Category Theory Course. John Baez March 28, 2018

Transcription

1 Cateory Theory Course John aez March 28,

2 Contents 1 Cateory Theory: Denton o a Cateory Cateores o mathematcal objects Cateores as mathematcal objects Don Mathematcs nsde a Cateory Lmts and Colmts Products Coroducts General Lmts and Colmts Equalzers, Coequalzers, Pullbacks, and Pushouts (Week 3) Equalzers Coequalzers Pullbacks Pullbacks and Pushouts Lmts or all nte darams Week Mathematcs etween Cateores Natural Transormatons Mas etween Cateores Natural Transormatons Examles o natural transormatons Equvalence o Cateores djunctons What are adjunctons? Examles o djunctons Daonal Functor Darams n a Cateory as Functors Unts and Counts o djunctons Cartesan Closed Cateores Evaluaton and Coevaluaton n Cartesan Closed Cateores Internalzn Comoston Elements Week Subobjects Symmetrc Monodal Cateores Guest lecture by Chrstna Osborne What s a Monodal Cateory? Gon back to the denton o a symmetrc monodal cateory

3 9 Week The subobject classer n Grah Set Theory, Toos, and Loc Where does toos theory o rom here?

4 1 Cateory Theory: Unes mathematcs. Studes the mathematcs o mathematcs (smlar to mathematcal loc). Moves towards hher-dmensonal alebra ( homotoyn mathematcs). Set Theory 0-dmensonal Cateory Theory 1-dmensonal 4

5 1.1 Denton o a Cateory cateory C conssts o: class Ob(C) o objects. I x Ob(C), we smly wrte x C. Gven x, y C, there s a set Hom C (x, y), called a homset, whose elements are called morhsms or arrows rom x to y. I Hom C (x, y), we wrte : x y. Gven : x y and : y z, there s a morhsm called ther comoste : x z. y x z Comoston s assocatve: (h ) = h ( ) ether sde s welldened. h h h ( )=(h ) For any x C, there s an dentty morhsm 1 x : x x 1 x We have the let and rht unty laws: Examles o Cateores x 1 x = or any : x x 1 x = or any : x x Cateores o mathematcal objects For any knd o mathematcal object, there s a cateory wth objects o that knd and morhsms ben the structure-reservn mas between the objects o that knd. Examle 1.1. Set s the cateory wth sets as objects and unctons as morhsms. 5

6 Examle 1.2. Gr s the cateory wth rous as objects and homomorhsms as morhsms. Examle 1.3. For any eld k, Vect k s the cateory wth vector saces over a eld k as objects and lnear mas as morhsms. Examle 1.4. Rn s the cateory wth rns as objects and rn homomorhsms as morhsms. These are cateores o alebrac objects, namely, a set (stu) wth oeratons (structure) such that a bunch o equatons hold (roertes), wth morhsms ben unctons that reserve the oeratons. ll ths s ormalzed n unversal alebra, usn alebrac theores. There are also cateores o non-alebrac adets: Examle 1.5. To s the cateory wth toolocal saces as objects and contnuous mas as morhsms. Examle 1.6. Met s the cateory wth metrc saces as objects and contnuous mas as morhsms. Examle 1.7. Meas s the cateory wth measurable saces as objects and measurable mas as morhsms Cateores as mathematcal objects There are lots o small, manaebable cateores: Denton 1.1. monod s a cateory wth one object. Remark. Hom C (, ) or ths object, s a set wth assocatve roduct and unt. 1 Examle The multlcaton table above tells us how to comose morhsms. The resultn monod s usually called Z/2Z. Now, consder the same daram but wth ths multlcaton table nstead: Here we et another amous monod: = true 1 = alse = alse or alternatvely = true = or = and 6

7 Denton 1.2. morhsm : x y s an somorhsm t has an nverse : y x, that s, a morhsm wth: = 1 x = 1 y I there exsts an somorhsm between two objects x, y C, we say they re somorhc. Denton 1.3. cateory where all morhsms are somorhsms s called a rouod. Examle 1.9. "The rouod o nte sets" s obtaned by takn FnSet, wth nte sets as objects and unctons as morhsms, and then thrown out all morhsms excet somorhsms (.e. bjectons). Denton 1.4. monod that s a rouod s called a rou. Remark. the usual "elements" o a rou are now the morhsms. Denton 1.5. cateory wth only dentty morhsms s a dscrete cateory. Remark. So any set s the set o objects o some dscrete cateory n a unque way. So a dscrete cateory s "essentally the same" as a set x x Denton 1.6. reorder s a cateory wth at most one morhsm n each homset. I there s a morhsm : x y n a reorder, we say x y ; not, we say x y. For a reorder, the cateory axoms just say: Comoston: x y and y z = x z. ssocatvty s automatc. Identtes: x x always. Let and rht unt laws are automatc. We re not ettn antsymmetry: x y and y x = x = y. Denton 1.7. n equvalence relaton s a reorder that s also a rouod. Prooston 1.1. reoder s a rouod and only ths extra law holds or all x, y C: x y = y x 7

8 Here we have transtvty, relexvty, and symmetry o. So we usually call ths relaton. Prooston 1.2. reorder s skeletal,.e. somorhc objects are equal, and only ths extra law holds or all x, y C: In ths case we say that C s a oset. (x y) (y x) = x = y Examle Preorder that s a rouod but not a oset: Examle Preorders that are osets but not rouods: Examle Preorder that s both a oset and a rouod: Snce cateores can be seen as mathematcal objects, we should dene mas between them: Denton 1.8. Gven cateores C and D, a unctor F : C D conssts o: a uncton called F rom Ob(C) to Ob(D): x C then F(x) D. unctons called F rom Hom C (x, y) to Hom C (F(x), F(y)), or all objects x, y C: : x y then F( ) : F(x) F(y) such that: F( ) = F() F( ) whenever ether sde s well dened. F(1 x ) = 1 F(x) or all x C. So a unctor looks lke ths: y F F( ) F(y) F() x z F(x) F() F( ) F(z) 1 x 1 F(x) 8

9 Examle There s a cateory called "1". It looks lke ths: 1 What s a unctor F : 1 C where C s any cateory? 1 F F( ) 1 The answer s: an object n C, snce or any object x C, there exsts a unque unctor F : 2 C such that F( ) = x. Examle There s a cateory called "2". It looks lke ths: Remark. lso a oset. x 1 x y 1 y What s a unctor F : 2 C where C s any cateory? It s just a morhsm or arrow n C! For any morhsm : n C, there exsts a unque unctor F : 2 C such that F( ) =. Prooston 1.3. I F : C D and G : D E are unctors, then you can dene a unctor G F : C E and (H G) F = H (G F). lso, or any cateory C there s an dentty unctor 1 C : C C wth: 1 C (x) = x or all x C 1 C ( ) = or all : x y n C F 1 C = F or all F : C D 1 C H = H or all H : D C Denton 1.9. Cat s the cateory whose objects are "small" cateores and whose morhsms are unctors. Remark. "small" cateory s one wth a set o objects. For examle, Set s not a small cateory because Set has a class o objects. Gr and Rn are also not small cateores or the same reason as Set. The cateores 1 and 2 on the other hand, are small cateores. 9

10 1.2 Don Mathematcs nsde a Cateory lot o math s done nsde Set, the cateory o sets and unctons. Let s try to eneralze all that stu to other cateores by relacn Set wth a eneral cateory C. In Set, we have onto and one-to-one unctons. In a cateory C, we eneralze these concets to emorhsms or es and monomorhsms or monos resectvely. Denton morhsm : s a mono or all, h : Q we have: = h = = h Q Remark. lso known as ben a let-cancellatve morhsm h Prooston 1.4. In Set, a morhsm s monc and only t s a one-to-one uncton. Turnn around the arrows n the denton o mono, we et: Denton morhsm : s a e or all, h : Q we have: = h = = h Q h Remark. lso known as ben a rht-cancellatve morhsm Prooston 1.5. In Set, a morhsm s an e and only t s an onto uncton. Denton morhsm : s an so there exsts 1 : that s a let nverse 1 = 1 and a rht nverse 1 = 1 Prooston 1.6. In Set, : s a mono and only t has a let nverse, and an e and only t has a rht nverse (usn the axom o choce). Thus, s an somorhsm and only t s mono and e. Prooston 1.7. In Rn (rns and rn homomorhsms) : Z Q (n n) s a mono and an e, but not an so. In act, t has nether a let nor a rht nverse. Proo. There sn t a rn homorhsm : Q Z, snce t would send 1 2 to some multlcatve nverse o 2. Why s mono? We need: = h = = h R Z Q h I ( )(r) = ( h)(r) r R, snce s one-to-one (r) = h(r) r (as a uncton), ths mles = h. Why s e? We need: 10

11 = h = = h Z Q R h The man dea s that any morhsm rom Q s comletely determned by ts values on the nteers. We know () = h() and (q) = h(q). So (1) = ( q q ) = (q)( 1 q ), so we can wrte ( 1 q ) = 1 (q). So ( q ) = ()( 1 q ) = (). So (and smlarly or h) s determned by ts (q) values on the nteers; snce they aree on Z, they re equal. Puzzle: In To, nd : that s e and mono, but not an so. 1.3 Lmts and Colmts These are ways o buldn new objects n a cateory C rom darams n C Products Denton Gven objects, C, a roduct o them s an object Z equed wth morhsms, and q called rojectons to and. Z q such that or any canddate Q Q q there exsts a unque ψ : Q Z such that the ollown daram commutes ψ Z q Q The denton o coroduct s just the same but wth all arrows reversed. 11

12 Prooston 1.8. In Set, the roduct o and, denoted, s: Proo. Gven = {(x, y) : x, y } Q Let ψ : Q be ψ(q) = ( (q), (q)). We ndeed et ψ =, q ψ =, and ψ s the unque ma obeyn these equatons. We could also take as our roduct any set S that s somorhc to, va some so α : S α S q α α q Use α and q α as rojectons; then you can check that α S q α s also a roduct o and. So any object somorhc to a roduct can also be a roduct. Prooston 1.9. Suose W q and Z q are both a roduct o and. Then W and Z are somorhc. That s, roducts are unque u to somorhsm. Proo. Snce W s a roduct. There exsts a unque ψ : Z W makn ths daram commute: 12

13 !ψ Z q W lso, snce Z s a roduct, There exsts a unque ϕ : W Z makn ths daram commute:!ϕ Z q W It suces to show ϕ and ψ are nverse. Why s ψ ϕ : W W the dentty? I we can show ths, the same arument wll show ϕ φ = 1 Z. Snce There s a unque arrow makn ths daram commute:! W q W 1 W : W W does the job, but so does ψ ϕ : W W. nd so by unqueness, 1 W = ψ ϕ. Prooston I a morhsm s an so, t s both a mono and an e. Remark. We ve seen that the converse s alse Proo. I : has a let nverse 1, t s a mono: = h = 1 = 1 h = = h, h Smlarly, I : has a rht nverse 1, t s an e: = h = 1 = h 1 = = h, h 13

14 Denton morhsm wth a let nverse s called a slt monomorhsm; a morhsm wth a rht nverse s called a slt emorhsm. Remark. In Set, every mono (or e) slts, but we saw that ths sn t true n Rn or To Coroducts Denton Gven objects and, a coroduct o and s an object Z equed wth morhsms, j called nclusons. Z j whch s unversal, whch means or any daram o the orm: Q There exsts a unque ψ : Z Q makn the ollown daram commute: j Z Q!ψ That s, = ψ and = ψ j. Prooston In Set, a coroduct o and s ther dsjont unon. wth morhsms: + = {0} {1} : + x (x, 0) j : + y (y, 1) Cateory PRODUCTS COPRODUCTS + Set cartesan roduct S T dsjont unon S T To cartesan roduct wth roduct tooloy dsjont unon Gr roduct o rous G H ree roduct G H bgr (abelan cateory) roduct o abelan rous Vect k (abelan cateory) V W drect sum o vector saces V W 14

15 The ree roduct G H conssts o equvalence classes o words x 1 x 2... x n where x G H, wth the ollown relatons: x 1 x 2... x 1 1x x n x 1 x 2... x 1 x x n x 1 x 2... x x x n x 1 x 2... x 1 yx x n where 1 s the dentty n G or H, and x, x +1 G or x, x +1 H, and y = x x General Lmts and Colmts Gven any daram n a cateory C: U Z cone over the daram s a choce o morhsms rom Z to each object n the daram, such that all newly ormed tranles commute: Z U Z lmt o the daram s a cone that s unversal,.e. ven any comettor Q (another canddate), another cone over the same daram, there exsts a unque ψ : Q Z such that all tranles ncludn ψ commute. I U s any object n the daram and : Z U s the morhsm n the unversal cone, and : Q U s the morhsm n the comettor, then = ψ!ψ Z Q U U Z cocone s lke a cone but wth arrows reversed. colmt s a unversal cocone. 15

16 Darams LIMITS COLIMITS bnary roduct bnary coroduct equalzer coequalzer ullback C C C C ushout termnal object 1 ntal object 0 What s a lmt o the emty daram? It s an object Z such that or all objects Q there exsts a unque ψ : Q Z. Ths s called a termnal object. In Set, any 1-element set s a termnal object. In Vect k, any 0-dmensal vector sace s a termnal object. In Rn, the zero rn, whch s the unque rn (u to somorhsm) consstn o one element s a termnal object. Smlarly, an ntal object Z s one such that or any object Q, there exsts a unque ψ : Z Q In Set, the emty set s an ntal object. In Vect k, any 0-dmensonal vector sace s an ntal object. In Rn, the rn o nteers Z s an ntal object. In any abelan cateory, ntal objects are termnal and vce-versa. 2 Equalzers, Coequalzers, Pullbacks, and Pushouts (Week 3) 2.1 Equalzers Denton 2.1. n equalzer s a lmt o ths daram: Prooston 2.1. In Set, the equalzer o s 16

17 Z q q = = wth Z = {a (a) = (a)}. where : Z has (a) = a or all a Z (It s an ncluson), and q s orced to be =. Remark. Snce q s determned by, we usually don t draw t, and wrte an equalzer lke Z colmts.. Smlarly, or lots o other lmts and Proo. We need to check that ths cone s unversal, so take a comettor: Q Z We want to show there exsts a unque ψ : Q Z makn everythn commute: ψ =. Snce (a) = a or all a, ( ψ)(q) = ψ(q) or all q Q. Thus, ψ = smly says ψ(q) = (q) or all q Q. Thus, there exsts a unque ψ makn everythn commute, namely ψ =. Prooston 2.2. In Gr, bgr, or Vect k, the equalzer o ker( ). Remark. ker( ) = {a (a) = (a)} s Proo. The same as beore. Prooston 2.3. I Z s an equalzer then s monc. Moral: moncs and lmts et alone well; ecs and colmts do too. Proo. ssume we have an equalzer. To check that s monc, we consder: h Z k and show h = k = h = k. s a comettor to Z. Snce Z s unversal, there exsts a unque ψ : Z makn everythn commute, so ψ = h = k. 17

18 2.2 Coequalzers Denton 2.2. coequalzer o daram..e. Z s a unversal cocone over ths (commutes) s.t. we have a comettor Z there exsts a unque ψ : Z Q makn everythn commute. Q Prooston 2.4. In Set, the coequalzer o s Z where Z = / where s the nest equvalence relaton s.t. (a) (a) or all a and mas b to ts equvalence class [b]. Proo. = wth ths denton, so ths s a cocone. Why s t unversal? Why does there exst a unque ψ : Z Q makn ths daram commute? Z Q!ψ To commute, we need: ψ = ψ((b)) = (b) ψ([b]) = (b) b Ths shows ψ s unque t exsts; to show t exsts, we need to check t s well-dened: I [b] = [b ] we need to show (b) = (b ). Snce [b] = [b ], ether b = b, or (a) = b and (a) = b or some a. Snce = or all a, the ma s well-dened. Prooston 2.5. In bgr or Vect k, the coequalzer o coker( ) = /m( ). s Prooston 2.6. I Z s a coequalzer, s ec. Proo. Same as roo o the dual rooston or equalzers. 18

19 2.3 Pullbacks Denton 2.3. The lmt o ths daram: s called a ullback, and denoted: C The object here, tmes over C, or the bered roduct, and we only need to draw ts morhsms to and called rojectons. We wrte: q C C Z when Z s a ullback. C Prooston 2.7. In Set, the ullback o C wth C = {(a, b) (a) = (b)} : C q : C s (a, b) a (a, b) b Proo. Ths s clearly a cone: to show t s unversal, use the next Pro. Prooston 2.8. Gven C the equalzer exsts:, the roduct exsts and Z π 1 π 2 C π 1 where : Z s the equalzer o C π 2, then ths s a ullback: π 1 Z π 2 C 19

20 2.4 Pullbacks and Pushouts Prooston 2.9. To comute a ullback o C a roduct o and : t suces to take π 1 π 2 C and then orm the equalzer o: Z π 1 π 2 C ullback: vn the desred π 1 Z π 2 C Proo. Note the last square commutes snce π 1 = π 2, so t s a canddate or ben the ullback. To show t s unversal, consder a comettor: Q!ψ q Z only lttle square does not commute. π 1 π 2 C How do we show there exsts a unque ψ : Q Z makn the newly ormed tranle commute? y the unversal roerty o the roduct, we et: Q q π 1 π 2 C makn ths commute. Why s Q a comettor? We need to show π 1 ψ = π 2 ψ. π 1 ψ = = q = π 2 ψ (by varous comm. darams) 20

21 y the unversal roerty o the equalzer, there exsts a unque ψ : Q Z makn ths daram commute: ψ Z C Q ϕ π 1 π 2 In artcular, ϕ = ψ. Why does ths mly: 1. π 1 ψ = 2. π 2 ψ = q 3. a unque ψ makn (1) and (2) true. For (1) and (2), t suces to show π 1 ψ = and π 2 ϕ = q, but we already had ths by the unversal roerty o the roduct. Exercse 1. check (3). Cateory theory makes trval thns trvally trval. - Mchael arr I m content to let them be trval. - Tmothy Gowers 2.5 Lmts or all nte darams cateory has lmts or all nte darams and only t has: roducts equalzers termnal object 1 Prooston I ths s a ullback: C and s a mono, then s a mono too. q C Proo. ssume s a mono. Show s a mono: 21

22 h C k q C Need: h = k = h = k h = k = h = k = q h = q k (by assocatvty and commutatvty o daram.) = q h =q k (snce s mono.) Note s a comettor to the ullback:!ψ q h=q k q h = q h = q k h= k C So there exsts a unque ψ : C makn ths commute. oth h and k do make t commute, so h = k. Prooston Gven: C D E F 1. I and are ullbacks, so s the combned square. 2. I and are ullbacks, so s. 3 Week Mathematcs etween Cateores Recall that ven cateores C and C a unctor F : C D s a ma sendn objects c C to objects F(c) D, morhsm : c c n C to morhsm F( ) : F(c) F(c ) n D reservn comoston F( ) = F( ) F( ), and denttes F(1 c ) = F(1 F(c) ). There are many "oretul unctor" on rom cateores o "ancy" mathematcal adets to cateores o less ancy ones, orettn some extra roertes, structure or stu. 22

23 Rn Vect k U 3 U 4 bgr U 2 Gr To Set 2 U 1 U 5 U 6 Examle 3.1. U 1 : Gr Set sends any rou G to ts underlyn set, and any homomorhsm : G G to ts underlyn uncton. Examle 3.2. Gven cateores C and D, there s a cateory C D, where objects are order ars (c, d) wth c C, d D, and morhsm are order ars (, ) wth a morhsm n C and a morhsm n D: ven : c c n C and : d d n D then (, ) : (c, d) (c, d ). We dene (, ) (, ) = (, ). In act C D s the roduct o the objects C, D Cat, whch s the cateory wth (small) cateores as objects unctors as morhsms Set mon other thns ths means we have rojectons C D q C D Set s a lare cateory but we can stll dene Set 2 = Set Set wth ars o sets as objects. In the chart, let U 6 : Set 2 Set, (S, T) S be the rojecton onto the rst comonent. Functons can be nce n two ways: one-to-one and onto. Functors can be nce n three ways: Denton 3.1. unctor F : C D s athul or any c, c C, F : hom(c, c ) hom(f(c), F(c )) s one-to-one. Denton 3.2. unctor F : C D s ull or any c, c C, F : hom(c, c ) hom(f(c), F(c )) s onto. Denton 3.3. unctor F : C D s essentally surjectve or any d D, there exsts c C such that F(c) = d, meann there exsts an somorhsm : F(c) d n D. Examle 3.3. Comare FnVect R (nte dmensonal vector saces) to ths cateory C, wth 23

24 {0}, R, R 2,... as objects, all lnear mas between these as morhsms There s a unctor F : C FnVect R, dened n objects as and smlarly or morhsms R n R n : R n R n : R n R n Ths s athull and ull, not surjectve on objects, but essentally surjectve. Later we ll dene "equvalent" cateores and see that F : C FnVect R s athull, ull and essentally surjectve then C and D are equvalent. Denton 3.4. We say: unctor U : C D orets nothn t s athull, ull, and essentally surjectve. unctor U : C D orets (at most) roertes t s athull and ull. unctor U : C D orets (at most) structure t s athull. In eneral we say U orets (at most) stu. Examle 3.4. U 1 : Gr Set orets (at most) structure. It s athull: ven, : G G n Gr, U 1 ( ) = U 1 ( ) =. It s not ull: there are usually unctons : U 1 (G) U 1 (G ) that don t come rom rou homomorhsm, e. : (h) = () (h) or (1) = 1. Examle 3.5. U 2 : bgr Gr orets (at most) roertes: the commutatve law s orotten. Ths s athull and also ull: you have any rou homomorhsm : U 2 () U 2 ( ) then U 2 ( ) = or some homomorhsm o abelan rous :. ut t s not esentally surjectve, G s nonabelan, G U 2 () or any bgr. Examle 3.6. U 6 : Set 2 Set orets stu: U 6 (S, S ) = S (t oret the second set n the ar). Techncally t s not athull: we can have 2 derent morhsms (, ), (, ) : (S, S ) (T, T ) wth U 6 (, ) = = U 6 (, ). In our chart, every oretul unctor U : C D has a "let adjont" F : D C whch "reely creates" stu, structure or roertes that U orets. Examle 3.7. S, F 1 (S). F 1 : Set Gr takes a set S and orm the ree roduct on F 2 : Gr bgr abelanzes any rou G, ormn F 6 : Set Set 2, S (S, ) F 2 (G) = G < xyx 1 y 1 > To dene adjont unctors (and many other thns) we need... 24

25 3.2 Natural Transormatons Gven two unctors F, G : C D, we can dene a natural transormaton α : F G. F(x) F( ) F(y) F x y α x α y G G(x) G( ) G(y) Denton 3.5. Gven unctors F, G : C D a transormaton α : F G s a uncton sendn each object x C to a morhsm α x : F(x) G(x). We say α : F G s a natural transormaton or each morhsm : x y n C ths square commutes: F(x) α x G(x) F( ) G( ) F(y) α y G(y) Prooston 3.1. Gven cateores C and D there s a cateory, the unctor cateory D C wth: objects ben unctors F : C D morhsms ben natural transormaton α : F G. In D C we comose α : F G, β : G H to et β α : F H as ollows: (β α) x : F(x) H(x) or all x C s ven by β x α x. In D C the dentty 1 F : F F, (1 F ) x : F(x) F(x) s ven by 1 F(x). Proo: We ll check that the comoste β α s natural. Gven : x y n C, we want the ollown daram to commute: (β α)x F(x) H(x) F( ) H( ) F(y) H(y) (β α) y We have F(x) F( ) F(y) α x α y (β α) x G(x) G( ) G(y) (β α) y β x H(x) H( ) β H(y) 25

26 Snce the to and botton commutes (α and β are natural), the whole daram commute. Remark. So just as ven two sets and, there s a set o all unctons :, ven two cateores, there s a cateory o all unctors F :. Gven two sets and they have a roduct: = {(x, y) : x, y } Notce = but we want to be honest = and there s a secc "ood" somorhsm α, :, ((x, y) (y, x)). It s ood because t s natural n the sense we just dened. There are two unctors rom Set 2 Set, F : (, ) G : (, ) and α s a natural transormaton rom F to G. In act t s a "natural somorhsm": Denton 3.6. I F, G : C D are unctors and α : F G s a natural transormaton, we say α s a natural somorhsm α x : F(x) G(x) s an somorhsm or all x C. Prooston 3.2. α : F G s a natural somorhsm t have and nverse α 1 : G F n D C. Proo: Key Idea:(α 1 ) x = (α x ) 1. Prooston 3.3. Suose C s a cateory wth bnary roduct :any ar o object have a roduct. Then we can choose, or any ar x, y C, a secc roduct:, q, and then there s a unctor : C 2 C, (, ). In act there are two unctors: = F : C 2 C, (, ) G : C 2 C, (, ) and ths are naturally somorhc. We say "roducts are commutatve u to natural somorhsm" Remark. lso roducts are assocatve u to natural somorhsms. α,,z : ( ) Z ( ) Z C 3 α,,z C (Just kee usn unversal roertes o roduct.) 26

27 Denton 3.7. cartesan cateory s a cateory wth bnary roducts and a termnal object. (I.e. t s a cateory where any nte set o objects have a roduct- a nte roduct cateory ) One can show that n a cartesan cateory we have natural somorhsms. l : 1. r : 1. ll ths work smlarly n a cat wth nte coroducts β, : + +. α,,z : ( + ) + Z + ( + Z). l : 0 +. r : + 0. In case C = FnSet (nte sets and unctons) ths ves laws o arthmetc: N s the somorhsm clases o objects n FnSet. nother examle: Examle 3.8. rou s a cateory G wth one object and all morhsms nvertble: 1= 3 2 What s a unctor F : G Set? 2 1= 3 x F F( 2) Z 3 F(1) F(x) F( ) G Set F cks out a set F(x) = and or each rou element t cks out a uncton F( ) : such that F( ) = F( )F( ) and F(1) = 1. So s a set acted by the rou G, or a G-set. So: a unctor F : G Set s a G-set. What s a natural transormaton between 2 such unctors?. 27

28 4 Mas etween Cateores 4.1 Natural Transormatons Examles o natural transormatons Examle 4.1. We saw that a 1-object cateory G wth all morhsms nvertble s a rou. We saw that a unctor F : G Set s a G-set: a set F( ) wth unctons F() : S S or all G such that F( ) = F() F( ) and F(1 ) = 1 F( ) Gven two unctors F, F : G Set, what s a natural transormaton α : F F? It s called a ma o ma o G-sets or G-equvarant ma, but let s draw one. F(1 ) 1 F F() F( ) F( ) ( ) α F ( ) F F () F ( ) F (1 ) It s a uncton α : F( ) F ( ) such that or all morhsms G, we have F () α = α F(). F( ) F ( ) F() α F( ) F ( ) Examle 4.2. Two sets are somorhc there are unctons F : and G : such that G F = 1 and F G = 1. Gven F, when can you nd such a G? I and only F s one-to-one and onto. 4.2 Equvalence o Cateores α F () Denton 4.1. n equvalence o cateores C and D conssts o: unctors F : C D and G : D C. natural somorhsms α : G F 1 C and β : F G 1 D. 28

29 We say that F and G are weak nverses. We say C and D are equvalent there exsts an equvalence between them. Theorem 4.1. unctor F : C D s art o an equvalence (F,G,α,β) and only F s athul, ull, and essentally surjectve. I such a G exsts, t may not be unque, but G was another one, t s naturally somorhc to G. 4.3 djunctons What are adjunctons? Recall an examle: U : Gr Set sendn each rou G to ts underlyn set U(G). F : Set Gr sendn each set S to the ree rou on t F(S). We say that U s the rht adjont o F, or synonymously, F s the let adjont o U. The basc dea s that morhsms rom the object F(S) to the object G n Gr are n 1-1 corresondence wth morhsms rom the object S to the object U(G) n Set. Gven a uncton : S U(G), we et a homomorhsm : F(S) G, the unque one such that (s) = (s) or all s S F(S). nd conversely, ven a homomorhsm h : F(S) G, we et h : S U(G) by restrctn h to S F(S). The usual cture looks lke ths: ncluson S F(S)! G We reer to say that there s a bjecton Hom Gr (F(S), G) = Hom Set (S, U(G)). Note that F s on the let o Hom Gr (F( ), ) and G s on the rht o Hom Set (, G( )). To dene adjont unctors, we need to say that ths knd o bjecton s natural. What unctors ve Hom Gr (F(S), G)? They must be two unctors rom Set Gr to Set. On objects, these do: What s the hom don here? (S, G) Hom Gr (F(S), G) (S, G) Hom Set (S, U(G)) Prooston 4.1. For any cateory, there s a unctor, called the hom unctor; Hom : C o C Set whch sends each object (, ) to the set Hom C (, ) Remark. Here, C o s the ooste o C: the cateory wth one morhsm o : or each : n C, and o o = ( ) o wth the same dentty morhsms. Proo. Sketch o roo: We need to dene Hom : C o C Set on morhsms. Gven a morhsm n C o C, ϕ : (, ) (, ). That s, a ar o morhsms: o : n C o and : n C. We need to dene a morhsm,.e. a uncton, Hom(ϕ) : Hom C (, ) Hom C (, ) n Set. 29

30 Gven h Hom C (, ), what s Hom(ϕ)(h) Hom C (, )? It s h. Thus, the hom unctor Hom : C o C Set wll not only descrbe hom sets, but also comoston n C. Then check t s really a unctor: For examle, check t reserves comoston. h h =Hom(ϕ)(h) Gven unctors F : C D and U : D C, how can we say that the somorhsm Hom D (F(), ) = Hom C (, U()) s natural? D o D Hom Set F o 1 D s α Hom C o D 1 C U C o C Examles o djunctons Let s at rst downlay the naturalty condton and look at examles ocusn on bjectons. Examle 4.3. The oretul unctor U : Gr Set sends each rou G to ts underlyn set U(G). The ree unctor F : Set Gr sends each set S to the ree rou on t F(S). Snce these two unctors orm an adjuncton between the cateores Gr and Set, we have bjectons or every G Gr and S Set: Hom Gr (F(S), G) = Hom Set (S, U(G)) These bjectons let us turn any uncton : S U(G) nto a homomorhsm = α 1 S,G ( ) : F(S) G. nd conversely; any homomorhsm h : F(S) G comes rom a uncton h = α S,G (h) : S U(G). Examle 4.4. Does the oretul unctor U : Vect k Set sendn each vector sace V over a eld K to ts underlyn set U(V) have a let adjont? es, or any set S, there s a vector sace F(S) whose bass s S, where the sums are ormal exressons: F(S) = { c s c K, only ntely many nonzero} s S 30

31 What does F : Set Vect k do to a morhsm : S T n Set? It should ve a lnear ma F( ) : F(S) F(T). What s t? It s: F( )( c s ) = c (s ) s S s S Check F s a unctor: That s, check that denttes F( ) = F() F( ) and F(1 S ) = 1 F(S) hold. Exercse 2. Why s the unctor F o the last examle, let adjont to U? Frst, or all V Vect K and Set, we need the ollown bjectons to hold (and check they re natural): Hom Vectk (F(S), V) = Hom Set (S, U(V)) Gven a uncton : S U(V), we need a lnear ma : F(S) V n some natural way. Try ( s S c s ) = s S c (s ). Conversely, ven a lnear ma l : F(S) V, we need a uncton l : S U(V). Try l(s) = l(s). Check these mas are nverses: ( ) = and ( l) = l, so that we have a bjecton: Hom Vectk (F(S), V) = Hom Set (S, U(V)) Examle 4.5. To dream u a let adjont o the oretul unctor U : To Set sendn each toolocal sace to ts underlyn set U(), we need to thnk o ways to turn a set S nto a toolocal sace. One way we can do ths s to ve ths set the dscrete tooloy, where you ve S as many oen sets as ossble, so every subset s oen. nother way we can do ths s to ve ths set the ndscrete tooloy, where you ve S as ew oen sets as ossble. The let adjont o U : To Set, say L : Set To, must have have the ollown bjectons or every To and S Set: Hom To (L(S), ) = Hom Set (S, U()) That s, contnuous mas : L(S) are the same as unctons : S U(). To make ths true, L(S) should have as many oen sets as ossble, so L(S) s S wth the dscrete tooloy. The rht adjont o U : To Set, say R : Set To, must have have the ollown bjectons or every To and S Set: Hom Set (U(), S) = Hom To (, R(S)) That s, contnuous mas h : R(S) are the same as unctons h : U() S. To make ths true, R(S) should have as ew oen sets as ossble, so R(S) s S wth the ndscrete tooloy Daonal Functor Suose C s any cateory. There s always a unctor : C C C called the daonal wth: () = (, ) or all objects C ( ) = (, ) : (, ) (, ) or all objects, C 31

32 Prooston 4.2. I C has bnary roducts, then the unctor : C C C s the rht adjont o : C C C. Remark. In act, the converse s true: has a rht adjont and only C has bnary roducts, and the rht adjont s. Proo. Sketch o roo: For starters, we need bjectons or all objects,, Z C: Hom C C ( (), (, Z)) = Hom C (, Z) snce a morhsm rom (, ) to (, Z) s a ar: :, : Z, or the let sde we have: Hom C C ( (), (, Z)) = Hom C C ((, ), (, Z)) = Hom C (, ) Hom C (, Z) So what we need to show s: Hom C (, ) Hom C (, Z) = Hom C (, Z) Indeed, the unversal roerty o the roduct says:!ψ Z q Z So (, ) ves ψ and conversely ψ ves = ψ and = q ψ, wo we have a bjecton: Hom C (, ) Hom C (, Z) = Hom C (, Z) (, ) ψ Prooston 4.3. I C has bnary coroducts, then the unctor + : C C C s the let adjont o : C C C. Remark. In act, the converse s true: has a let adjont and only C has bnary coroducts, and the let adjont s +. Proo. Sketch o roo: For starters, we need bjectons or all objects,, Z C: Hom C ( + Z), ) = Hom C C ((, Z), ()) snce a morhsm rom (, ) to (, Z) s a ar: :, : Z, or the rht sde we have: Hom C C ((, Z), ()) = Hom C C ((, Z), (, )) = Hom C (, ) Hom C (Z, ) 32

33 So what we need to show s: Hom C ( + Z, ) = Hom C (, ) Hom C (Z, ) Indeed, the unversal roerty o the coroduct says:!ψ + Z j Z So (, ) ves ψ and conversely ψ ves = ψ and = j ψ, wo we have a bjecton: Hom C ( + Z, ) = Hom C (, ) Hom C (Z, ) ψ (, ) roduct (an examle o a lmt) s an examle o a rht adjont - t s easy to descrbe morhsms on nto t. coroduct (an examle o a colmt) s an examle o a let adjont - t s easy to descrbe morhsms on out o t. 5 Darams n a Cateory as Functors Last tme, we saw that C has roducts, the unctor : C 2 C s a rht adjont to the daonal unctor : C C 2 c (c, c). Smlarly, the unctor + : C 2 C, C has coroducts, s a let adjont to. Thus, : Vect 2 k Vect k s both let and rht adjont to : Vect 2 F Vect F. In act, a cateory has lmts, these lmts ve a rht adjont to some unctor: lmts are rht adjonts colmts are let adjonts We oten thnk about the lmt o a daram n a cateory C. What s a daram n C, really? k c c c c Namely, t s a collecton o objects and morhsms between them. We can make t nto a subcateory o C: 33

34 k 1 c 1 c c c c c k 1 c 1 c We re oten nterested n darams o some shae, lke ullbacks: These shaes can be nterreted as cateores: Let D be any cateory: we ll take ths as our daram shae. What s a D-shaed daram n some cateory C? It s a unctor F : D C: F F( ) F( ) F( ) F( ) When we take the lmt o ths daram, we et an object lmf C (dened u to somorhsm). What s the rocess that takes us rom F : D C to lmf C? The key s that there s a cateory C D wth: objects ben unctors F : D C. G morhsms ben natural transormatons α. D α C These morhsms look lke: F 34

35 F( ) F( ) F F(x) F( ) x α x G( ) G( ) G G(x) G( ) When we take a lmt o F : C D, we study dones over F. Denton 5.1. cone over F s a natural transormaton α : G F where G sends every object o D to some object o C, and G sends every morhsm o D to the dentty morhsm o that object. F( ) F( ) F F(x) F( ) α x α x G( ) G( ) G G(x) G( ) Here, G : D C was determned by the object x va the above rece. It turns an object x C nto an object G C D. So ths rece should be a unctor D : C C D. C (x) s the daram: x x D (x) 1 x 1 x 1 x x x So a cone over F wth aex x C s a natural transormaton α : C (x) F. What s the lmt o a daram? I F C D, t s a unversal cone over that daram. 35

36 !ψ lmf x Remember U s the rht adjont o F : Hom D (F(x), y) = Hom C (x, U(y)) So adjont unctors are about convertn one knd o morhsms nto another n a bjectve way, and that s what we re don when we re statn the unversal roerty: morhsms ψ : q lmf n C. cones over F wth aex q,.e. natural transormatons α : D (q) F. (morhsms α rom D (q) to F n C D.) So: Hom C D( D (q), F) = Hom C (q, lmf) So t looks lke we have lm : C D C whch s rht adjont to D : C C D. Ths s true, you need to check that the bjecton above s natural to nsh the roo o: Theorem 5.1. I C has all lmts or D-shaed darams, then we have a unctor lm : C D C whch s rht adjont to D : C C D. Conversely, D : C C D has a rht adjont, then ths ves lmts o D-shaed darams n C. What choce o D ves the case o bnary roducts (a secal case o lmts)? D (C C ) C C C C α q G C C Here, D has two objects and only dentty morhsms, so we could call t 2, so C D = C 2 and : C 2 C s rht adjont to 2 = : C C 2. Smlarly, Theorem 5.2. I a cateory C has colmts o all D-shaed darams, there s a unctor colm : C D C whch s let adjont to D : C C D. Conversely, D : C C D has a let adjont, then ths ves lmts o D-shaed darams n C. Note: α Hom C D(F, D (q)) s a cocone: 36

37 q Theorem 5.3. Let adjonts reserve colmts; rht adjonts reserve lmts. Proo. Sketch o roo: Let s show that F : C D s a let adjont to U : D C, then F reserves colmts. For concreteness, let s show F reserves ushouts - eneral case s analoous. So suose we have a ushout n C: a b c Here, x s the aex o a cocone on the daram we re takn a colmt o, and the unversal roerty holds. The clam s that alyn F to ths unversal cocone ves a unversal cocone n D: x F(a) F(b) F(c) F(x)!ψ Q Choose a comettor cocone wth aex Q. We need to show!ψ : F(x) Q makn the newly ormed tranle commute. We can look at U(Q) : C: a b c x U(Q) 37

38 Snce F s let adjont to U, we have: Hom D (F(x), Q) = Hom C (x, U(Q)) So to et ψ : F(x) Q, let s nd ϕ : x U(Q). U(Q) becomes a comettor due to the adjontness o F and U, e.. Hom D (F(a), Q) = Hom C (a, U(Q)) For some reason, the tranles nvolvn U(Q) commute snce those nvolvn Q commute. So U(Q) s a comettor. Thus,!ϕ : x U(Q) makn the newly ormed tranles commute. a b c x!ϕ U(Q) Ths ves us ψ : F(x) Q, check t makes ts newly ormed tranle commute and s unque (snce ϕ s). Examle 5.1. F : Set Gr reserves colmts, e.. coroducts, so F(S + T) = F(S) + F(T). Here, S + T s the dsjont unon o S and T, F(S + T) s the ree rou wth elements o S + T as enerators, and F(S) + F(T) = F(S) F(T) s the ree roduct o F(S) and F(T). Examle 5.2. U : Gr Set reserves lmts, e.. roducts, so U(G H) = U(G) U(H) where G H s the usual roduct o rous G H. Theorem 5.4. The comoste o let adjonts s a let adjont. The comoste o rht adjonts s a rht adjont. F F Proo. Suose we have unctors C D E and F and F are let adjont o unctors U and U C D E U U. We ll show that F F : C E s the let adjont o U U : E C. We want a natural somorhsm: Here s how we et t: Hom E (F F(c)), e) = Hom C (c, U U (e)) Hom E (F F(c), e) = Hom D (F(c), U (e)) Snce F s let adjont to U Hom D (F(c), U (e)) = Hom C (c, U U (e)) Snce F s let adjont to U Examle 5.3. F F s let adjont to the oretul unctor U U rom Rn to Set. 38

39 Rn F U F F bgr U U F U Set Startn rom the emty set (the ntal set) we et F( ) = {0} (the trval abelan rou, whch s the ntal abelan rou) and then F (F( )) = Z (the rn o nteers, whch s the ntal rn). Startn rom a one-element set {x}, we et F({x}) = {..., x, 0, x, x + x,... } = Z and then F (F(x)) = Z[x], the rn o olynomals n x wth nteer coecents. 5.1 Unts and Counts o djunctons F Suose we have C D U and d D, we have: wth F let adjont to U. So that or all c C Hom D (F(c), d) = Hom C (c, U(d)) We can aly ths bjecton to an dentty morhsm and et somethn nterestn. We can do ths d = F(c). Hom D (F(c), F(c)) ϕ Hom C (c, U(F(c))) 1 F(c) ϕ(1 F(c) ) ϕ(1 F(c) ) s called the unt, ι c : ι c : c U(F(c)) We can also aly ϕ 1 to an dentty c = U(d). Hom D (F(U(d)), d) ϕ 1 Hom C (U(d), U(d))) ϕ 1 (1 F(c) ) 1 F(c) ϕ 1 (1 U(d) ) s called the count, ɛ d : ɛ d : F(U(d)) d 39

40 These ve varous amous morhsms. Examle 5.4. Gven any set S, we et a unt: F : Set Gr U : Gr Set ι S : S U(F(S)) Ths s the ncluson o the enerators : elements o S are enerators o F(S). Gven a rou G, we et a count: ɛ G : F(U(G)) G 1 ±1 2 ±1 n ±1 1 ±1 ±1 2 n ±1 ormal roduct n F(U(G)). actual roduct n G. The counts convert ormal exressons nto actual ones. 6 Cartesan Closed Cateores ny cateory has a set Hom(, ) o morhsms rom one object to another object, but n a cartesan closed cateory (or ccc) you also have an object o morhsms rom to. Examle 6.1. I C = Cat, Hom C (, ) s the set o unctors F :, whle s the cateory o unctors F : and natural transormatons between them. In eneral, you can et Hom C (, ) rom but not vce versa. We call Hom C (, ) the homset or external hom (t lves outsde o C, n Set), and the exonental or nternal hom (snce t lves nsde C). Internalzaton s the rocess o takn math that lves n Set and movn t nto some cateory C. Examle 6.2. In Set you can dene a rou to be an object G Set wth morhsms: m : G G G nv : G G : 1 G Multlcaton Inverses The dentty-assnn ma. It mas the one element o 1 to the dentty element n G. assocatve law: 40

41 G G let and rht unt laws: m 1 G m G G G G 1 G m m G G 1 G G G 1 (, ) (, ) 1 G m 1 G 1 G G G G G nverse laws: nv 1 G m G G G G G m! G G G G G m (1, ) 1 = = 1 (, 1) 1 G nv ll these darams make sense n any cartesan cateory (=cateory wth nte roducts = cateory wth bnary roducts and termnal object). So we can dene a rou nternal to C or rou n C usn these axoms whenever C s cartesan. For examle: I C = To, a rou n C s called a toolocal rou. I C = D, a rou n C s called a Le rou. I C s the cateory o alebrac varetes, a rou n C s called an alebrac rou. Puzzle: I C = Gr, a rou n C s a very amous thn. What s t? 6.1 Evaluaton and Coevaluaton n Cartesan Closed Cateores Recall a cartesan cateory C s a ccc or any C, the unctor C has a rht adjont: Hom C (, Z) = Hom C (, Z ) F ny adjuncton C D U has a unt and count: ι : UF ɛ : FU C D Now we have an adjuncton C C ι : ( ) 41

42 ɛ : C The second one s called evaluaton: n Set ɛ : (, y) (y) The rst one s called coevaluaton: n Set ι : ( ) x (x)(y) = (x, y) So we have analoous o these n any ccc Internalzn Comoston In any cateory, we have comoston: : Hom(, Z) Hom(, ) Hom(, Z) (, ) In a ccc, we can nternalze ths and dene nternal comoston : : Z Z Hom(Z, Z ) = Hom(Z, (Z ) () ) = Hom(Z, Z) So we et orm a morhsm: whch we ndeed have n any ccc: : Z Z Z 1 Z ɛ Z ɛ Z Ths s just an nternalzed way o sayn the old denton o comoston: ( )(x) = ((x)) Emly Rehl, Cateores n Context, Dover Pub. ree on the web. 42

43 6.2 Elements Sets have elements, but what about objects n other cateores? Elements o a set are n 1 1 corresondence wth unctons : 1, where 1 s a termnal object n Set (1 = a one element set). So: Denton 6.1. I C s a cateory wth a termnal object, an element o an object C s a morhsm 1. We dene the set elt() to be Hom(1, ). Examle 6.3. I C = To, elt() = {contnuous mas : { }, where { } s the one-ont sace,.e. the termnal object n To}. In act, elt() s n 1 1 corresondence wth the underlyn set o : Gven x, : { } where x, and conversely any such ( ). Examle 6.4. I C = Gr, elt(g) = {homomorhsms : 1 G, where 1 s the trval rou,.e. the termnal object n Gr}. So elt(g) has just one element: there s just one homomorhsm : 1 G, snce 1 s also ntal! Examle 6.5. I C = Cat, elt(d) = {unctors : 1 D, where 1 s the termnal cateory n Cat}. unctors : 1 D are n 1 1 corresondence wth the objects o D. So elt(d) = {objects n D} F 1 F( ) 1 F( ) Here, as n the revous examle, elt orets a lot o normaton: elt( ) = elt( ) 7 Week 9 Prooston 7.1. Suose C s a cateory wth termnal object 1 C. Then there s a unctor elt : C Set wth elt() = Hom(1, ), C and ven any morhsm : nc, elt() : elt() elt() s dened as ollows: elt() = 1 43

44 h Proo: elt reserve comoston: ven Z we need Gven elt() we have elt(h ) = elt(h) elt() 1 elt(h ) = (h ) = h ( ) = h (elt() ) = elt(h)(elt()( )) Smlarly elt(1 x ) = 1 x =, or all elt(). So elt(1 x ) = 1 elt(). Examle 7.1. elt : C Set may not be ahtull,.e we can have two derent morhsms, : n C wth elt() = elt( ). I C = Gr, we saw elt(g) = 1 Set or all G, so any homomorhsm h : G G wll be et sent to a uncton elt(h) : 1 1, but there s only one o these. Prooston 7.2. I C s a cartesan cateory elt : C Set reserve nte roducts. Proo: I easy to show elt reserve the termnal object: 1 C then elt(1) = { : 1 1} s one-element set, so t s termnal n Set. Why does elt reserve bnary roducts? Suose, C, then ther roduct s a unversal cone Z h q To show elt reserve roducts, we need ths cone s unversal n Set: Choose a comettor: elt() elt() elt( ) elt(q) elt() 44

45 Q ψ elt( ) q elt() elt() Want!φ : Q elt( ) makn the newly ormed tranles commute. : Q elt() sends any a Q to a ont (a) elt() = h : 1, so (a) : 1. Smlarly (a) : 1. We want to dene ψ : Q elt( ); ths wll send any a Q to ψ(a) : 1. y the unversal roerty o, or each a Q!ψ(a) : 1 so that ths commutes 1 (a)!ψ (a) q Dene ψ ths way, check that (*) commutes, and moreover (*) commutn orces us to choose ths ψ, so ψ s unque. What C s a ccc? then Snce 1 = so: hom(, ) = hom(1, ) = hom(1, ) = elt( ) 1 1 α ve us a bjecton α α 1 α 1 hom( ) = hom(1, ) α α 1 The moral: we can convert the hom-object C nto the hom-set hom(, ) Set by takn elements. 45

46 Gven : n hom(, ) we can convert t nto an element o called the name o : : 1. Conversely, any elemnet o s the name o a unque morhsm :. In unctonal rorammn, objects are data tyes, morhsms are rorams and any roram : have a name elt( ). 7.1 Subobjects Denton 7.1. In a cateory C s an equvalence class o monomorhsms :, where monos :, j : are equvalent there s an somorhsms : so that ths commutes: j Examle 7.2. I C = Set, subobjects o Set corresonds to subsets o. Gven a monomorhsm : we et a subset m(). ny subset S arse n ths way va the ncluson: : S s s ths has m() = S. Fnally, ven monos : and j : that dene the same subset m() = m(j), then there exsts a bjecton : so that j coomutes, namely = (j m(j) ) 1. e Examle 7.3. In Grah,how many subobjects does ths rah?: v w Here they are e 46

47 v v v v e w w w w v e e w e ny object ve a subobject o tsel:1 : s a monomorhsm. ( rah s a ar o unctons E V s t Prooston 7.3. In Set, subobjects o S Set are n 1-1 corresondence wth unctons : S 2 where 2 = {F, T}. Proo: Subobjects o S are just subsets S. ny such subset has a characterstc uncton : S 2 ven by { F s / χ(s) = T s Conversely, ven χ : S 2,let = χ 1 (T) = {s S : (s) = T} Rouhly, a "subobject classer" n a cateory C s an object Ω C that lays the role o 2 = {F, T}, n that subobjects o any subset S C are on to be n 1-1 corresondence wth morhsms χ : S Ω. Set has the "subobject classer" 2 = {F, T}. What does ths really means?. Frst,there s a uncton called true:t : 1 2 rom 1 = { } to 2 ven by t( ) = T 2. For any set there s a unque uncton! : 1 snce 1 s termnal. I clam that or any monomorhsm : (that s a 1-1 uncton), there exsts a unque uncton S : 2 called the characterstc uncton o, such that:! E 1 E 2 t 47

48 s a ullback. χ,n more amlar terms, wll be the characterstc uncton o the subset m(), but we call t the characterstc ucnton o the monomorhsm. Frst Let s show that ths χ : { T x m() χ (x) = F x / m() Let Q be a comettor Q!Q φ!q! 1 t χ 2 Then show!ψ : Q makn the newly ormed tranles commute. Snce Q s a comettor: χ ( (q)) = t(! Q (q)), q Q = t( ) = T (usn the denton o χ ) (q) m(). So snce s one-to-one, or each q Q,!a wth (q) = (a). So dene φ : Q by φ(q) = a. Ths makes = ß ψ and t s the unque φ : Q that does so (snce s one-to-one). The other newly ormed tranle automatcally commutes: Q φ!q χ 1 you can also check that χ : 2 s the unque morhsm rom to 2 that makes the square a ullback. So eneralzn: Denton 7.2. Gven a cateory C wth a termnal object, a subobject classer s an object Ω C wth a morhsm t : 1 Ω such that : or any monomorhsm : there exsts a unque χ : Ω such that ths square s a ullback:! 1 t Ω Denton 7.3. (elementary) toos s a cartesan closed cateory wth nte lmts (lmts o nte szed darams) and a subobject classer. 48

49 Grothendeck n the 1960 s ntroduced a concet o toos, now Grothendeck toos, whch s a secal case o alementary toos, as art o rovn the Wel hyothess n number theory. Later n the late 60 s and early 70 s Lawrence and Trerney smled the concet o toos to dene an "elementary toos". Examle 7.4. Examles o elementary toos 1. Set: cateory o sets and unctons. 2. FnSet: cateory o nte sets and unctons, ths doesn t have all lmts only nte lmts, so toos theory ncludes ntest mathematcs. 3. Set : cateory o sets and unctons as dened usn ZF=Zermelo-Fraenkel axoms wthout axom o choce. The axom o choce s aquvalent to: there exsts a monomorhsms : so that = 1.I ths true we say the emorhsm slts. In a eneral toos, not every emorhsms slts so the axom o choce need not hold. 4. Grahs: The cateory o rahs: E s t V 5. Prevous examle s ana secal case o a cateory Set C, where C s any cateory. These are called reshea cateores when we wrte them as Set Do (e.d = C o so D o = C) I C = 1x x y 1y then Set C = Grah x y α F F(x) αx G( ) F(y) αy G G(x) G(y) C G() Set unctor F : C Set s a rah wth E = F(x), V = F(y), s = F( ), t = F(). So a rah s an object n Set C. Smlarly, a morhsm n Set C s a morhsm between rahs. 49

50 6. nother examle o a reshea cateory s the cateory o smlcal sets: These are undamental to alebrac tooloy. 7. Preshea cateores are closely connected to cateores o sheaves, whch are also too. Sheaves are undamental to alebrac eometry. 8 Symmetrc Monodal Cateores 8.1 Guest lecture by Chrstna Osborne cateory theorst s sort o lke a socolost. He looks at mathematcal objects - he doesn t ry t oen and see how t works - but sees how t behaves n relaton to all other thns. - Chrs Heunen What s a Monodal Cateory? Denton 8.1. monod s a nonemty set G toether wth a bnary oeraton on G whch s: assocatve: (xy)z = x(yz) x, y, z G and contans a (two-sded) dentty element e G such that xe = ex = x x, y, z G Remark..e. take the denton o a rou and dro the requrement o nverses Denton 8.2. monodal cateory s a cateory C whch s equed wth: 1. tensor roduct unctor : C C C where the mae o a ar o objects (x, y) s denoted by x y. 2. unt object I. 3. For every x, y, z Ob(C), and assocatvty somorhsm a x,y,z : (x y) z x (y z), natural n the objects x, y, and z. 4. For every x Ob(C), a let unt somorhsm l x : I and a rht unt somorhsm r x : x I x, both natural n x. We urther assume the ollown darams commute or any objects w, x, y, and z: the entaon dentty: 50

51 ((w x) y) z a w,x,y d z a w x,y,z (w (x y)) z (w x) (y z) a w,x y,z a w,x,y z w ((x y) z) w (x (y z) the tranle dentty: d z a x,y,z a x,i,y (x I) y x (I y) r x d y d x l y x y Remark. When we want to emhasze the tensor roduct and unt, we denote a monodal cateory by (C,, I). Examle 8.1. (Set,, { }) Examle 8.2. (Set,, { }) Examle 8.3. (Gr,, {e}) Examle 8.4. (Hlb,, C), where the cateory Hlb has Hlbert saces as objects and short lnear mas (lnear mas o norm at most 1) as morhsms. Why s a x,y,z : (x y) z x (y z) an somorhsm and not an equalty? Let s consder the examle (Set,, { }) : ( ) Z = {(w, z) w, z Z} = {((x, y), z) x, y Z, z Z} ( Z) = {(x, w) x, w Z = {(x, (y, z)) x, y, z Z} These sets are not equal - but we can easly construct an somorhsm. Examle 8.5. How can we take a monod G and construct a monodal cateory? Frst we need a cateory C: objects: elements o G. 51

52 morhsms: dentty morhsms. We et a monodal cateory (C,, e) where s the bnary roduct o G and e s the dentty element o G. Note: In eneral: I C has roducts, we et a monodal cateory (C,, 1). I C has coroducts, we et a monodal cateory (C, +, 0). Denton 8.3. monodal cateory (C,, I) s symmetrc t addtonally s equed wth an somorhsm s x,y : x y y x or any objects x and y o C, natural n x and y, such that the ollown darams commute or all objects x, y, and z: s x,y d z (x y) z (y x) z a x,y,z a y,x,z x (y z) y (x z) s x,y z dy s x,z (y z) x y (z x) a y,z,x x I s x,i I x r x lx x x y s x,y y x d x y s y,x x y 52

53 Most o the examles o monodal cateores we have talked about are symmetrc. So what s an examle o a monodal cateory that s not symmetrc? Examle 8.6. Let R be a non-commutatve rn. The cateory R-R-bmodules wth R R as the tensor and R as the unt s an examle o a monodal cateory that s not symmetrc. Note: Let (C,, e) be the monodal cateory ven by the monod G. I G s an abelan rou, then (C,, e) s symmetrc Gon back to the denton o a symmetrc monodal cateory... Q: Why s the hexaon commutn daram sucent? There are 6 derent ways to order 3 elements. There are 2 ways o assocatn 3 elements. So there are 12 ossbltes (we would exect all o these to be somorhc). : reeat! (x y) z (y x) z x (y z) y (x z) (y z) x y (z x) a y,z,x (y z) x (z y) x y (z x) z (y x) (z x) y z (x y) 53