Category Theory 2. Eugenia Cheng Department of Mathematics, University of Sheffield Draft: 31st October 2007

Transcription

1 Cateory Theory 2 Euenia Chen Department o Mathematics, niversity o Sheield echen@sheieldacuk Drat: 31st October 2007 Contents 2 Some basic universal properties 1 21 Isomorphisms 2 22 Terminal objects 3 23 Initial objects 5 24 Products 6 25 Coproducts Pullbacks Pushouts Equalisers Coequalisers Exercises 21 2 Some basic universal properties Many important concepts in Cateory Theory are determined or deined by a universal property In act, one o the eneral principles o Cateory Theory is that universal properties are somehow an ideal way o describin siniicatn structure The idea o a universal property is to ormalise our intuition about somethin bein canonical the best, the biest or smallest, the most extreme, eneric, ree Examples o thins with universal properties include: cartesian products disjoint unions quotients equivalence classes ree roups, ree rins, etc completions

2 21 Isomorphisms 2 closures abelianisation tensor products and direct sums o vector spaces direct and inverse limits simplicial sets eometric realisation nerves sinular complex sheaiication One very important idea is that when we deine somethin by a universal property it isn t exactly uniquely determined it is unique up to uniqu isomorphism This means that any other object havin this particular universal property will be isomorphic to the irst one, via a canonical isomorphism Here canonical will have a very precise meanin This touches on another eneral principle o Cateory Theory that our preerred notion o sameness is canonically isomorphic We ll see many examples o this; irst o all we d better deine isomorphism 21 Isomorphisms You miht have already uessed what an isomorphism is Deinition 21 morphism : X Y C is an isomorphism i and only i there exists a morphism : Y X such that = 1 X and = 1 y In this case is an inverse or s you miht expect, inverses are unique i a morphism has an inverse at all, then it has only one The proo o this is just the same as the proo that inverses are unique in a roup we suppose we have two inverses, use the act that one o the maps is an inverse, and then the act that the other is an inverse, and conclude that they must be the same Proposition 22 I 1 and 2 are both inverses or : X Y then 1 = 2 Proo 1 = 1 1 Y by the deinition o identities = 1 ( 2 ) since 2 is an inverse or = ( 1 ) 2 by associativity = 1 X since 1 is an inverse or = 2 by the deinition o identities

3 22 Terminal objects 3 Proposition 23 i) Identity maps are always isomorphisms ii) The composite o two isomorphisms is an isomorphism Proo Exercise or the reader 22 Terminal objects We now come to our irst universal property Deinition 24 terminal object in C is an object T such that or any object X C there is a unique morphism X T The part o that sentence ater the words such that is the universal property that deines a terminal object ***pic? Example: terminal objects in Set In the cateory o sets and unctions, any 1-element set is terminal there is a unique unction rom any other set because everythin simply has to be mapped to the sinle element For this reason we oten call terminal objects 1 in any cateory Note that the set {0} is terminal, but so are {7}, {64835}, and {x} ll these sets are isomorphic in a dull and straihtorward way Indeed, in any cateory, all the terminal objects are uniquely isomorphic The proo is similar to the proo that inverses are unique we assume we have two, and then we use the universal property o each in turn to show that they are in act isomorphic Proposition 25 Suppose T and T are both terminal in C Then there exists a unique isomorphism : T T Proo We proceed in steps: 1 Since T is terminal, there is a unique morphism : T T 2 Since T is terminal, there is a unique morphism : T T 3 Since T is terminal, there is a unique morphism T T ie 1 T 4 The composite is a morphism T T, so by (3) we must have = 1 T 5 Similarly, we must have = 1 T Hence is the unique isomorphism as required

4 22 Terminal objects 4 More examples It is instructive to think about terminal objects in other cateories and see whether or not they resemble 1-element sets at all: In Gp, the trivial roup is terminal Its only element is its unit, so it is indeed a 1-element set rearded as a roup In ect, the zero vector space is terminal; this is rather similar to the case o roups In Top, the point is terminal This is a 1-element set rearded as a space In a poset (rearded as a cateory as in ***re), a terminal object is an element t such that x t or all x in the poset so t is the maximum (i there is one) The cateory o ields and ield homomorphisms does not have a terminal object We can t have the 1-element set rearded as a ield, because a ield must have at least two elements 0 and 1 (we usually impose an axiom to ensure that 0 1) O course, this isn t really a proo that there isn t a terminal object, it s just pointin out that there isn t an obvious one In act, there really isn t one ***why?? The last example showed us that not every cateory has a terminal object Here are some very small cateories with no terminal object: Non-example 1: Non-example 2: any discrete cateory Non-example 3: Non-example 4: i we enerate a cateory rom a non-identity morphism : x x, we cannot have a terminal object x

5 23 Initial objects 5 23 Initial objects Deinition 26 n initial object in C is an object I such that or any object X C there is a unique morphism I X You may have noticed that there s somethin rather similar about the deinitions o initial and terminal object, and you re riht We et one rom the ohter by turnin around the arrow in the deinition This is an important principle in Cateory Theory: the principle o duality We ll discuss it ormally later ***re but here s the idea The principle o duality The idea is that we could consider all our arrows to be oin in the opposite direction It s quite obvious when you think about posets we declared there to be a morphism x y whenever x y but that was slihtly arbitrary We could just as easily have said: whenever x y It s the same with any cateory we could have declared all the morphisms x y to be morphisms y x instead Now, iven any deinition usin morphisms, we can simply turn all the arrows around and et what is called the dual notion Then any result we prove or the oriinal notion is automatically proved in dual orm as well because we can turn all the arrows around in the proo as well! There is a completely precise, ormal way o sayin this which we ll come to a bit later We quite oten (but not always) indicate that we ve taken a dual by stickin the preix co onto the name o the thin in question Limits become colimits, monads become comonads, alebras become coalebras This does result in some slihtly strane words thouh, such as cocone, cooperad, corin (how is that pronounced?), counit, countable 1 Examples o initial objects Here are some examples In Set, the empty set is initial there is a unique unction rom the empty set to any other set 2 We oten call initial objects 0, no matter what cateory they re in In Gp, the trivial roup is initial and recall that it s also terminal This is a special (and useul) sort o situation, and is very dierent rom the case o sets, where the initial and terminal objects are very dierent 1 This is Richard Garner s joke, and you know you re ettin into the spirit o Cateory Theory i you ind it unny 2 I you have trouble with this idea, remember that a unction : X Y is deined by ivin or every element x X an element (x) Y I X is empty, the or every element in X bit is vacuously satisied It s just like i someone ave you a box and said told you to run a mile or every elephant you ind in the box This task is rather easy i there are in act no elephants in the box

6 24 Products 6 Other cateories in which initial and terminal objects coincide include Top (based spaces) and Set (pointed sets) Remember that the empty set is not an object o Set ; the smallest possible pointed set is { }, the set which has just one element, which is its basepoint This has a unique morphism to any other pointed set, because the basepoint simply has to be mapped to the basepoint o the other pointed set, and there s nothin else to deine In Field there is no initial object ***proo? In eneral initial objects can be thouht o as the smallest possible thin with all the structure in question; this, like so many thins, will be made precise later In act, the act that Field has no initial object ives the result that we can t orm a ree ield on a set This is somethin very siniicant that comes o considerin initial objects 24 Products Our next universal property is the notion o product Deinition 27 product o objects and in C is an object equipped with morphisms as shown below such that iven any diaram p q there exists a unique morphism h makin the ollowin diaram commute 3 : p!h q 3 The exclamation mark in this diaram is notation that means unique, so!h is to be read there exists a unique h

7 24 Products 7 Here is or niversal s beore, the part o the deinition ater the words such that is ivin the universal property in question We can think o a product o and as bein a universal diaram o the orm It is universal amon all such diarams, the best possible such diaram We also say that every diaram o this orm actors throuh it uniquely The morphism h is the actor in question, and the last diaram exhibits the actorisation the bendy part o the diaram is a product (or rather, composite) o the universal diaram and the actor h We also say that the morphism h is induced by the universal property Note that althouh it s the object that is usually reerred to as a product, it s really important that comes equipped with the maps p and q, as we ll soon see Example: products in Set This deinition miht seem a bit obscure to you at the moment, so let s work out what products are in our prototype cateory Set You miht be rather hopin that products in Set will turn out to be products, that is, cartesian products nd you re riht! Phew ut can we actually show that this is true? To do that, we have to take two sets and and show that the cartesian product has the required universal property Recall that 4 Do we have unctions = { (a, b) a, b } p q that are somehow obvious enouh to ive a universal property? The answer is yes: we have the projection maps that send the pair (a, b) to a on the one hand, and to b on the other, as shown below: 4 The vertical line is to be read such that, so this notation is to be read the set o all pairs (a, b) such that a is an element o and b is an element o

8 24 Products 8 a (a, b) p 1 p 2 b We call these p 1 or project onto the irst component and p 2 or project onto the second component Now, does this have the riht universal property? Given unctions can we deine a unction h : makin the correct diaram commute? Well, let s consider an element v, and suppose we have a unction h with h(v) = (a, b) Now in order or the diaram to commute, we must have h p 1 p 2 (v) = (p 1 h)(v) = p 1 (h(v)) by the deinition o composition o unctions = p 1 (a, b) by the notation we introduced above = a by the deinition o p 1 and similarly (v) = (p 2 h)(v) = p 2 (h(v)) = p 2 (a, b) = b So we deine a unction h : by h(v) = ((v), (v)) and it is indeed the unique unction makin the required diaram commute

9 24 Products 9 Incidentally, inspired by this example, we write this product as and the induced actor as (, ), and we enerally reer to the maps p and q as projections niqueness o products re products unique? You can probably uess that the answer is: they are unique up to unique isomorphism Let s think about this in Set or a second I we were eelin a bit perverse, we miht decide that the product o and should be Would this work? The diaram exhibitin this product would then be: and the actorisation would be p 2 p 1 (,) p 2 p 1 O course, and are isomorphic in a very very canonical way There are even more perverse ways o ormin speciic products in Set For example, iven the most obvious product is = {0, 1} = {0, 1} but the set = {(0, 0), (0, 1), (1, 0), (1, 1)} {1, 2, 3, 4} can also be iven the structure o a (cateorical) product o and ll we have to do is deine the projects p and q For example, we could deine p by

10 24 Products 10 and q by I you think about why this ives a product, you will (hopeully) see that all we ve really done here is imaine that the elements 1,2,3,4 are actually the elements (0, 0), (1, 0), (0, 1), (1, 1) in disuise, and we ve deined the projection maps accordinly In act, deinin maps corresponds precisely to tellin the secret o which element is disuised as which It should now be clear that any 4-element set could be iven the structure o in this case Moreover, once the projection maps are deined, these products all become uniquely isomorphic In eneral, 4-element sets are all isomorphic but in many possible ways; however, here we are only interested in isomorphisms that respect the product structure, that is, that maps (0, 0)-in-disuise in one product, to (0, 0)-in-disuise in the other, and likewise or all the other elements eore this analoy oes much too ar (i it hasn t already), we d better make that precise Proposition 28 Given products and p p there exists a unique isomorphism h : commute q q makin the ollowin diaram p p h q q Note that there may be many isomorphisms, but only one makin the diaram commute

11 24 Products 11 Proo The universal property o induces a unique morphism h makin the above diaram commute, so it only remains to show that h is an isomorphism The universal property o induces a unique morphism h : makin the ollowin diaram commute: p p h and we can now show that h is inverse to h The ollowin diaram commutes: p h h q q p q p q q which means in particular that the morphism h h ollowin diaram commute : makes the p p q but 1 : also makes this diaram commute y the universal property o there is a unique morphism makin this diaram commute, and so we must have h h = 1 Similarly, we must have h h = 1, so h is indeed inverse to h, and h is the unique isomorphism required You may notice that this proo ollowed the exact same structure as our proo o the analoous uniqueness result or terminal objects This is no coincidence! Terminal objects and products are both examples o limits, and there is a eneral proo that limits are unique up to unique isomorphism in this very way; we have now seen two speciic examples o this It is useul to warm up to the eneral q

12 24 Products 12 case usin these small examples, because when we inally see the ormal eneral deinition o limit it will help to have some intuition about what it s supposed to be sayin Here s another useul result about products, that ormalises somethin we know already about cartesian products o sets Given sets,,, and unctions : and :, we et a unction deined simply by : (a, b) ((a), (b)) We will now see that althouh we have deined this on elements 5, it in act ollows just rom the cateorical deinition o product Proposition 29 Given products and and morphisms : and :, there is a unique morphism makin the ollowin diaram commute: p q p q Proo The required morphism is immediately induced rom the universal property o Note that we didn t use an awul lot o the product structure o ; the point is that the induced actorisation only deserves to be called i it s oin rom a product to a product in this way You may wish to check that in Set the morphism induced in this way really is the one we irst thouht o above More examples o products In Gp, products are direct products, that is, we take the cartesian product o the underlyin sets and induce the obvious pointwise roup operation In case it s not obvious to you: iven roups and we deine a roup operation on the cartesian product o their underlyin sets by (a, b) (a, b ) = (a a, b b ) 5 To turn a act about sets and unctions into a act about objects an morphisms in any cateory, we have to express the act without ever reerrin to elements o sets

13 25 Coproducts 13 O course, anythin isomorphic to this will also do In Top, products are product spaces; aain the underlyin set is the cartesian product o the underlyin sets nd aain, anythin isomorphic to this will also do In ect, products are direct products nd yet aain, the underlyin set o is the cartesian product o the underlyin sets, anythin isomorphic to this will also do side: the property that the underlyin set o the product is the product o the underlyin sets is a useul one, and will later be honoured with a eneralisation, and a name In an ordered set (rearded as a cateory), the product o x and y is the minimum o x and y It is actually unique To see that this is true, suppose x y, so the claim is that x y = x We must check it has the required universal property, which translates into the ordered set as: iven v such that v x and v y, then v x This is clearly true! In a poset (rearded as a cateory), the product o x and y is the reatest lower bound, or meet, o x and y (I you don t know what a meet is, you now do: it s the cateorical product o x and y!) Here s an oh-so-witty example Let C be the cateory whose objects are the natural numbers For morphisms: there is precisely one morphism n m i n divides m, and none otherwise This is in act a poset nyway, in this example we et to write silly thins like 6 8 = 2, and = 12, and everyone lauhs Seriously thouh the product o n and m in this cateory is their hihest common actor, which is really quite satisyin Cateories with products In eneral we say that a cateory has (all) binary products i any two objects have a product There is a more eneral notion o products o more than two objects which we ll come to later Thus, Set, Gp, Top and ect have binary products n ordered set rearded as a cateory has binary products poset rearded as a cateory miht not have binary products; it depends on the poset 25 Coproducts Coproducts are the dual o products, as the co preix indicates In a way there s nothin more to say i you understand the principle o duality you ll be able to write down the deinition immediately However, we will now spell it out, or the record 6 6 nd because it s pretty easy to just turn all the arrows around in the LTEXcode! ***xypic

14 25 Coproducts 14 Deinition 210 coproduct o objects and in C is an object equipped with morphisms as shown below p q such that iven any diaram there exists a unique morphism h makin the ollowin diaram commute: Reassurance about directions!h p q You miht be wonderin how you re ever oin to remember which direction h oes in products as opposed to coproducts Now, there are a lot o times in Cateory Theory when it really is hard to remember which way round thins o 7 but this really isn t one o them You just have to remember that the universal property o your universal thiny says that any other thiny actors throuh it Then as lon as you understand what a actor is, you can t et your directions wron Examples o coproducts In Set, coproducts are disjoint unions The maps p and q are the insertions p q 7 Distributive laws are my current avourite example o this

15 26 Pullbacks 15 and we oten use this terminoloy or the analoous morphisms or coproducts in other cateories as well; we also sometimes say coprojections since they are the dual o projections In Top, coproducts are also disjoint unions In Gp, coproducts are ree products, sometimes written You basically take all the elements o and and then enerate a roup reely while preservin the roup operations o and (so the ree part is when you re multiplyin an element o a with an element o b) Deinin this riorously is quite hard, and indeed it is oten the case in alebraic structures that the products are obvious and the coproducts are much less obvious 8 However, the idea is clear take the disjoint union o the underlyin sets and then orm the smallest possible structure o the kind you want For spaces the disjoint union was already a structure o the kind we wanted For roups, it wasn t In ect the situation is rather similar to the case or roups; ater all vector spaces are special kinds o roups nyway here, coproducts are direct sums This conlation o sum/product/coproduct terminoloy is not very helpul, I m sure you ll aree In Set, we make coproducts by basically takin the disjoint union o two sets, but we then identiy their basepoints to make the new basepoint This is also the case in Top but it ets the ancy name wede Hence in topoloy you hear people talkin about the wede o two circles or a wede o spheres 26 Pullbacks Here s a slihtly more complicated sort o universal property Deinition 211 pullback square is a commutative square such that iven any commutative square C s t C 8 This actually has a precise ormulation in terms o creation o limits

16 26 Pullbacks 16 there is a unique actorisation h makin the ollowin diaram commute s!h t C In this case we also say that is a pullback o over (or alon), or equivalently that is a pullback o over (or alon) You will notice that we ve called the top let corner or niversal This is because it s at that corner that all the actorin happens, and it s the object at that corner that is the new one we miht well start with the data and produce the pullback square rom it So we sometimes even say is a pullback i we re eelin particularly sloppy To indicate a pullback square, we oten put a little sin in the corner like this: Pullbacks are o course unique up to unique isomorphism; that is, iven any two pullbacks o the same morphisms, there exists a unique isomorphisms between them makin the relevant diaram commute Pullbacks in Set We can construct a pullback in Set by takin to be the ollowin subset o : C C { (a, b) (a) = (b) } and then and are the restricted projections onto the irst and second component respectively For this reason we sometimes write a pullback as C

17 26 Pullbacks 17 which is ine i and are obvious and canonical, but severely insuicient i they re not Pullbacks are also called also called a ibred product or cartesian square Products as pullbacks I C has a terminal object, then a product can be expressed as a pullback: 1 We check that this has the correct universal property Now to ive a commutative square 1 we just have to ive a pair o morphisms s t because the other two morphisms are uniquely determined by the act that 1 is terminal; urthermore because 1 is terminal the square must commute (since there is a unique morphisms 1) Now the universal property o the product induces a unique morphism h makin the ollowin diaram commute s h p q ut this diaram commutes i and only i the ollowin diaram commutes t

18 27 Pushouts 18 s h p t q 1 and so the universal property is the correct one 27 Pushouts Pushouts are dual to pullbacks ain, we ll spell it out althouh it is usual in this case not just to reverse the arrows but also to turn the diaram around on the pae Deinition 212 pullback square is a commutative square!! C such that iven any commutative square C t there is a unique actorisation h makin the ollowin diaram commute s C s t!h In this case we also say that is a pushout o alon, or equivalently that is a pushout o alon

19 28 Equalisers 19 nions as pushouts We know that pullbacks are somehow related to pushouts, so dually we expect pushouts to be somehow related to coproducts In Set this can maniest itsel in the orm o unions as opposed to disjoint unions ie coproducts The ollowin is a pushout square in Set, where the morphisms are the obvious inclusions: There are more eneral pushouts than this as well ***say? 28 Equalisers Deinition 213 Given morphisms an equaliser or them is a ork that is morphism e as below with e = e e such that iven any ork s there is a unique actorisation Example: equalisers in Set e s In Set we et the subset o deined by: h { a (a) = (a) }; the map e is then just the inclusion o this subset into Note that we would still have a ork i we threw in some more elements to this subset, as lon as e mapped them all to elements in the oriinal subset o However, this would

20 29 Coequalisers 20 no loner be a universal ork we would have existence o actorisations, but not uniqueness On the other hand i we removed some elements o this subset, we miht not have existence any more, but i a actorisation did exist it would be unique 29 Coequalisers Coequalisers are the dual o equalisers Deinition 214 Given morphisms a coequaliser or them is a ork c that is, c = c, such that iven any ork s there exists a unique actorisation e s h Examples Coequalisers in Set are quotients With the above notation we et / where we are quotientin out by the smallest equivalence relation enerated by (a) (a) or all a The sloan coequalisers are quotients isn t ar wron Coequalisers ive quotient spaces in Top and quotient roups in Gp 9 In act, it s usin coequalisers that we eneralise the act that every roup is a quotient o a ree roup Coequalisers seem to crop up more than equalisers, even thouh they re the ones with the co in ront o them 9 Technically we can only quotient out by a normal subroup, so or eneral roups we ll end up quotientin out by a normal subroup enerated by the non-normal subroup in question

21 210 Exercises Exercises 1 Show that or any object X in a cateory C, 1 X is an isomorphism 2 Show that i : and : C are isomorphisms then is an isomorphism 3 What is an isomorphism in Rel, the cateory o sets and relations? 4 Let C be the cateory as in the oh-so-witty example That is, objects are natural numbers and there is precisely one morphism n m i n divides m, and none otherwise What are coproducts in this cateory? 5 Write down the relevant uniqueness property or pushouts, and prove that it holds 6 Guess the deinition o 3-old product, that is, product o three objects,, C Show that in Set the cartesian product C is a 3-old product, as are ( ) C and ( C) 7 Show how to express a coproduct as a pushout Hint: just dualise the whole arument or expressin a product as a pullback 8 In Set there is a useul diaonal map : a (a, a) Show how to induce this usin the universal property o the product, thus ivin us a notion o diaonal in any cateory with products What is the dual notion? 9 Let, : Show that the ollowin pushout provides an equaliser or and : (,) Here : is the diaonal unction deined above, that maps b to (b, b) 10 i) What are products in Set? ii) What are products in Top?

22 210 Exercises Suppose that is a commutative diaram i) Show that i both small squares are pullbacks then so is the lare rectanle ii) Show that i the lare rectanle and the riht hand square are pullbacks, then so is the let hand square iii) Deduce rom the above (or prove directly) that the pullback o a pullback square is a pullback square, statin clearly what you take this to mean