Category Theory PDF
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Category Theory LecturesbyPeterJohnstone NotesbyDavidMehrle [email protected] CambridgeUniversity MathematicalTriposPartIII Michaelmas2015 Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 TheYonedaLemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3 Adjunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4 LimitsandColimits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 5 Monads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 6 FilteredColimits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 7 AbelianandAdditiveCategories . . . . . . . . . . . . . . . . . . . . . 63 LastupdatedJune5,2016. 1 Contents by Lecture Lecture1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Lecture2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Lecture3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Lecture4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Lecture5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Lecture6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Lecture7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Lecture8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Lecture9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Lecture10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 Lecture11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 Lecture12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 Lecture13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 Lecture14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Lecture15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 Lecture16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 Lecture17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 Lecture18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 Lecture19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 Lecture20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Lecture21 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 Lecture22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 Lecture23 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 Lecture24 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 2 1 Introduction 1 Introduction Categorytheoryhasbeenaroundforabouthalfacenturynow,inventedinthe 1940’sbyEilenbergandMacLane. Eilenbergwasanalgebraictopologistand MacLanewasanalgebraist. Theyrealizedthattheyweredoingthesamecalcu- lationsindifferentareasofmathematics,whichledthemtodevelopcategory theory. Categorytheoryisreallyaboutbuildingbridgesbetweendifferentareas ofmathematics. 1.1 Definitionsandexamples Thisisjustaboutsettinguptheterminology. Therewillbenotheoremsinthis chapter. Definition1.1. AcategoryCconsistsof (i) acollectionobCofobjects A,B,C,... (ii) acollectionmorCofmorphisms f,g,h... (iii) twooperations,calleddomp´qandcodp´q,frommorphismstoobjects. f We write A ÝÑ B or f: A Ñ B for f P morC and dompfq “ A and codpfq“ B; (iv) anoperation AÞÑ1 fromobjectstomorphisms,suchthat AÝ1ÑA A; A (v) an operation ˝: pf,gq ÞÑ f ˝g from pairs of morphisms (so long as we have dom f “ codg) to morphisms, such that dompfgq “ dompgq and codpfgq“codpfq. Thesedatamustsatisfy: (vi) forall f: AÑ B, f1 “1 f “ f; A B (vii) compositionisassociative. If fgandgharedefined,then fpghq“pfgqh. Remark1.2. (a) Wedon’trequirethatobCandmorCaresets. (b) Iftheyaresets,thenwecallCasmallcategory. (c) Wecangetawaywithouttalkingaboutobjects,sinceAÞÑ1 isabijection A fromobCtothecollectionofmorphisms f satisfying fg“ gandhf “h whenevertehcompositesaredefined.Essentially,wecanrepresentobjects bytheiridentityarrows. Example1.3. (a) ThecategorySetwhoseobjectsaresetsandwhosearrowsarefunctions. Technically,weshouldspecifythecodomainforthefunctionsbecausere- allythedefinitionofafunctiondoesn’tspecifyacodomain. Somorphisms arepairspf,Bq,whereBisthecodomainofthefunction f. Lecture1 3 9October2015 1 Introduction 1.1 Definitionsandexamples (b) Gpisthecategoryofgroupsandgrouphomomorphisms; (c) Ringisthecategoryofringsandringhomomorphisms; (d) R-ModisthecategoryofR-modulesandR-modulehomomorphisms; (e) Topisthecategoryoftopologicalspacesandcontinuousmaps; (f) Mfisthecategoryofsmoothmanifoldsandsmoothmaps; (g) ThehomotopycategoryoftopologicalspacesHtpyhasthesameobjects asTop,butthemorphisms X Ñ Y arehomotopyclassesofcontinuous maps; (h) foranycategoryC,wecanturnthearrowsaroundtomaketheopposite categoryCop. Example(h)leadstothedualityprinciple,whichisakindof“twoforthe priceofone”dealincategorytheory. Theorem1.4(TheDualityPrinciple). Ifφisavalidstatementaboutcategories, soisthestatementφ˚obtainedbyreversingallthemorphisms. Example(g)abovegivesrisetothefollowingdefinition. Definition 1.5. In general, an equivalence relation „ on the collection of all morphismsofacategoryiscalledacongruenceif (i) f „ g ùñ dom f “domgandcod f “codg; (ii) f „ g ùñ fh„ ghandkf „kgwheneverthecompositesaredefined. There’sacategoryC{„withthesameobjectsasCbut„-equivalenceclassesas morphisms. Example1.6. ContinuedfromExample1.3. (i) A category C with one object ˚ must have dom f “ cod f “ ˚ for all f PmorC. Soallcompositesaredefined,and(ifmorCisaset),morCis justamonoid(whichisasemigroupwithidentity). (j) In particular, a group can be considered as a small category with one object,inwhicheverymorphismisanisomorphism. (k) Agroupoidisacategoryinwhichallmorphismsareisomorphisms. For atopologicalspaceX,thefundamentalgroupoidπpXqisthe“basepoint- lessfundamentalgroup;”theobjectsarepointsofXandthemorphisms x Ñ y are homotopy classes paths from x to y. (Homotopy classes are requiredsothateachpathhasaninverse). (l) a category whose only morphisms are identites is called discrete. A categoryinwhich,foranytwoobjects A,Bthereisatmostonemorphism A Ñ Biscalledpreorder,i.e. it’sacollectionofobjectswithareflexive andtransitiverelation. Inparticular,apartialorderisapreorderinwhich theonlyisomorphismsareidentities. Lecture1 4 9October2015 1 Introduction 1.1 Definitionsandexamples (m) The category Rel has the same objects as Set, but the morphisms are relations instead of functions. Precisely, a morphism A Ñ B is a triple pA,R,BqwhereRĎ AˆB.ThecompositepB,S,CqpA,R,BqispA,R˝S,Cq where R˝S“tpa,cq|DbP Bs.t. pa,bqP Randpb,cqPSu. NotethatSetisasubcategoryofRelandRel–Relop. Let’scontinuewiththeexamples. Example1.7. ContinuedfromExample1.3. (n) Let K beafield. ThecategoryMat hasnaturalnumbersasobjects. A K morphism n Ñ p is a pˆn matrix with entries in K. Composition is op just matrix multiplication. Note that, once again, Mat – Mat , via K k transpositionofmatrices. (o) Anexamplefromlogic. SupposeyouhavesomeformaltheoryT. The categoryDetTofderivationsrelativetoThasformulaeinthelanguage ofTasobjects,andmorphismsφÑψarederivations φ ψ and composition is just concatenation. The identity 1 is the one-line φ derivationφ. Definition1.8. LetCandDbecategories. AfunctorF: CÑDconsistsof (i) anoperation AÞÑ FpAqfromobCtoobD; (ii) anoperation f ÞÑ FpfqfrommorCtomorD, satisfying (i) domFpfq“ Fpdom fq,codFpfq“ Fpcod fqforall f; (ii) Fp1 q“1 forall A; A FpAq (iii) andFpfgq“ FpfqFpgqwhenever fgisdefined. Let’sseesomeexamplesagain. Example1.9. (a) TheforgetfulfunctorGp Ñ Setswhichsendsagroupto itsunderlyingset,andanygrouphomomorphismtoitselfasafunction. Similarly,there’soneRingÑSet,andRingÑAb,andTopÑSet (b) Therearelotsofconstructionsinalgebraandtopologythatturnoutto be functors. For example, the free group construction. Let FA denote the free group on a set A. It comes equipped with an inclusion map Lecture2 5 12October2015 1 Introduction 1.1 Definitionsandexamples η : AÑ FA,andany f: AÑG,whereGisagroup,extendsuniquelyto A ahomomorphismFAÑG. FA G ηA f A FisafunctorfromSettoGp,andgiveng: AÑ B,wedefineFgtobethe uniquehomomorphismextendingthecomposite AÝÑg BÝηÑB FB. (c) TheabelianizationofanarbitrarygroupGisthequotientG{G1 ofGby it’sderivedsubgroupG1 “xxyx´1y´1 | x,y P Gy. Thisgivesthelargest quotientof G whichisabelian. If φ: G Ñ H isahomomorphism, then itmapsthederivedsubgroupofGtothederivedsubgroupofH,sothe abelianizationisfunctorialGpÑAb. (d) Thepowersetfunctor. Foranyset A,letPAdenotethesetofallsubsetsof A. PisafunctorSetÑSet;given f: AÑ B,wedefinePfpA1q“tfpxq| xP A1ufor A1 Ď A. Butwealsomake Pintoafunctor P˚: Set Ñ Setop (orSetop Ñ Set)by settingP˚fpB1q“ f´1pB1qforB1 Ď B. Thislastexampleiswhatwecallacontravariantfunctor. Definition1.10. Acontravariantfunctor F: C Ñ Disafunctor F: C Ñ Dop (equivalently, Cop Ñ D). The term covariant functor is used sometimes to makeitclearthatafunctorisnotcontravariant. Example1.11. ContinuedfromExample1.9 (e) ThedualspaceofavectorspaceoverKdefinesacontravariantfunctor k-ModÑk-Mod. Ifα: V ÑW isalinearmap,thenα˚: W˚ ÑV˚isthe operationofcomposinglinearmapsW ÑKwithα. (f) Let Cat denote the category of small categories and functors between them. ThenCÞÑCopiscovariantfunctorCatÑCat. (g) If M and N are monoids, regarded as one-object categories, what is a functorbetweenthem? It’sjustamonoidhomomorphismfromMtoN: it preservestheidentityelementandcomposition. Inparticular,if M,Nare groups,thenthefunctorisagrouphomomorphism. Hence,wemaythink ofGpisasubcategoryofCat. (h) Similarly,ifPandQarepartiallyorderedsets,regardedascategories,a functorPÑQisjustanorder-preservingmap. (i) LetGbeagroup,regardedasacategory. AfunctorF: GÑSetspicksout asetastheimageoftheoneobjectinG,andeachmorphismofGisan isomorphismsogetsmappedtoabijectionofthisset. Sothisisagroup actionGœFpGq. IfwereplaceSetsbyk-Vectforkafield,wegetlinear representationsofG. Lecture2 6 12October2015 1 Introduction 1.1 Definitionsandexamples (j) In algebraic topology, there are many functors. For example, the fun- damental group π pX,xq defines a functor from Top (the category of 1 ˚ pointedtopologicalspaces,i.e.,thosewithadistinguishedbasepoint)to Gp. Similarly, homology groups are functors H : Top Ñ Ab (or more n commonly,HtpyÑAb). There’sanotherlayertoo. Therearemorphismsbetweenfunctors,called naturaltransformations. Definition1.12. LetCandDbecategoriesandF,G: CÑD. Anaturaltrans- formation α: F Ñ G is an operation A ÞÑ α from obC to morD, such that A dompα q “ FpAq,codpα q “ GpAqforall A,andthefollowingdiagramcom- A A mutes. Ff FA FB αA αB Gf GA GB Again,weshouldmentionsomeexamplesofnaturaltransformations. Example1.13. (a) There’s a natural transformation α: 1 Ñ ˚˚, where k-Mod ˚ is the dual space functor. This is the statement that a vector space is canonically isomorphic to it’s double dual. α : V Ñ V˚˚ sends r P V V tothe“evaluateatr”map V˚ Ñ k. Ifwerestricttofinite-dimensional spaces,thenαbecomesanaturalisomorphism,i.e. anisomorphismin thecategoryrk-fgMod,k-fgMods,whererC,Dsdenotesthecategoryof allfunctorsCÑDwithnaturaltransformationsasarrows. Remark 1.14. Note that if α is a natural transformation, and each α is an A isomorphism,thentheinversesβ oftheα alsoformanaturaltransformation, A A because β ˝Gf “ β ˝Gf ˝α ˝β “ β ˝α ˝Ff ˝β “ Ff ˝β . B B A A B B A A Example1.15. ContinuedfromExample1.13 (b) Let F: Sets Ñ Gp be the free group functor, and let U: Gp Ñ Set be the forgetful functor. The inclusion of generators η : A Ñ UFA is the A A-componentofanaturaltransformation1 ÑUF. Set (c) Foranyset A,themappinga ÞÑ tauisafunctiont´u : A Ñ PpAq. We A seethatt´uisanaturaltransformation1 Ñ P,sinceforany f: AÑ B, Set wehavePfptauq“tfpaqu. (d) Suppose given two groups G,H and two homomorphisms f, f1: G Ñ H. A natural transformation f Ñ f1 is an element h P H such that hfpgq“ f1pgqhforallgPG,orequivalently,hfpgqh´1 “ f1pgq. Sosucha transformationexistsifandonlyif f and f1areconjugate. Lecture3 7 14October2015 1 Introduction 1.1 Definitionsandexamples (e) For any space X with a base point x, there’s a natural homomorphism h : π pX,xq Ñ H pXq called the Hurewicz homomorphism. This pX,xq 1 1 is the pX,xq-component of the natural transformation h from π to the 1 composite Top ÝÑU TopÝHÝÑ1 AbÑÝI Gp, ˚ whereUistheforgetfulfunctorand I istheinclusion. It’snotoftenusefultosaythatfunctorsareinjectiveorsurjectiveonobjects. Generally,afunctormightoutputsomeobjectwhichisisomorphictoabunch of others, but might not actually be surjective – it could be surjective up to isomorphism. This is is similar to the idea that equality is not useful when comparinggroups,butratherisomorphism. Definition1.16. LetF: CÑDbeafunctor. WesayFis (1) faithful if, given f,g P morC, the three equations dompfq “ dompgq, codpfq“codpgq,andFf “ Fgimply f “ g; (2) fullif,giveng: FAÑ FBinD,thereexists f: AÑ BinCwithFf “ g. WesayasubcategoryC1ofCisfulliftheinclusionfunctorC1 ÑCisfull. Example1.17. (a) AbisafullsubcategoryofGp; (b) The category Lat of lattices (that is, posets with top element 1, bottom element 0, binary joint _, binary meet ^) is a non-full subcategory of Posets. Likewise,equalityofcategoriesisaveryrigididea. Isomorphismofcate- gories,aswell,isalittlebittoorigid.Wemighthaveseveralobjectsinacategory CwhichareisomorphicinCandallmappedtothesameobjectinD–inthis case,wewanttoconsiderthesecategoriessomehowthesame. Ifwerequire isomorphismofcategories,wecannotinsistoneventhenumberofobjectsbeing thesame. SeeExample1.20foraconcreterealizationofthis. Definition 1.18. Let C and D be categories. An equivalence of categories betweenCandDisapairoffunctorsF: CÑDandG: DÑCtogetherwith naturalisomorphismsα: 1 ÑGF,β: FGÑ1 . C D ThenotationforthisisC»D. Definition1.19. Wesaythatapropertyofcategoriesisacategoricalproperty ifwheneverChaspropertyPandC»D,thenDhasPaswell. Example1.20. (a) GivenanobjectBofacategoryC,wewriteC{Bforthecategorywhose f objectsaremorphisms AÝÑ BwithcodomainB,andwhosemorphisms Lecture3 8 14October2015 1 Introduction 1.1 Definitionsandexamples g: pAÝÑf BqÝÑpA1 ÝÑf1 Bqarecommutativetriangles g A A1 f f1 B The category Sets{B is equivalent to the category SetsB of B-indexed familiesofsets. Inonedirection, wesendpA ÝÑf Bqtopf´1pbq | b P Bq, andintheotherdirectionwesendpC |bP Bqto b ď C ˆtbuÝπÑ2 B. b bPB Composingthesetwofunctorsdoesn’tgetusbacktowherewestarted, butitdoesgiveussomethingclearlyisomorphic. (b) Let1{SetbethecategoryofpointedsetspA,aq,andletPartbethesubcat- egoryofRelwhosemorphismsarepartialfunctions,i.e. relationsRsuch thatpa,bqP Randpa,b1qP Rimpliesb“b1. Then1{Set»Part:inonedirectionwesendpA,aqtoAztauand f: pA,aqÑ pB,bqto tpx,yq| xP A,yP B, fpxq“y,y‰bu Intheotherdirectionwesend AtopAYtAu,Aqandapartialfunction f ¨A ã B (apparently that’s the notation for partial functions) to the function f definedby $ ’’&fpaq aPdom f fpaq“ B aP Azdom f ’’% B a“ A (c) ThecategoryfdMod offinitedimensionalvectorspacesoverkisequiva- k op lenttofdMod bythedualfunctors k ˚ op fdMod fdMod , k ˚ k andthenaturalisomorphism1 Ñ ˚˚. Thisisanequivalencebut fdModk notanisomorphismofcategories. (d) ThecategoryfdMod isalsoequivalenttoMat : inonedirectionsendan k k objectofnofMat tokn,andamorphism Atothelinearmapwithmatrix k Arelativetothestandardbasis. Intheotherdirection,sendavectorspace V todimV andchooseabasisforeachV tosendalineartransformation θ: V ÑW tothematrixrepresentingθwithrespecttothechosenbases. ThecompositeMat ÑfdMod ÑMat istheidentity;theothercompos- k k k iteisisomorphictotheidentityviatheisomorphismssendingthechosen basestothestandardbasisofkdimV. Lecture4 9 16October2015 1 Introduction 1.1 Definitionsandexamples There’sanothernotionslightlyweakerthansurjectivityofafunctor. Some callit“surjectiveuptoisomorphism.” Definition 1.21. We say a functor F: C Ñ D is essentially surjective if for everyobjectDofD,thereexistsanobjectCofCsuchthatD– FpCq. Thenextlemmasomehowusesamorepowerfulversionoftheaxiomof choiceandisbeyondusualsettheory. Lemma1.22. AfunctorF: CÑDispartofanequivalencebetweenCandDif andonlyifFisfull,faithful,andessentiallysurjective. Proofpùñq. SupposegivenG: DÑC,α: 1 Ñ GFandβ: FG Ñ1 asinthe C D definitionofequivalenceofcategories. ThenB– FGBforallB,soFisclearly essentiallysurjective. Let’s prove faithfulness. Now suppose given f,g: A Ñ B P C such that Ff “ Fg. ThenGFf “ GFg. Usingthenaturalityofα,thefollowingdiagram commutes: GFf“GFg GFA GFB αA αB (1) f A B Now f “α´1pGFfqα B A “α´1pGFgqα B A “ g, thelastlinebythenaturalityofαwithgalongthebottomarrowof(1)instead of f. Therefore, f isfaithful. Forfullness,supposegiveng: FAÑ FBinD. Define f “α´1˝pGgq˝α : AÑGFAÑGFBÑ B. B A ObservethatGg“α ˝ f ˝α´1. Similarly,thefollowingsquarecommutes: B A GFf GFA GFB αA αB f A B bythenaturalityofα. Therefore,GFf “α ˝ f ˝α´1aswell. Hence, B A GFf “α ˝ f ˝α´1 “Gg. B A Applyingtheargumentforfaithfulnessof FtothefunctorGshowsthatGis faithful. Therefore,GFf “GgimpliesthatFf “ g. Hence,thefunctorFisfull. pðùq. WehavetodefinethefunctorG: DÑC. Lecture4 10 16October2015
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