a composition unctor, g : C/C C/D

17 CONSTRUCTIONS ON CATEGORIES 17 deined by g () = g, X C g g D and similarly or arrows in C/C. Indeed, the whole construction is a unctor, C/( ) : C Cat as the reader can easily veriy. Compared to the Cayley representation, this unctor gives a representation o C as a category o categories and unctors rather than sets and uctions. O course, the Cayley representation was just this one ollowed by the orgetul unctor U : Cat Sets which takes a category to its underlying set o objects. I C = P is a poset category and p P, then P/p = (p) the slice category P/p is just the principal ideal (p) o elements q P with q p. We will have more examples o slice categories soon. The coslice category C/C o a category C under an object C o C has as objects all arrows o C such that dom() = C, and an arrow rom : C X to : C X is an arrow h : X X such that h =. The reader should now carry out the rest o the deinition o the coslice category by analogy with the deinition o the slice category. How can the coslice category be deined in terms o the slice category and the opposite construction? Example 1.8. The category Sets o pointed sets consists o sets A with a distinguished element a A, and arrows : (A, a) (B, b) are unctions : A B that preserves the points, (a) = b. This is isomorphic to the coslice category, Sets = 1/Sets o Sets under any singleton 1 = { }. Indeed, unctions a : 1 A correspond uniquely to elements, a( ) = a A, and arrows : (A, a) (B, b) correspond exactly to commutative triangles: 1 a A b B

18 18 CATEGORIES 1.7 Free categories Free monoid. Start with an alphabet A o letters a, b, c,..., i.e. a set, A = {a, b, c,...}. A word over A is a inite sequence o letters: thisword, categoriesare un, asddjbnzz j,... We write - or the empty word. The Kleene closure o A is deined to be the set A = {words over A}. Deine a binary operation on A by w w = ww or words w, w A. Thus, is just concatenation. The operation is thus associative, and the empty word - is a unit. Thus, A is a monoid called the ree monoid on the set A. The elements a A can be regarded as words o length one, so we have a unction i : A A deined by i(a) = a, and called the insertion o generators. The elements o A generate the ree monoid, in the sense that every w A is a -product o a s, that is, w = a 1 a 2 a n or some a 1, a 2,..., a n in A. Now what does ree mean here? Any guesses? One sometimes sees deinitions in baby algebra books along the ollowing lines: A monoid M is reely generated by a subset A o M, i the ollowing conditions hold. 1. Every element m M can be written as a product o elements o A m = a 1 M... M a n, a i A. 2. No nontrivial relations hold in M, that is, i a 1... a j = a 1... a k, then this is required by the axioms or monoids. The irst condition is sometimes called no junk, while the second condition is sometimes called no noise. Thus, the ree monoid on A is a monoid containing A and having no junk and no noise. What do you think o this deinition o a ree monoid? I would object to the reerence in the second condition to provability, or something. This must be made more precise or this to succeed as a deinition. In category theory, we give a precise deinition o ree capturing what is meant in the above which avoids such vagueness. First, every monoid N has an underlying set N, and every monoid homomorphism : N M has an underlying unction : N M. It is easy to see that this is a unctor, called the orgetul unctor. The ree

19 FREE CATEGORIES 19 monoid M(A) on a set A is by deinition the monoid with the ollowing so called universal mapping property, or UMP! Universal Mapping Property o M(A) There is a unction i : A M(A), and given any monoid N and any unction : A N, there is a unique monoid homomorphism : M(A) N such that i =, all as indicated in the ollowing diagram: in Mon: in Sets: M(A)... N M(A) i N Proposition 1.9. A has the UMP o the ree monoid on A. Proo. Given : A N, deine : A N by A ( ) = u N, the unit o N (a 1... a i ) = (a 1 ) N... N (a i ). Then is clearly a homomorphism with (a) = (a) or all a A. I g : A N also satisies g(a) = (a) or all a A, then or all a 1... a i A : So, g =, as required. g(a 1... a i ) = g(a 1... a i ) = g(a 1 ) N... N g(a i ) = (a 1 ) N... N (a i ) = (a 1 ) N... N (ai ) = (a 1... a i ) = (a 1... a i ). Think about why the above UMP captures precisely what is meant by no junk and no noise. Speciically, the existence part o the UMP captures the vague notion o no noise (because any equation that holds between algebraic combinations o the generators must also hold anywhere they can be mapped to,

20 20 CATEGORIES and thus everywhere), while the uniqueness part makes precise the no junk idea (because any extra elements not combined rom the generators would be ree to be mapped to dierent values). Using the UMP, it is easy to show that the ree monoid M(A) is determined uniquely up to isomorphism, in the ollowing sense. Proposition Given monoids M and N with unctions i : A M and j : A N, each with the UMP o the ree monoid on A, there is a (unique) monoid isomorphism h : M = N such that h i = j and h 1 j = i. Proo. From j and the UMP o M, we have j : M N with j i = j and rom i and the UMP o N, we have ī : N M with ī j = i. Composing gives a homomorphism ī j : M M such that ī j i = i. Since 1 M : M M also has this property, by the uniqueness part o the UMP o M, we have ī j = 1 M. Exchanging the roles o M and N shows j ī = 1 N ; in Mon: j ī M... N... M in Sets: M j N i j A ī M i For example, the ree monoid on any set with a single element is easily seen to be isomorphic to the monoid o natural numbers N under addition (the generator is the number 1). Thus, as a monoid, N is uniquely determined up to isomorphism by the UMP o ree monoids. Free category. Now, we want to do the same thing or categories in general (not just monoids). Instead o underlying sets, categories have underlying graphs, so let us review these irst. A directed graph consists o vertices and edges, each o which is directed, that is, each edge has a source and a target vertex. A x u z B y C D

21 FREE CATEGORIES 21 We draw graphs just like categories, but there is no composition o edges, and there are no identities. A graph thus consists o two sets, E (edges) and V (vertices), and two unctions, s : E V (source) and t : E V (target). Thus, in Sets, a graph is just a coniguration o objects and arrows o the orm: E s t V Now, every graph G generates a category C(G), the ree category on G. It is deined by taking the vertices o G as objects, and the paths in G as arrows, where a path is a inite sequence o edges e 1,..., e n such that t(e i ) = s(e i+1 ), or all i = 1... n. We ll write the arrows o C(G) in the orm e n e n 1... e 1. Put: v 0 e 1 v 1 e 2 v 2 e 3... dom(e n... e 1 ) = s(e 1 ) cod(e n... e 1 ) = t(e n ) and deine composition by concatenation: e n... e 1 e m... e 1 = e n... e 1 e m... e 1. e n v n For each vertex v, we have an empty path denoted 1 v, which is to be the identity arrow at v. Note that i G has only one vertex, then C(G) is just the ree monoid on the set o edges o G. Also note that i G has only vertices (no edges), then C(G) is the discrete category on the set o vertices o G. Later on, we will have a general deinition o ree. For now, let us see that C(G) also has a UMP. First, deine a orgetul unctor U : Cat Graphs in the obvious way: the underlying graph o a category C has as edges the arrows o C, and as vertices the objects, with s = dom and t = cod. The action o U on unctors is equally clear, or at least it will be, once we have deined the arrows in Graphs. A homomorphism o graphs is o course a unctor without the conditions on identities and composition, that is, a mapping o edges to edges and vertices to vertices that preserves sources and targets. We will describe this rom a slightly dierent point o view, which will be useul later on. First, observe that we can describe a category C with a diagram like this: C 2 C1 cod i dom C 0

22 22 CATEGORIES where C 0 is the collection o objects o C, C 1 the arrows, i is the identity arrow operation, and C 2 is the collection {(, g) C 1 C 1 : cod() = dom(g)}. Then a unctor F : C D rom C to another category D is a pair o unctions F 0 : C 0 D 0 F 1 : C 1 D 1 such that each similarly labeled square in the ollowing diagram commutes: cod C 2 C1 i dom C 0 F 2 D 2 F 1 F 0 cod D 1 i dom D 0 where F 2 (, g) = (F 1 (), F 1 (g)). Now let us describe a homomorphism o graphs, h : G H. We need a pair o unctions h 0 : G 0 H 0, h 1 : G 1 H 1 making the two squares (once with t s, once with s s) in the ollowing diagram commute: G 1 t s G 0 h 0 h 1 H 1 t s H 0 In these terms, we can easily describe the orgetul unctor, as sending the category to the underlying graph U : Cat Graphs cod C 2 C1 i dom C 0 C 1 cod dom C 0.

23 FOUNDATIONS: LARGE, SMALL, AND LOCALLY SMALL 23 And similarly or unctors, the eect o U is described by simply erasing some parts o the diagrams (which is easier to demonstrate with chalk!). Let us again write C = U(C), etc., or the underlying graph o a category C, in analogy to the case o monoids above. The ree category on a graph now has the ollowing UMP: Universal Mapping Property o C(G) There is a graph homomorphism i : G C(G), and given any category D and any graph homomorphism h : G D, there is a unique unctor h : C(G) D with h i = h. in Cat: in Graph: h C(G)... D C(G) i h D h G The ree category on a graph with just one vertex is just a ree monoid on the set o edges. The ree category on a graph with two vertices and one edge between them is the inite category 2. The ree category on a graph o the orm: e A B has (in addition to the identity arrows) the ininitely many arrows: e,, e, e, ee, e, ee, Foundations: large, small, and locally small Let us begin by distinguishing between the ollowing things: categorical oundations or mathematics, mathematical oundations or category theory. As or the irst: one sometimes hears it said that category theory can be used to provide oundations or mathematics, as an alternative to set theory. That is in act the case, but it is not what we are doing here. In set theory, one oten begins with existential axioms such as there is an ininite set and derives urther sets by axioms like every set has a powerset, thus building up a

24 24 CATEGORIES universe o mathematical objects (namely sets), which in principle suice or all o mathematics. Our axiom that every arrow has a domain and a codomain is not to be understood in the same way as set theory s axiom that every set has a powerset! The dierence is that in set theory at least as usually conceived the axioms are to be regarded as reerring to (or determining) a single universe o sets. In category theory, by contrast, the axioms are a deinition o something, namely o categories. This is just like in group theory or topology, where the axioms serve to deine the objects under investigation. These, in turn, are assumed to exist in some background or oundational system, like set theory (or type theory). That theory o sets could itsel, in turn, be determined using category theory, or in some other way. This brings us to the second point: we assume that our categories are comprised o sets and unctions, in one way or another, like most mathematical objects, and taking into account the remarks just made about the possibility o categorical (or other) oundations. But in category theory, we sometimes run into diiculties with set theory as usually practiced. Mostly these are questions o size; some categories are too big to be handled comortably in conventional set theory.we already encountered this issue when we considered the Cayley representation in Section 1.5. There we had to require that the category under consideration had (no more than) a set o arrows. We would certainly not want to impose this restriction in general, however (as one usually does or, say, groups); or then even the category Sets would ail to be a proper category, as would many other categories that we deinitely want to study. There are various ormal devices or addressing these issues, and they are discussed in the book by Mac Lane. For our immediate purposes, the ollowing distinction will be useul: Deinition A category C is called small i both the collection C 0 o objects o C and the collection C 1 o arrows o C are sets. Otherwise, C is called large. For example, all inite categories are clearly small, as is the category Sets in o inite sets and unctions. (Actually, one should stipulate that the sets are only built rom other inite sets, all the way down, i.e. that they are hereditarily inite.) On the other hand, the category Pos o posets, the category Groups o groups, and the category Sets o sets are all large. We let Cat be the category o all small categories, which itsel is a large category. In particular, then, Cat is not an object o itsel, which may come as a relie to some readers. This does not really solve all o our diiculties. Even or large categories like Groups and Sets we will want to also consider constructions like the category o all unctors rom one to the other (we will deine this unctor category later). But i these are not small, conventional set theory does not provide the means to do this directly (these categories would be too large ). So, one needs a more elaborate theory o classes to handle such constructions. We will not worry about this when it is just a matter o technical oundations (Mac Lane I.6

25 EXERCISES 25 addresses this issue). However, one very useul notion in this connection is the ollowing: Deinition A category C is called locally small i or all objects X, Y in C, the collection Hom C (X, Y ) = { C 1 : X Y } is a set (called a hom-set). Many o the large categories we want to consider are in act locally small. Sets is locally small since Hom Sets (X, Y ) = Y X, the set o all unctions rom X to Y. Similarly, Pos, Top, and Group are all locally small (is Cat?), and, o course, any small category is locally small. Warning Don t conuse the notions concrete and small. To say that a category is concrete is to say that the objects o the category are (structured) sets, and the arrows o the category are (certain) unctions. To say that a category is small is to say that the collection o all objects o the category is a set, as is the collection o all arrows. The real numbers R, regarded as a poset category, is small but not concrete. The category Pos o all posets is concrete but not small. 1.9 Exercises 1. The objects o Rel are sets, and an arrow : A B is a relation rom A to B, that is, a subset A B. The equality relation { a, a A A a A} is the identity arrow on a set A. Composition in Rel is to be given by g = { a, c A C b ( a, b & b, c g)} or A B and g B C. Show that Rel is a category. 2. Consider the ollowing isomorphisms o categories and determine which hold. (a) Rel = Rel op (b) Sets = Sets op (c) For a ixed set X with powerset P (X), as poset categories P (X) = P (X) op (the arrows in P (X) are subset inclusions A B or all A, B X). 3. (a) Show that in Sets, the isomorphisms are exactly the bijections. (b) Show that in Monoids, the isomorphisms are exactly the bijective homomorphisms. (c) Show that in Posets, the isomorphisms are not the same as the bijective homomorphisms. 4. Let X be a topological space and preorder the points by specialization: x y i y is contained in every open set that contains x. Show that this

26 26 CATEGORIES is a preorder, and that it is a poset i X is T 0 (or any two distinct points, there is some open set containing one but not the other). Show that the ordering is trivial i X is T 1 (or any two distinct points, each is contained in an open set not containing the other). 5. For any category C, deine a unctor U : C/C C rom the slice category over an object C that orgets about C. Find a unctor F : C/C C to the arrow category such that dom F = U. 6. Construct the coslice category C/C o a category C under an object C rom the slice category C/C and the dual category operation op. 7. Let 2 = {a, b} be any set with exactly 2 elements a and b. Deine a unctor F : Sets/2 Sets Sets with F ( : X 2) = ( 1 (a), 1 (b)). Is this an isomorphism o categories? What about the analogous situation with a one element set 1 = {a} instead o 2? 8. Any category C determines a preorder P (C) by deining a binary relation on the objects by: A B i and only i there is an arrow A B Show that P determines a unctor rom categories to preorders, by deining its eect on unctors between categories and checking the required conditions. Show that P is a (one-sided) inverse to the evident inclusion unctor o preorders into categories. 9. Describe the ree categories on the ollowing graphs by determining their objects, arrows, and composition operations. (a) a e b (b) e a b (c) a e b g c

27 EXERCISES 27 (d) e a b d h g 10. How many ree categories on graphs are there which have exactly six arrows? Draw the graphs that generate these categories. 11. Show that the ree monoid unctor exists, in two dierent ways: c M : Sets Mon (a) Assume the particular choice M(X) = X and deine its eect on a unction : A B to be M() : M(A) M(B) M()(a 1... a k ) = (a 1 )... (a k ), a 1,... a k A. (b) Assume only the UMP o the ree monoid and use it to determine M on unctions, showing the result to be a unctor. Relect on how these two approaches are related. 12. Veriy the UMP or ree categories on graphs, deined as above with arrows being sequences o edges. Speciically, let C(G) be the ree category on the graph G, so deined, and i : G U(C(G)) the graph homomorphism taking vertices and edges to themselves, regarded as objects and arrows in C(G). Show that or any category D and graph homomorphism : G U(D), there is a unique unctor with h : C(G) D U( h) i = h, where U : Cat Graph is the underlying graph unctor. 13. Use the Cayley representation to show that every small category is isomorphic to a concrete one, i.e. one in which the objects are sets and the arrows are unctions between them. 14. The notion o a category can also be deined with just one sort (arrows) rather than two (arrows and objects); the domains and codomains are taken to be certain arrows that act as units under composition, which is

28 28 CATEGORIES partially deined. Read about this deinition in section I.1 o Mac Lane s Categories or the Working Mathematician, and do the exercise mentioned there, showing that it is equivalent to the usual deinition.