TOPOLOGY FROM THE CATEGORICAL VIEWPOINT

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1 TOOLOGY FROM THE CATEGORICAL VIEWOINT KYLE ORMSBY One o the primary insihts o twentieth century mathematics is that obects should not be studied in isolation. Rather, to understand obects we must also understand relationships between obects. Topoloies oer one notion o relation, oranizin the points o a set into neihborhoods. As an abstraction o eometry, topoloy is immensely successul, but it ails to capture a second, more structural notion o relation. Cateory theory oranizes obects by their transormations. Once these notions o obect and relation are abstracted, it becomes possible to compare and contrast dierent ields o mathematics. Theorems about wildly dierent mathematical obects are oten identical in their cateorical content. In this manner, cateory theory becomes a meta-mathematical tool or both identiyin and conecturin structural results. These notes aim to introduce cateory theory in parallel with James Munkres s Topoloy. We will use this lanuae to motivate deinitions and interpret theorems. Movin into the alebrotopoloical portion o the course, this lanuae will become even more important as we use the undamental roup unctor to compare the cateories o topoloical spaces and roups. 1. CATEGORIES We bein with a motivatin eample. Sets are a type o mathematical obect. Sets are related by unctions. Each unction has a domain (source) and codomain (taret). In standard notation, : A Ñ B denotes a unction with domain A and codomain B. O course, unctions admit composition: iven : B Ñ C and : A Ñ B we can orm : A Ñ C by assinin ppaqq to a A. This composition is associative: i : A Ñ B, : B Ñ C, and h : C Ñ D are unctions, then h p q ph q. Moreover, each set (even the empty set!) admits an identity unction 1 A : A Ñ A. This unction takes each a A to a and satisies the ollowin property: or each : A Ñ B and each : C Ñ A, 1 A and 1 A. In the ollowin deinition, we will see that cateories consist o obects, morphisms (with source and taret obects), composition, and identity morphisms satisyin associativity and identity properties. As such, sets and unctions orm our irst eample o a cateory. Deinition 1.1. A cateory C consists o a collection o obects Ob C and a collection o morphisms Mor C alon with assinments s, t : Mor C Ñ Ob C (called the source and taret maps). Let C p, yq Ď Mor C denote the collection o morphisms with source Ob C and taret y Ob C. Then or each, y, z Ob C, C is also equipped with a composition : C py, zq ˆ C p, yq Ñ C p, zq p, q ÞÑ. Additionally, or each Ob C there is an identity morphism 1 C p, q. This data must satisy the ollowin properties: (associativity) For obects, y, z, w and morphisms C p, yq, C py, zq, and h C pz, wq, we have h p q ph q. 1

2 (identity) For each, y Ob C, C p, yq, and C py, q, we have 1 and 1. It is useul to think about cateories diarammatically. These diarams use letters to represent obects, and labelled arrows to represent morphisms. So i, y Ob C and C p, yq, we may draw ÝÑ y in order to represent that is a morphism in with source and taret y. (Note that we have dropped C rom our notation here: usually it will be clear rom contet which cateory we are workin in.) We may also write : Ñ y to represent C p, yq. Now suppose that in addition to we also have y ÝÑ z, a morphism with source y and taret z. Composition tells us that we then et a new morphism ÝÝÑ z. We can put all o this inormation into a sinle commutative diaram y z. When we say that a diaram y h commutes, we are sayin precisely that h. The eometric presentation o the diaram is unimportant, and we could ust as easily draw commutative diarams z. y z or y h to communicate that h. We can use diarams to epress the aioms or a cateory. Associativity becomes h z y h w z h so we can interpret this aiom as sayin that we can paste toether commutative trianles to produce commutative quadrilaterals. The identity aiom becomes 1 and y. 1 2

3 Without realizin it, you have been workin with cateories or a lon time. Consider the ollowin eamples. Eample 1.2. We have already mentioned that sets and unctions orm a cateory. We denote this cateory Set. Similarly, there is a cateory FinSet with obects inite sets and morphisms unctions between inite sets. Eample 1.3. Let k be a ield and let Vect k have obects k-vector spaces and morphisms k-linear transormations. Since linear transormations compose (in the set-theoretic sense) to ive new linear transormations, Vect k is also a cateory. (It is obvious that the identity unction 1 V : V Ñ V is linear.) We can also consider the cateory FinVect k o inite-dimensional k-vector spaces and k-linear transormations. Eample 1.4. The empty cateory has no obects (i.e., Ob ) and no morphisms (Mor ). The source, taret, and composition unctions are all the empty unction Ñ and all properties are satisied vacuously! Eample 1.5. The trivial cateory has a sinleton set t u or its obects and a sinle morphism (necessarily the identity on ), 1 : Ñ. The only composition to deine is and this is enouh to check the associativity and identity properties as well. Eample 1.6. Not every cateory has special classes o unctions as morphisms. Consider Mat k whose obects are the natural number N t0, 1, 2,...u and whose morphisms Matpm, nq are n ˆ m matrices with entries in a ield k. Composition is iven by matri multiplication (check compatibility!) and the n ˆ n identity matri is 1 n. Since matri multiplication is associative, this orms a cateory. I you are suspicious that this cateory is eerily similar to FinVect k, worry not: it is! Ater studyin equivalences o cateories and skeleta, you will understand eactly how similar. The ollowin two eamples should seem completely obvious i you have already taken a course in abstract alebra. I you have not, there is no harm in skippin them. Eample 1.7. There are cateories Gp, FinGp, and AbGp o roups, inite roups, and abelian roups, respectively. In each case, morphisms are roup homomorphisms. Eample 1.8. There are cateories Rin and CommRin o rins and commutative rins, respectively. In both cases, morphisms are rin homomorphisms. There is also a cateory Field o ields and ield homomorphisms. Finally, we come to the cateory o primary interest in this course, the cateory o topoloical spaces and continuous unctions. Eample 1.9. The cateory Top has topoloical spaces as its obects and continuous unctions as its morphisms. In order to check this, we must see that the composition o continuous unctions is continuous and that identity unctions are continuous. The ormer condition is the content o Theorem 18.2(c) in Munkres. I X is a topoloical space and 1 X : X Ñ X is the identity unction, then or any U Ď X open, 1 1 X puq U is open in X, so 1 X is continuous. The associativity and identity aioms hold because they hold or unctions. Remark O course, there is also a cateory whose obects are topoloical spaces and morphisms are arbitrary unctions between underlyin sets. And while we are ree to deine such a cateory, it is not o particular interest. A cateory is a tool or studyin relationships between obects. I we choose the wron set o relationships (i.e., the wron morphisms), then we end up with an uninterestin or, worse yet, misleadin cateory. 3

4 2. ISOMORHISMS The morphisms in a cateory ive us a way to relate or compare obects. When are two obects the same" in liht o these comparisons? O course, this notion o sameness is dierent rom equality. We would like to know when two obects are indistinuishable via morphisms" rather than when they are literally equal. The ollowin deinition provides this notion. Deinition 2.1. A morphism C p, yq is an isomorphism i there eists a morphisms C py, q such that 1 and 1 y. We call a (two-sided) inverse to. Obects, y Ob C are isomorphic i there eists an isomorphism C p, yq. In this case we write y, and i C p, yq is an isomorphism we write : y. roposition 2.2. Isomorphism is an equivalence relation on Ob C. roo. We irst check that is releive. Indeed, 1 : or all Ob C because by the identity aiom. We now check that is symmetric. Suppose : y. Then there eists C py, q such that 1 and 1 y. This tells us that : y as well. Finally, we check that is transitive. Suppose : y and : y z. There eist 1 C py, q and 1 C pz, yq such that 1 1, 1 1 y, 1 1 y, and 1 1 z. Observe that p 1 1 q p q p 1 p 1 qq p 1 1 y q 1 1 and p q p 1 1 q p p 1 qq 1 p 1 y q z. Thus : z, as desired. In the ollowin sequence o propositions, we identiy which morphisms are isomorphisms in some o the cateories we introduced in 1. roposition 2.3. A unction SetpA, Bq is an isomorphism i and only i it is a biection. roo. First suppose that is an isomorphism, in which case there eists SetpB, Aq such that 1 A and 1 B. Suppose that paq pa 1 q. Then, applyin to both sides, we et that a a 1, so is inective. Given b B we see that ppbqq b, so is surective as well, and hence a biection. Now suppose that is a biection. Given b B, surectivity o implies that there eists an a A such that paq b. By inectivity o, this a is unique. We may thus deine : B Ñ A by the rule pbq a (or a the unique element o A such that paq b). It is simple to check that 1 A and 1 B, so : A B. roposition 2.4. A linear transormation L Vect k pv, W q is an isomorphism i and only i it is biective as a unction. roo. First suppose that our linear transormation L : V Ñ W is a biective unction. As we saw in the previous proo, there eists a unction (but is it a linear transormation?) M : W Ñ V such that M L 1 V and L M 1 W. For w, w 1 W suppose that Mpwq v and Mpw 1 q v 1. Then Lpvq w and Lpv 1 q w 1, so or a scalar λ k we have Lpv ` λv 1 q w ` λw 1 by linearity o L. By the deinition o M it ollows that Mpw ` λw 1 q v ` λv 1 Mpwq ` λmpw 1 q. We conclude that M Vect k pw, V q (i.e., that M is k-linear) so L : V W. Now suppose that L Vect k pv, W q is an isomorphism. Then, since linear transormations are ust special sorts o unctions, we also have that L SetpV, W q is an isomorphism. The previous proposition then implies L is a biection. 4

5 It is temptin to now uess that isomorphisms in Top are continuous biective unctions. This is alse! Let S 1 tz C z 1u be the unit circle with basic opens the open arcs" te iθ θ pa, bq, a ă bu. Let r0, 2πq denote the hal-open interval o nonneative real numbers less than 2π with basic opens r0, 2πq X pa, bq. The reader may check that the unction w : r0, 2πq Ñ S 1 θ ÞÑ e iθ is a continuous biection. Do we still hope that continuous biections are isomorphisms o topoloical spaces? Should an interval and a circle really be the same"? Thankully, w is not an isomorphism. In the net pararaph we will check that its inverse, which takes the arument o a unit comple number, is not continuous. Since inverse unctions are unique, it ollow that there are no maps in ToppS 1, r0, 2πqq which are inverses to w. Let ar denote the inverse to w. The set U r0, πq is open in r0, 2πq, and ar 1 puq wpuq te iθ θ r0, πqu. It is readily checked that this set is not open, whence ar is not continuous. Otentimes, isomorphisms in a particular cateory et a special name; this is the case in the cateory Top. Deinition 2.5. An isomorphism in Top (i.e., a continuous unction : X Ñ Y or which there eists a continuous unction : X Ñ Y such that 1 X and 1 Y ) is called a homeomorphism. I two spaces X, Y are isomorphic in Top, then we call them homeomorphic and write X Y. Deinition 2.6. A continuous unction ToppX, Y q is called open i puq Ď Y is open or all open U Ď X. roposition 2.7. A map ToppX, Y q is a homeomorphism i and only i it is an open biection. roo. Suppose is an open biection. To prove that is a homeomorphism, it suices to show that its inverse unction is continuous. For U Ď X open, 1 puq puq is open in Y by hypothesis, so is continuous. Now suppose is a homeomorphism and let ToppY, Xq be its inverse. By roposition 2.3, is a biection. For any open U Ď X, puq 1 puq is open in Y because is continuous. We conclude that is open as well. In practice, we will only sometimes use roposition 2.7 to check that a iven map is a homeomorphism. Frequently, it is ust as easy or easier to directly construct a continuous inverse. 3. BINARY RODUCTS Given obects, y in a cateory C we can consider the collection o diarams z in C. In other words, we are considerin obects z which map to" both and y (alon with the mappins"). A typical sort o cateorical question is the ollowin: Is there a universal z mappin to and y? Such a universal obect is reerred to as the product o and y in C. We make this notion precise in the ollowin deinition. 5 y

6 Deinition 3.1. Suppose Ob C is equipped with morphisms p C p, q and p y C p, yq such that or any C pz, q and C pz, yq, there is a unique morphism h C pz, q such that p h and p y h. Then the data p, p, p y q is called the product o and y in C and is denoted ˆ y. We can succinctly summarize the product aioms in the ollowin commutative diaram: z D! y. Here the dotted arrow labelled D!" indicates that there eists (D) a unique (!) morphism which makes the diaram commute. Frequently, we will abuse notation and write ˆ y, leavin the morphisms p and p y implicit. In this case, the diaram reads Our deinition raises two obvious questions: (1) Does ˆ y eist? (2) I it does eist, is ˆ y unique? z D! ˆ y y. The answer to (1) is no in eneral. There are many cateories which do not have products. Nonetheless, many cateories o interest do have products, includin Set and Top. We will study these eamples in a moment. As or (2), eistence o a product does imply a very ood sort o uniqueness. Namely, when it eists, ˆ y is unique up to unique isomorphism. By this, we mean that i there eist Ð Ñ y and Ð Q Ñ y in C both satisyin Deinition 5.1, then there is a unique morphism Ñ Q such that the diaram y Q commutes, and this morphism is an isomorphism. In the cateorical contet, this is the best sort o uniqueness we can hope or. roposition 3.2. I ˆ y eists in C, then it is unique up to unique isomorphism in the above sense. roo. Suppose that Ð Ñ y and Ð Q Ñ y in C both satisy Deinition 5.1. Then the diaram Q 6 y

7 and Deinition 5.1 tell us that there is a unique morphism : Ñ Q such that Q y commutes. Our proo will be complete i we can show that is an isomorphism. To this end, consider the diaram Q y and use the deinition o product to produce a unique morphism : Q Ñ such that Q y commutes. astin our diarams toether, we et the commutative diaram Q y which tells us that y commutes. But observe that 1 : Ñ in place o also makes this diaram commute. Since such a morphism is unique by deinition, 1. The reader should work out a similar arument to show that 1 Q. Hence : Q, and we have thus proven that ˆ y is unique up to unique isomorphism. We now investiate several cateories that have binary products. We bein with Set, the cateory o sets and unctions. roposition 3.3. For sets A and B, the cartesian product A ˆ B tpa, bq a A, b Bu alon with the proection maps p A : A ˆ B Ñ A, pa, bq ÞÑ a and p B : A ˆ B Ñ B, pa, bq ÞÑ b is the cateorical product o A and B. 7

8 roo. Given a set C and unctions A ÐÝ C ÝÑ B, deine h : C Ñ A ˆ B takin c ÞÑ hpcq ppcq, pcqq. It is easy to check that A p A C h A ˆ B commutes and moreover that h is the only unction makin the diaram commute. We conclude that A ˆ B is indeed the cateorical product o A and B. roposition 3.4. I k is a ield and Mat k is the cateory o natural numbers and k-matrices, then the cateorical product o n and m is the natural number n ` m alon with the morphisms p n `I n 0 nˆm and p m `0 mˆn I m where I is the ˆ identity matri and 0 ˆl is the ˆ l matri o 0 s. roo. Given a natural number l, an n ˆ l matri N, and an m ˆ l matri M, deine to be the pn ` mq ˆ l matri ˆN. M This is precisely the data o a diaram in the cateory Mat k. Observe that p n `I ˆN n 0 nˆm M n N p n l n ` m p B p m M B m N and p m `0 ˆN mˆn I m M M so the diaram commutes. We leave it to the reader to check that is the unique matri havin this property. We now turn to the study o products in Top, the cateory o topoloical spaces and continuous unctions. We certainly suspect that the product o spaces X and Y has the cartesian product X ˆ Y as its underlyin set. But is there a topoloy on X ˆ Y which will make it satisy Deinition 5.1? To approach this question, observe that we will need both o the proection maps X p X ÐÝÝ X ˆ Y p Y ÝÑ Y to be continuous. For U Ď X, we have p 1 X U U ˆ Y, and or V Ď Y, we have p 1 Y V X ˆ V. Thus or U Ď X open and V Ď Y open, we must have U ˆ Y and X ˆ V open in X ˆ Y. Moreover, open sets are closed under inite intersection, so U ˆ V pu ˆ Y q X px ˆ V q must be open in X ˆ Y. Since Deinition 5.1 does not seem to put additional restrictions on the topoloy, our best uess is that the topoloy on X ˆ Y is enerated by the sets U ˆ V. (The reader should check that tu ˆ V U Ď X and V Ď Y openu is a basis.) Theorem 3.5. For topoloical spaces X and Y, let τ XˆY be the topoloy on the cartesian product X ˆ Y enerated by the basis o sets U ˆ V where U Ď X and V Ď Y are open. Then px, τ XˆY q equipped with the standard proection maps p X and p Y is the cateorical product o X and Y. 8

9 roo. First note that p X and p Y are continuous by the discussion precedin the statement o the theorem. Given a space Z and continuous unctions X ÐÝ Z ÝÑ Y, let h : Z Ñ X ˆ Y be the unction takin z ÞÑ ppzq, pzqq. By roposition 5.5, h is the unique unction makin the diaram X p X Z h X ˆ Y commute (as a diaram o sets and unctions). I we prove h is continuous, then we will be done with our proo. Since τ XˆY is enerated by the basis o sets U ˆ V (U Ď X and V Ď Y open), it suices to check that h 1 U ˆ V Ď Z is open. By an easy computation, p Y Y h 1 U ˆ V 1 U X 1 V. Since and are continuous and inite intersections o open sets are open, we see that h 1 U ˆ V is open, completin our proo. We conclude this section by mentionin that there are many cateories which do not have binary products. For instance, suppose C is a cateory in which C p, yq whenever y. An eample o such a cateory is Σ 8 which has the sets n t0, 1,..., n 1u as obects and Σ 8 pn, mq equal to the set o biections n Ñ m. In order or ˆ y to eist in C, we must have morphisms Ð ˆ y Ñ y, and in a cateory such as Σ 8, we won t have any such morphisms when y! 4. MONOMORHISMS AND SUBOBJECTS In 2 we studied isomorphisms, the morphisms in a cateory which identiy when obects are cateorically indistinuishable. In this section, we turn to monomorphisms, another special class o morphisms. Deinition 4.1. A morphism i : Ñ y in a cateory C is a monomorphism i or all morphisms 1, 2 : z Ñ in C, the equality i 1 i 2 implies that 1 2. (In other words, i is letcancellative with respect to composition in C.) When i is a monomorphism we say that it is monic and write i : ãñ y. We can think o this deinition as sayin that whatever the composites i 1 and i 2 can see" o y, in act happened in beore postcomposition with i. A ew eamples will help us hone our intuition about monomorphisms. roposition 4.2. The monomorphisms in Set are precisely the inective unctions. roo. Suppose that i : A ãñ B is a monomorphism. For elements a 1 and a 2 o A, consider the unctions 1, 2 : t u Ñ A takin ÞÑ a 1 and ÞÑ a 2, respectively. Suppose that ipa 1 q ipa 2 q. We may reinterpret this condition as sayin that i 1 i 2 since the irst composite takes ÞÑ ipa 1 q while the second takes ÞÑ ipa 2 q. Since i is monic, we have 1 2, which is equivalent to a 1 a 2. Now suppose that i : A Ñ B is inective and that there are unctions 1, 2 : C Ñ A such that i 1 i 2. Then or each c C, ip 1 pcqq ip 2 pcqq. Since i is inective, we learn that 1 pcq 2 pcq, whence 1 2, so i is monic. roposition 4.3. For k a ield, the monomorphisms in Vect k are the inective k-linear maps. roo. Suppose that i : V ãñ W is a monomorphism in Vect k. Given a nonzero element v V, let v : k Ñ V be the k linear map takin 1 ÞÑ v and let 0 : k Ñ V be the map takin everythin in 9

10 k to 0 V. Suppose or contradiction that ipvq 0. Then it is easy to check that i v i 0, so v 0, whence v 0, a contradiction. We conclude that ker i 0, whence i is inective. Now suppose that i : V Ñ W is an inective linear map and that there are linear maps L 1, L 2 : U Ñ V such that i L 1 i L 2. Since everythin in siht is a unction, the arument rom the second pararaph o the proo o roposition 4.2 oes throuh, and we may conclude that L 1 L 2 so i is monic. We now consider monomorphisms in the cateory o topoloical spaces and continuous unctions. roposition 4.4. The monomorphisms in Top are precisely the inective continuous unctions. roo. Suppose that i : X ãñ Y is a monomorphism in Top. Endow the sinleton set t u with the discrete topoloy (which is in act the unique topoloy on this set). Notin that every unction t u Ñ X is continuous (check this!), we may directly adapt the proo o roposition 4.2 to determine that i is inective. It is similarly easy to see that inective continuous unctions are monomorphisms. Given a subset A o a topoloical space X, we may now ask when the inclusion i : A Ñ X takin a ÞÑ a is a monomorphism. Obviously, this map is inective, so we are really askin what topoloies on A make i continuous. Given U Ď X, we have i 1 U U X A, so it suices that U X A be open in A or all open U Ď X. In act, it is easy to check that τ A tu X A U Ď X openu is a topoloy on A. We conclude that or any topoloy on A iner than τ A, i : A ãñ X is a monomorphism. Equivalently, τ A is the coarsest topoloy on A such that i : A ãñ X is continuous. Since τ A satisies such a special property, we will ive it a name. Deinition 4.5. For a subset A o a topoloical space X, the topoloy τ A tu X A U Ď X openu is called the subspace topoloy on A. We use the subspace topoloy in makin the ollowin deinition. Deinition 4.6. For topoloical spaces X, Y, an embeddin : X Ñ Y is a continuous unction which is a homeomorphism onto its imae. More precisely, the induced unction : X Ñ pxq is a homeomorphism where the imae pxq is iven the subspace topoloy in Y. It is ood to think o embeddins as ways o identiyin spaces with subspaces o other spaces. Note that the embeddin is strictly more inormation than its domain: it tells us not only the homeomorphism type o the domain, but also how the domain sits in the codomain. Knots probably orm the most amous eample o a class o embeddins. Let S 1 denote the unit circle in R 2 with the subspace topoloy. To a topoloist, a knot is an embeddin S 1 ãñ R 3. One can think o this as a knotted strin with its ends used in three-dimensional euclidean space. The ollowin proposition ehibits that the notion o an embeddin is not the same as that o a monomorphism in Top. roposition 4.7. In the cateory Top, every embeddin is a monomorphism, but not every monomorphism is an embeddin. roo. Suppose that : X Ñ Y is an embeddin. Then : X Ñ pxq is a homeomorphism and hence a biection, whence must be inective. The unction is continuous by hypothesis, so is a monomorphism by roposition. To see that the converse is alse, let X t0, 1u with the discrete topoloy and let Y t0, 1, 2u with the indiscrete topoloy. (Thus all subsets o X are open, while only and Y are open in Y.) Let : X ãñ Y be the obvious inclusion, which is clearly continuous and hence a monomorphism. 10

11 Note, thouh, that : X Ñ pxq is not open since t0u is open in X but not open in pxq with the subspace topoloy. Hence, by roposition 2.7, is not an embeddin. 5. GENERAL RODUCTS In 3 we studied cateorical products ˆ y o two obects and y in a cateory C. What i we have a more eneral collection o obects i Ob C where i runs throuh some indein set I? Is there a universal obect which maps to all the i? I there is, this obect (alon with its morphisms to the i ) is called the product o the i. We develop this notion in this section. Deinition 5.1. Suppose that I Ñ Ob C is a unction takin i ÞÑ i, Ob C, and I Ñ Mor C is a unction takin i ÞÑ p i C p, i q such that or any z Ob C and morphisms i C pz, i q, there is a unique morphisms h C pz, q such that p i h i or each i I. Then the data p, tp i i Iuq is called the product o the i in C and is denoted ź ii We can epress this deinition succinctly via the ollowin commutative diaram: Some comments on the diaram are in order: i p i p i. z D!h p k k i k.» We read the diaram as sayin that iven z and the morphisms i, i I, there is a unique morphism h makin the diaram commute.» We have permitted ourselves the liberty o placin morphism labels within" the arrows they are labelin. This is strictly or aesthetic reasons. Havin the arrow pass under" p is an aesthetic choice as well.» Each o i,, and k are elements o I, and the ellipses in the bottom row represent the additional l belonin to t i i Iu. The morphisms l and p l are implicit in the diaram as well.» Make sure that you can spot the I -many commutin trianles p i h i in the above diaram. (O course, I 3 o them are implicit.) In act, we could reinterpret the above diaram as sayin that there is a unique h : z Ñ makin the diarams z h p commute or each I. As with binary products in an arbitrary cateory C, eneral products may or may not eist. It is the case, thouh, that when a eneral cateorical product eists, it is unique up to unique isomorphism. 11

12 Theorem 5.2. Suppose p, tp i i Iuq and pq, tq i i Iuq both satisy Deinition 5.1. Then there is a unique morphism Ñ Q makin the diaram commute, and this morphism is an isomorphism. p i p p k i k q i q roo. The proo is essentially identical to the proo o roposition 3.2, ust with I 2 more morphisms in play. The reader should convince hersel that this is the case, careully writin out the diaram chase i necessary. Remark 5.3. We could ust as easily have deined ź A ź or A any subset o Ob C. Such a product is equipped with maps p : ś A Ñ or each A. Q A Certain products have special names and special notation:» I A t, yu, then, when it eists, ś A is typically denoted ˆ y. This matches our notion o binary product rom Section 3. nź» I A t 1, 2,..., n u, then, when it eists, we write 1 ˆ 2 ˆ ˆ n or i or ś A.» I A tu is a sinleton set, it is easy to check that ź tu where the morphism p : Ñ is the identity morphism 1. This product always eists.» I A is the empty subset o Ob C, then, when it eists, ś is called the terminal obect o C, and is typically denoted (or sometimes 1). I we careully sort out the quantiiers in Deinition 5.1, we see that is an obect in C which admits a unique morphism rom each z Ob C. In other words, C pz, q 1 or each z Ob C. The cateory Set o sets and unctions has terminal obect any sinleton set. Similarly, Top has terminal obect any sinle point topoloical space. The cateory Vect k o k-vector spaces has terminal obect 0 t0u. We now undertake the task o identiyin products in several amiliar cateories. We bein with Set. Deinition 5.4. Given a amily tx i i Iu o sets X i (in other words, a unction" I Ñ Ob Set, inorin the act that Ob Set is not a set), deine the cartesian product ą ii to be the set o unctions : I Ñ Ť ii X i such that piq X i. In a cartesian product, we will typically write i or piq. An element o Ś ii X i is called a tuple, and we will requently write p i q or p i q ii or. Given I, we deine the proection p : Ś ii X i Ñ X by the ormula p pq. 12 X i q k i 1

13 roposition 5.5. The cartesian product Ś ś ii X i alon with its proection unctions p i orm the product ii X i in Set. roo. Given a set Z and unctions i : Z Ñ X i or i I, deine h : Z Ñ Ś X i by hpzq p i pzqq ii. Then p i phpzqq i pzq or all i I and all z Z, whence p i h i or all i I. The reader may check that the equations p i h i completely speciy h, so Ś X i is in act the product in Set o the X i. Remark 5.6. Now that we have proven that cartesian and cateorical products are identical in Set, we will adopt the more common notation ś X i or Ś X i when the X i are sets. Eample 5.7. For k a ield, the product in Vect k o k-vector spaces V i, i I, has underlyin set the cartesian product o the V i with linear structure iven by pv i q ` pw i q pv i ` w i q and λpv i q pλv i q. The proection maps are usual ones. The reader may easily check the correctness o these assertions. Note, thouh, that FinVect k, the cateory o inite-dimensional k-vector spaces, does not have arbitrary products. Indeed, ś ii V i eists i and only i ř ii dim k V i ă 8; in this case, the product is the same as that or Vect k. As we shall presently see, arbitrary products do eist in Top, the cateory o topoloical spaces and continuous unctions. Rather than leap directly to the deinition/theorem (deineorem?) ivin these products, we will radually develop the necessary concepts. For topoloical spaces X i, i I, it is reasonable to uess that the set underlyin ś ii X i is the cartesian product o the sets underlyin each X i. I this is the case, then we need each o the proection unctions p : ś X i Ñ X to be continuous. For U Ď X, the preimae p 1 U is eactly ś Xi but with the X actor replaced by U. In other words, p 1 U tp i q i X i or i I tu and Uu Ď ź X i is the set o tuples with -th coordinate in U. When U is open in X, this set must be open in ś X i. Let S tp 1 U U Ď X openu be the set o such subsets o ś X i, and let S ď I S. Supposin that τ is the topoloy on ś ś X i which we seek (i.e., the topoloy which will make Xi a product in Top), we see that we must have S Ď τ. Note that S is a subbasis. (Indeed, p 1 X ś X i is already an element o S, and the subbasis condition only mandates that the union o all elements o S be ś X i.) Since little else seems to be hidin in Deinition 5.1, it is reasonable to uess that τ is the topoloy τ S enerated by the subbasis S. This is in act the case. We will develop some properties o τ S and then prove that p ś X i, τ S q alon with the proection maps p : ś X i Ñ X orm a product in Top. roposition 5.8. For J Ď I and U Ď X open or each J, let U tu Ju and let B U ź V i ii where V U i J and V i X i i i I J. We will reer to J as the inde subset o U. Usin this terminoloy, the basis B enerated by S is equal to B tb U U has inite inde setu. As such, the topoloy enerated by S consists o arbitrary unions o elements o B. 13

14 Remark 5.9. Note that the basis sets B U are precisely products o open subsets U i o each X i where all but initely many U i X i. Later, we will comment on bo topoloy, a dierent topoloy on the cartesian product o the X i in which we permit the U i to be arbitrary open subsets o X i. roo. The basis B enerated by S consists o inite intersections o sets in S. Note that so it suices to consider inite intersections p 1 U X p 1 V p 1 pu X V q, B p 1 1 U 1 X p 1 2 U 2 X X p 1 n U n in which the indices k are all distinct. A point p i q is in B i and only i 1 U 1, 2 U 2,..., n U n while the other i are allowed to rane reely throuh X i. Hence B B U where U tu 1,..., U n u, completin our proo. We now come to our main theorem or this section, which identiies products in Top. Theorem For topoloical spaces X i, i I, the cartesian product ś ii X i equipped with the topoloy τ S enerated by the subbasis S is the product o the X i in Top. roo. We aim to prove that or any space Z and collection t i : Z Ñ X i i Iu o continuous unctions, there is a unique continuous unction h : Z Ñ ś X i such that the diarams Z h ś Xi X p commute or each I. Given such data, deine h by the ormula hpzq p i pzqq ii. In our proo o roposition 5.5, we have already seen that h is the unique unction makin the diaram commute. It remains to show that h is continuous relative to τ S. Since τ S is enerated by the subbasis S, it suices to check that each o the subbasic opens U or I and U Ď X open are taken to open subsets o Z by h 1. Since p h, we have 1. The unction is continuous by hypothesis, so h 1pp 1 Uq 1 U is open in Z, as desired. p 1 h 1 p 1 The ollowin theorem is really ust a reinterpretation o the statement that ś X i is a cateorical product in Top. In order to state it, we introduce a small amount o terminoloy. Given a unction (not necessarily continuous) : Z Ñ ś X i, deine the component unctions o to be i p i or i I. Also note that iven unctions i : Z Ñ X i, i I, roposition 5.5 tells us that there is a unique unction : Z Ñ ś X i whose component unctions are i. (It makes no dierence to call the unction instead o h.) Theorem A unction : Z Ñ ś X i is continuous i and only i each o its component unctions i p i is continuous. roo. First suppose that is continuous. Since each p i is continuous and compositions o continuous unctions are continuous, we learn that i p i is continuous or each i I. 14

15 Now suppose that each i is continuous. By roposition 5.5, is the unique unction makin i p i or each i I, and the act that ś X i is a product in Top now uarantees that is continuous. Given Theorem 5.10, it is reasonable to make the ollowin deinition. Deinition The topoloy on ś ii X i enerated by the subbasis S is called the product topoloy on ś X i. I we make no urther comment, we will always assume that ś X i has been iven the product topoloy. There is another, more naïve and less useul topoloy on ś X i which we will sometimes consider. Deinition Given a cartesian product ś ii X i o topoloical spaces, let B bo denote the collection o sets B U (deined in roposition 5.8) in which the inde set o U is allowed to be any subset o I. The bo topoloy on ś X i is the topoloy enerated by this basis. Remark We leave it to the reader to check that B bo is in act a basis. Also note that we could ust as well mandate that the inde set o U be all o I. Remark I the inde set I is inite, then the bo and product topoloies are the same. The ollowin eample is one o those eamples that every topoloist knows. You should commit it to memory so that you never make the mistake o conusin the product and bo topoloies. Eample Let R N denote the countably ininite product o R with itsel, i.e., Let denote the diaonal unction R N ź nn R. : R ÝÑ R N t ÞÝÑ pt, t, t,...q. Each component unction n o is the identity unction 1 R, hence is continuous i we ive R N the product topoloy. Now consider R N with the bo topoloy. The set B ź ˆ 1 n ` 1, 1 p 1, 1q ˆ p 1{2, 1{2q ˆ p 1{3, 1{3q ˆ n ` 1 nn is an element o B bo and hence is a bo-open subset o R N. Note, thouh, that 1 B č ˆ 1 n ` 1, 1 t0u. n ` 1 nn (Eercise: check that the above equalities hold!) Hence 1 B is not an open subset o R, and we conclude that is not continuous relative to the bo topoloy on R N. We can draw the ollowin morals rom this eample: (a) The product and bo topoloies are in act dierent topoloies in eneral (despite the act that they are the same when the inde set is inite). We know this because we have ust seen that they have dierent sets o continuous unctions. From this, we can conclude that there are instances in which the bo topoloy is strictly iner than the product topoloy. (b) The act that the product topoloy makes ś X i into a cateorical product should convince us o its superiority to the bo topoloy. But even i we do not trust cateory theorists, continuity o is clearly a desirable property, and we must use the product topoloy to uarantee this. 15

16 6. QUOTIENTS Recall the notion o an equivalence relation on a set X. Speciically, is a relation which is releive ( or all X), symmetric ( y ùñ y ), and transitive ( y, y z ùñ z). For instance, the relation y ðñ y Z is an equivalence relation on R. Given an equivalence relation on X, we may partition X into equivalence classes (relative to ). The equivalence class o X is the set o y X such that y. Writin X{ or the set o equivalence classes, we see that X ž A AX{ where > denotes disoint union. In the case o the eample rom the previous pararaph, we see that R{ ta ` Z a r0, 1qu and the partition into equivalence classes takes the orm R ž a ` Z. ar0,1q Not wishin to be circular in our reasonin, we will postpone commentin on the topoloy o R{ until we have introduced the quotient topoloy and quotient spaces. Given a partition X ž ii X i o a set X, there is an equivalence relation iven by y i and only i there is an inde i I such that, y X i. For this equivalence relation, X{ tx i i Iu. There is a natural quotient unction q : X Ñ X{ takin to the equivalence class o. Note that q is surective. Given any surective unction p : X Ñ Y, we can partition X into the ibers o p, p 1 tyu or y Y. This partition yields an equivalence relation whose associated quotient unction is p. As such, we see that equivalence relations, partitions, and surective unctions are three sides o the same hypercoin. Now suppose that X is a topoloical space with an equivalence relation on its underlyin set. How should we topoloize X{? Our search should be uided by the quotient map q : X Ñ X{, which ouht to be continuous. Consider the set τ tu Ď X{ q 1 U is open in Xu. The reader may check that τ is a topoloy, and is in act the inest topoloy on X{ such that q is continuous. We call τ the quotient topoloy on X{. It is not the case that every surective continuous unction p : X Ñ Y arises rom an equivalence relation on X. Indeed, the quotient map q : X Ñ X{ has the special property that U Ď X{ is open i and only i q 1 U is open in X. Meanwhile, or p to be continuous, we only need that U open in Y implies that p 1 U is open in X. But this crucial distinction is the only one. I we want to recover our hypercoin in the topoloical settin, we must replace surective unction" with the ollowin notion o quotient map. Deinition 6.1. A continuous unction q : X Ñ Y is a quotient map i it is surective and U Ď Y is open i and only i q 1 U Ď X is open. I q is a quotient map, then Y is called a quotient o X or a quotient space. I q is the quotient unction o an equivalence relation, then we will always ive Y X{ the quotient topoloy, the unique topoloy that makes q a quotient map. Note that this is not a strictly cateorical deinition. While it is possible to recast quotients in purely cateorical terms, there is little beneit to doin so (at least currently), and we ind the above 16

17 derivation suicient or our purposes. Nonetheless, it is the case that quotient maps q : X Ñ Y satisy a certain universal property. Theorem 6.2. Given a quotient map q : X Ñ Y and y Y, let X y : q 1 tyu denote the iber o q over y. Suppose : X Ñ Z is a continuous unction which is constant on X y or each y Y. Then there is a unique map : Y Ñ Z such that q. I, additionally, is a quotient map with 1 tpqu X qpq or all X, then is a homeomorphism. We may diarammatically represent the irst part o the theorem as ollows: X y q X D! Z. D Here denotes a one point space, X y Ñ X is the inclusion mappin, and the commutativity o the upper riht paralleloram says that is constant on X y (since the imae o Ñ Z is necessarily a sinle point). The and D (on the riht-hand vertical arrow) are meant to indicate that or every y Y there is a map Ñ Z such that the upper riht paralleloram commutes, recapitulatin the hypothesis that is constant on each iber. In this case, there is a unique map : Y Ñ Z makin the bottom trianle commute. roo o Theorem 6.2. C. Theorem 22.2 in Munkres or the irst part o the theorem. It remains to show that is a homeomorphism under the additional hypotheses on. We bein by showin that the hypotheses o the irst part o the theorem hold with and q swapped. Since both q and are surective, it is clear that the ibers o q are in biective correspondence with the ibers o. Since q is constant on its own ibers, we see that it is also constant on the ibers o. Hence there is a unique map h : Z Ñ Y such that h q. We now check that is a homeomorphism by provin that h 1 Y and h 1 Z. Consider the commutative diaram Y q h X q Z Y. Thus, by the irst part o the theorem (with q playin the roles o both q and ), we see that h is the unique map Y Ñ Y such that p hq q q. Since 1 Y is a map satisyin 1 Y q q, we learn that h 1 Y. A similar arument with the commutative diaram X Z Y shows that h 1 Z, concludin our proo. q h Z We now return to R{ where y ðñ y Z. Note that each equivalence class is o the orm a ` Z or eactly one a r0, 1q, and 1 ` Z Z, so it seems like R{ circles back on itsel when a oes past 1. This is precisely the case, as we shall now prove via Theorem

18 Let q : R Ñ R{ denote the quotient map, and consider : R Ñ S 1 takin a ÞÑ e 2πia. Here S 1 is the unit circle in C R 2 with the standard (e.. subspace) topoloy. The reader may check that is a continuous map. The ibers o q are precisely the equivalence classes a ` Z, and we see that or b Z, pa ` bq e 2πipa`bq e 2πia. Thus is constant on the ibers o q and we conclude that there is a unique map : R{ Ñ S 1 such that q. We now check the additional hypotheses uaranteein that is a homeomorphism. The map is clearly surective, and the reader may check that is in act a quotient map. (The details are painul to write down, but more or less obvious.) Moreover, 1 tpaqu a ` Z q 1 tqpaqu or all a R. Thus the second part o Theorem 6.2 tells us that is a homeomorphism; i.e. R{ is a circle. Remark 6.3. I you have seen roup-theoretic quotients beore, you may reconize R{ as R{Z where Z is considered as a subroup o the additive roup R. I you eel comortable with this, I hihly encourae you to do the supplementary eercises on topoloical roups on pp o Munkres. 7. FUNCTORS Cateory theory tells us that in order to understand mathematical obects, we must understand the relations (i.e. morphisms) between those obects. I cateories themselves are mathematical obects, then cateory theory demands that we must study how cateories are related. O course, this is both possible and useul: a morphism o cateories is called a unctor. Deinition 7.1. I C and D are cateories, then a unctor F : C Ñ D consists o assinments» F Ob : Ob C Ñ Ob D, and» F Mor : C pa, bq Ñ C pf Ob a, F Ob bq or all a, b Ob C satisyin the ollowin properties:» F Mor 1 a 1 FOb a or all a Ob C, and» F Mor p q pf Mor q pf Mor q or all composable morphisms, Mor C. Remark 7.2. In practice, the distinction between F Ob and F Mor is obvious rom contet, and we simply write F a F Ob a when a Ob C and F F Mor when Mor C. Abusin notation in this ashion, we see that properties satisied by F take the simpler orms:» F 1 a 1 F a, and» F p q pf q pf q. It is nice to think o the second property as sayin that F takes commutative diarams in C to commutative diarams in D: a b c ÞÝÑ F a F F b F p q We have encountered many unctors in the past, and have even used unctorial properties in aruments. Here are a ew pertinent eamples. Eample 7.3. Given sets A and B and a unction : A Ñ B, recall the direct imae unction : 2 A Ñ 2 B and preimae unction : 2 B Ñ 2 A. You proved that these satisied the ollowin compatibilities: p q and p q whenever the composition o unctions makes sense. Additionally, p1 A q 1 2 A p1 A q. Thus we may deine a unctor F : Set Ñ Set via F A 2 A and F, the direct imae unctor. 18 F F c

19 We almost have a preimae unctor G as well, iven by deinin GA 2 A and G, but G reverses the direction o morphisms, and Gp q G G instead o G G. (In act, the latter composition does not even make sense in eneral.) Such an assinment ets a name as well. Deinition 7.4. I C and D are cateories, then G is a contravariant unctor rom C to D i it consists o assinments» G : Ob C Ñ Ob D, and» G : C pa, bq Ñ DpGb, Gaq or all a, b Ob C such that» G1 a 1 Ga or all a Ob C, and» Gp q G G or all composable morphisms, Mor C. We write G : C op Ñ D when G is a contravariant unctor. In contradistinction, unctors F : C Ñ D are sometimes called covariant unctors. Clearly, the assinment G : Set op Ñ Set iven by GA 2 A, G is a contravariant unctor rom Set to Set. Remark 7.5. Secretly, C op is notation or the opposite cateory o C. The cateory C op has the same obects and morphisms as C, but the source and taret unctions are swapped. The cateorically inclined reader should spend some time veriyin that C op is a cateory, and checkin that a contravariant unctor rom C to D is the same thin as a covariant unctor C op Ñ D. Eample 7.6. Many amiliar cateories C have a oretul unctor U : C Ñ Set. For instance, the oretul unctor U : Top Ñ Set takes a space X to its underlyin set UX X and takes a continuous unction : X Ñ Y to the unction U : UX Ñ UY. It should be clear that this satisies the unctor properties. The letter U stands or underlyin, and we have such a oretul unctor basically whenever the obects in a cateory are sets equipped with etra structure and the morphisms o a cateory a special types o unctions. There are oretul unctors rom roups to sets, rom rins to sets, rom vector spaces to sets, etc. There are also intermediate oretul unctors. For instance, every rin has an underlyin (additive) abelian roup. Since rin homomorphisms are special types o roup homomorphisms on the underlyin abelian roups, we et a oretul unctor U : Rin Ñ AbGp. I we then oret all the way to sets, we et the bizarre equality o unctors U U U (where each U is denotin a dierent oretul unctor). Eample 7.7. Consider the cateory o vector spaces Vect k over a ield k. Each k-vector space V has a linear dual V _ Vect k pv, kq (where k is iven the standard k-linear structure). Each k-linear map : V Ñ W has a dual _ : W _ Ñ V _ iven by pαq α. It is a standard eercise in linear alebra to check that p q _ is a unctor Vect op k Ñ Vect k. Eample 7.8. The dual vector space unctor is our irst eample o a representable unctor. Given a (locally small 1 ) cateory C and obect Ob C, there is a unctor C p, q : C op Ñ Set takin obects a to the set o morphisms C pa, q and morphisms : a Ñ b to C p, q : C pb, q Ñ C pa, q such that C p, qpαq α. We call C p, q the unctor reprsented by. The dual vector space unctor is the unctor represented by the one-dimensional k-vector space k. Eample 7.9. The cateorically inclined reader should deine and investiate the corepresentable unctors C p, q or Ob C. 1 Locally small cateories are cateories C or which C pa, bq is a set (as opposed to a class or some other set-theoretic monster) or each a, b C. 19

20 Eample Recall that or a space X, π 0 X denotes the path-connected components o X. In your homework, you will prove that π 0 is a unctor Top Ñ Set. The cateorically inclined reader may deine the cateory Hot o topoloical spaces and homotopy classes o maps between spaces, and then prove that π 0 is the unctor Hot Ñ Set corepresented by the unit interval I r0, 1s The undamental roup unctor. We now turn to our motivation or introducin the lanuae o unctors: the undamental roup unctor. We have already noted that the undamental roup depends on a choice o basepoint, so we will need to introduce a cateory o topoloical spaces with basepoint" in order to develop our story. We leave it as an easy eercise to the reader to check that the ollowin choice o obects and morphisms does indeed deine a cateory. Deinition The cateory Top o based topoloical spaces has obects which are pairs px, 0 q where X is a topoloical space and 0 X is a point o X. A morphism px, 0 q Ñ py, y 0 q in Top is a continuous unction : X Ñ Y such that p 0 q y 0. Theorem Given a based space px, 0 q, let π 1 px, 0 q denote the undamental roup o X based at 0. Given a based map : px, 0 q Ñ py, y 0 q, let π 1 denote the assinment Then π 1 is a unctor Top Ñ Gp. π 1 px, 0 q ÝÑ π 1 py, y 0 q rγ : I Ñ Xs ÞÝÑ r γs. roo sketch. We must check the ollowin: (1) The assinment π 1 : rγs ÞÑ r γs is well-deined (i.e. it does not depend on our choice o path homotopy class representatives). (2) The unction π 1 is in act a roup homomorphism. (3) I 1 px,0 q is the identity map, then π 1 1 px,0 q is the identity homomorphism on π 1 px, 0 q. (4) I : px, 0 q Ñ py, y 0 q and : py, y 0 q Ñ pz, z 0 q are based maps, then π 1 p q π 1 π 1. In order to check (1), irst note that p γqp0q pγp0qq p 0 q y 0, and similarly p γqp1q, so r γs π 1 py, y 0 q. Now suppose that H : γ» p γ 1. The reader may check that H is a path homotopy rom γ to γ 1, whence r γs r γ 1 s. This proves that π 1 is well-deined. To check (2), suppose that rγs and rδs are elements o π 1 px, 0 q. The reader may check that p γq p δq pγ δq, whence π 1 pγq π 1 pδq π 1 pγ δq. We conclude that π 1 is a roup homomorphism rom π 1 px, 0 q to π 1 py, y 0 q. The easiest o the properties, (3) holds because pπ 1 1 px,0 qqrγs r1 γs rγs. Finally, we check (4), which asserts that π 1 respects composition o based maps. Suppose that : px, 0 q Ñ py, y 0 q and : py, y 0 q Ñ pz, z 0 q are based maps. Then π 1 p q : rγs ÞÝÑ rp q γs while π 1 π 1 : rγs ÞÝÑ r p γqs. Since p q γ p γq, we conclude that π 1 p q π 1 π 1. 20