UMS 7/2/14. Nawaz John Sultani. July 12, Abstract

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1 UMS 7/2/14 Nawaz John Sultani July 12, 2014 Notes or July, UMS lecture Abstract 1 Quick Review o Universals Deinition 1.1. I S : D C is a unctor and c an object o C, a universal arrow rom c to S is a pair r, u consistin o an object r o D and an arrow u : c Sr o C, such that to every pair d, with d an object o D and : c Sd an arrow o C, there is a unique arrow : r d o D with S u =. In other words, every arrow to S actors uniquely throuh the universal arrow u, as in the commutative diaram: c Sr r u S c Sd d This is the basic notion o a universal arrow rom an object to a unctor. The entire chapter will ocus on dierent aspects o these universals. I ind it best to think o these universals as a deinin trait o the object and a correspondin unctor that you can rely on, especially at this level o abstract nonsense. Maclane ives some ood examples o universals on pae 56, the bases o vector space one in particular bein one that Kenny did last lecture. There is also a notion o a universal element, as ollows: Deinition 1.2. I D is a cateory and H : D Set a unctor, a universal element o the unctor H is a pair r, e consistin o an object r D and an element e Hr such that or every pair d, x with x Hd, there is a unique arrow : r d o D with (H)e = x. However, the notion o the universal element is just a special case o a universal arrow, as you can see discussed in the middle o pae 58. Later, we will also see that universal elements and arrows subsume each other, so unless otherwise stated, we will work with universal arrows. Finally, we arrive at the dual concept o the universal arrow deined above. Deinition 1.3. A universal arrow rom S to c is a pair r, v consistin o an object r D and an arrow v : Sr c with codomain c such that to every pair d, with : Sd c there is a unique : d r with = v S, as in the commutative diaram: d Sd c S r Sr v c Notice that the dierence between the examples is that one is to a unctor, while the other is rom a unctor, a distinction that will allow us to deine dierences between limits and colimits later. There is a particularly simple example o universals on the bottom o pae 58 which introduces the diaonal unctor, a unctor that will be important later. The kernel o a homomorphism in roups, or example, can also be seen as a universal or a suitable contravariant unctor. 1

2 2 The Yoneda Lemma This section will introduce some o the irst non-trivial propositions, as well as some useul properties o universals. First, we discuss the notion o universality in terms o hom-sets, an oten easier way o lookin at thins. Proposition 2.1. For a unctor S : D C a pair r, u : c Sr is universal rom c to S i and only i the unction sendin each : r d into S u : c Sd is a bijection o him sets D(r, d) = C(c, Sd). (1) This bisection is natural in d. conversely, iven r and c, any natural isomorphism as above is determined in this way by a unique arrow u : c Sr such that r, u is universal rom c to S. The proo o the proposition can be seen on pae 59, but or the dedicated student, we leave it as an exercise in deinitions and diaram chasin. But rom this proposition, we do et a new concept. I C and D have small hom-sets, then (1) tells us that the unctor C(c, S ) to Set is naturally isomorphic to the hom-unctor D(r, ) (It s important that they are small, and small in the cateorical sense, or else it would not be a unctor to the cateory Set. Ask yoursel, why?) These isomorphisms are called representations: Deinition 2.2. Let D have small him-sets. A representation o a unctor K : D Set is a pair r, ψ, with r an object o D and ψ : D(r, ) = K a natural isomorphism. The object r is called the representin object. The unctor K is said to be representable when such a representation exists. Thus, representable unctors are related up to isomorphism to the unctor D(r, ). related to universal arrows? Well... How are these Proposition 2.3. Let denote any one point set and let D have small him-sets. i r, u : Kr is a universal arrow rom to K : D Set, then the unction ψ which or each object d o D sends the arrow : r d to K( )(u ) Kd is a representation o K. Every representation o K is obtained in this way rom exactly one such universal arrow. Proo. For any set X, a unction : X rom the one-point set to X is determined by the element ( ) X. Thus, this creates a bijection Set(, K ) X iven by ( ), natural in X Set. I you compose with K, you et a natural isomorphism Set(, K ) K. But as seen in the deinition o ψ, we et that Set(, K ) = K = D(r, ) So it suices to ind a natural isomorphism Set(, K ) = D(r, ). But this ollows rom proposition 2.1. Thereore, we have that the notions universal element, universal arrow, and representable unctor subsumes the other two. However, or those dedicated students who read the proo o proposition 2.1, there was a act used, namely that each natural transormation o φ : D(r, ) K is completely determined but eh imaed under ψ r o the identity 1 r : r r. What ollows is potentially the reatest lemma o all time. Lemma 2.4. (Yoneda) I K : D Set is a unctor rom D and r an object in D (For D a cateory with small hom-sets), there is a bijection y : Nat(D(r, ), K) = Kr which sends each natural transormation α : D(r, ) K to α r1 r, the imae o the identity r r. The proo is evident by observin the ollowin commutative diaram: D(r, r) α r K(r) r =D(r,) K() D(r, d) α d K(d) d 2

3 Corollary 2.5. For objects r, s D, each natural transormation D(r, ) D(s, ) hast the orm D(h, ) or a unique arrow h : s r. To Yoneda map, y, is a natural map, which is can be observed i we view K as an object in Set D, with both the domain and codomain o y as unctors on the pair K, r, which is an object in the cateory Set D D. The codomain o y would b eht evaluation unctor E, which sends K, r to Kr, while the domain is the unctor N which sends K, r to the set Nat(D(r, ), K) o all natural transormations. Now we can add to the Yoneda Lemma. Lemma 2.6. The bijection y is a natural isomorphism y : N E between the unctors E, N : Set D D Set. The object unction r D(r, ) and the arrow unction ( : s r) d(, ) : D(r, ) D(s, ) or an arrow o D toether deine a ull and aithul unctor Y : D op Set D called the Yoneda unctor. It s dual is another unctor that is also aithul, Y : D Set Dop. Make sure D has small hom-sets. I not, you can replace every Set in this section with Ens and the results will apparently still hold. 3 Coproducts and Colimits This is where stu ets unky. Let s bein with coproducts. Deinition 3.1. For any Cateory C, the diaonal unctor : C C C is deined by (c) = c, c, () =,, where c is an object and is a unction. A universal arrow rom an object a, b o C C to the unctor is called a coproduct diaram. Basically, you need an object c C and two arrows i : a c, j : b c. O course, we must discuss the universal property: or any pair o arrows : a d, : b d there is a unique h : c d with = h i, = h j. When the diaram exists, we have c unique up to isomorphism, and we write c = a b or c = a+b. c in this case is called the coproduct object. The universality can be seen in another diaram as ollows: a a b i h j d b Let s do an example. consider the cateory Set. Then take a b to be the disjoint union o the sets a and b, while i and j are the inclusions maps o each respectively. Now a unction h is determined by the composites hi and hj. This ulills the diaram above. Disjoint unions are not necessarily unique, but they are up to a bijection, which beits a universal. Some more examples ollow rom these coproduct deinitions in certain cateories Set : thedisjointunionosets Top : disjointunionospaces Group : F reeproduct Ab, R mod : directsuma B and many more which I trust the reader to ind on their own. Ininite coproducts work as you would expect. Replace C C = C 2 with C X where X is any set. Reardin the set S as a discrete cateory, the unctor cateory C X has as its objects the X-indexed amily a = {a x x X}, and the diaonal unctor sends c to the constant amily all c x = c. The rest works as you would imaine, except there are multiple (ininite) injections rom each C in the index. I you consider Set, then the ininite coproduct would be the X-old disjoint union... Copowers is just havin all your a x equal i.e. a x = b x. Now let s talk about more, o what you will soon learn, are special colimits... 3

4 Deinition 3.2. Suppose that C has a null object z, such that or any two objects b, c C there is a zero arrow 0 : b z c. The cockerel o : a b is then an arrow u : b e such that or (i) u = 0 : a e (ii) i h : b c has h = 0, then h = h u or a unique arrow h : e c. The picture is as ollows: a u b e d h In Ab, the cokernel o : A B is the projection B B/A. But what i we have no null object in our cateory. No problem. ABSTRACTION! Deinition 3.3. Given in C a pair, : a b o arrows with the same domain a and the same codomain b, a coequalizer o, is an arrow u : b e such that (i) u = u (ii) i h : b c has h = h, then h = h u or a unique arrow h : e c. a, u b e d h You can also think o coequalizers as universal arrows! Denote as the cateory with two objects and two non-identity arrows rom the irst object to the second. (Because o the diiculty o LateX, I will denotes this cateory as Q; the notation in Maclane is two dots with double riht arrows between them.) Form the unctor cateory C whose objects are unctors rom Q to C, i.e. a pair, : a b o parallel arrows. An arrow between pairs in this cateory is a natural transormation h, k, with h : a a and k : b b in C. This makes the square diaram below commute: or both the -square and -square. One can also deine a diaonal unctor : C C by (c) = 1 c, 1 c, (r) = r, r, where c and r are objects and arrows o C respectively. It ollows that a coequalizer o the pair, in C is the universal arrow rom,, to the unctor. Now somethin new: Deinition 3.4. Given in C a pair : a b, : a c o arrows with a common domain a, a pushout o, is a commutative square as shown on 66 and let a b a b such that add diaram v k c r c s to every other commutative square similar to the riht one, there is a t : r s with tu = h, tv = k. Basically, its the universal way to ill out a commutative square on the sides,. To view it as a universal, one must ask use the diaonal unctor which takes an object c C to 1 c, 1 c, i.e. = 1 c, 1 c. I would advise readin pae 66 or a more in-depth discussion on how this is a universal. Also, there is a section on the cokernel pair which I will not touch on reatly in these notes. The section is short so or the dilient student, I recommend readin it. For others, here is the basic deinition: Deinition 3.5. Given an arrow : a b in C, the pushout o with is the cokernel pair o. This is illustrated in the ollowin diaram: a u,v b r h,k s t u u = v, h = k I you haven t noticed, all these cases deal with particular unctor cateories, and can be eneralized as such. Let C and J be cateories(j or the index cateory). The diaonal unctor : C C J sends each object c to the constant unctor c- the unctor which has the value c at each object i J and the value 1 c at each arrow o J. For an arrow : c c o C, is the natural transormation h 4

5 : c c, which has the value o at each object i J. Each unctor F : J C is an object o C J by deinition. Now a universal arrow r, u rom F to, with r an object o C and u : F c a natural transormation, is called a colimit diaram or the unctor F. Deinition 3.6. A universal arrow r, u rom F to, with r an object o C and u : F c a natural transormation, is called a colimit or the unctor F. We denote r = ColimF, and the natural transormation u to be universal amon natural transormations τ : F c. τ consists o arrows τ i : F i c, c C or each i J and satisies the property τ j F u = τ i or each arrow u : i j: F 1 τ 1 F F 2 τ 2 F F 3 c c c τ 3 (diaram keeps oin or all i J, just wrote a portion o it) This... is a crazy notion. But it s best not to question it and uddle around: take it as it is and et amiliar with it. This will be your bread and butter or the next ew weeks. One can also identiy the diaram o a colimit as a cone, by takin the vertex o the cone to be the common point c as in the diaram above. This results in the cone-like diaram: F i F u F j τ i τ j F τ k c A cone with the vertex ColimF and the cone µ : F ColimF is called the universal cone or limitin cone. For any cone τ : F c, one can ind a unique arrow t : ColimF c such that τ i = t µ i. Note that this c is dierent ront the colimit o F, which is akin to the r deined above. I would advise oin back throuh this section to make sure you have everythin down, especially with which cateory each object and arrow is, like how ColimF C. It will help clear thins up and make this lare abstraction more comortable. F k 4 Products and Limits As we deined a dual notion to a universal arrow, it comes to no surprise that we can also deine a dual notion to colimits and coproducts, namely limits and products. Deinition 4.1. Given cateories C, J and the diaonal unctor : C C J, a limit or a unctor F : J C is a universal arrow r, v rom tof. r = limf is an object in C, called the limit object or the unctor F, and v : r F is a natural transormation universal amon natural transormations τ : c F. τ consists o one arrow τ i : c F i or each i J such that or every arrow u : i j, o J, one has τ j = F u τ i. When drawn, this is the cone rom the vertex LimF to the base F. That it is a universal t can be seen in the diaram on pae 68, and below: c LimF τ j τ i v j v i 1 F u F i F j As beore, limitin cones are unique up to isomorphism. The ollowin diaram sums up how Colimits τ σ and Limits interact: c F c LimF v F µ ColimF Additionally, when the limits exist, we have natural isomorphisms C(c, Lim) = Nat( c, F ) = Cone(c, F ) Cone(F, c) = Nat(F, c) = C(ColimF, c) Now we may talk about the duals o those special colimits. Deinition 4.2. I J is the discrete cateory {1, 2}, a unctor F : 1, 2 C is a pair o objects a, b o C. The limit object object is called a product o aandb, denoted by a b or aπb. 5

6 From each a b, there are projection arrows, denoted p, q such that p : a b a and q : a b b. This constitutes a cone rom the vertex a b (its a very simple cone so don t strain yoursel too much), and also ives us a bijection o sets C(c, a b) = C(c, a) C(c, b) natural in c, sendin each h : c a b to the pair ph, qh. Conversely, when iven arrows : c a and : c b, there is a unique h : c a b with ph =, qh =. This should all be very amiliar, and I don t think it is diicult to construct an example in Set with your imaination. I will skip Ininite Products and Powers because I don t think they are very interestin. I you want a quick overview o them, return to the correspondin Colimit subsections and lip all the arrows. These two behave exactly as you would expect as duals to the Colimit subsections. Deinition 4.3. I J =, a unctor F : C is a pair, : b a o parallel arrows o C. I it exists, e the limit object d o F is called an equalizer with diaram as ollows: d b, a (e = e) With e : d b the limit arrow, called the equalizer o and, or any h : c b with h = h, there is a unique h : c d with eh = h. Consider the cateory Set. Then d is the set {x b x = x} with e : d b the injection map. Even better, the equalizer always exists in Set, which is not necessarily true o all cateories. Finally, let s talk about the best special case o a Limit: Deinition 4.4. I J = ( )(the cateory I denoted as Q earlier), a unctor F : ( ) C is a pair o arrows : b a, : a d o C with common codomain a. A commutative square is then o the k orm shown on the riht: c d b a d q d h b a b a Such that or any square with vertex c similar to the square on the let, there is a unique r : c b a d with k = qr, h = pr. This square ormed by the universal cone is called the pullback square and the vertex b a d o the universal cone is called a pullback. Similar to pushouts, the pullback o a pair o equal arrows : b a b : is called the kernel pair o. it works similar to cokernel pairs, and i you want to know more, I would suest readin pae 71. Final notes include the limits on the empty cateory, i.e. J = 0. it shouldn t be much to convince you that the limit o the empty unctor to C is a terminal object o C. In addition to this, one can also talk about limits o diarams, but you essentially et the same inormation, so I would recommend stickin to the methods illustrated above. I, or some reason, you need this inormation thouh, the bottom o pae 71 contains what Maclane says on the subject matter. p 6