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ReprintsinTheoryandApplicationsofCategories,No.3,2003.
ABELIANCATEGORIES
PETERJ.FREYD
Foreword
Theearly60swasagreattimeinAmericaforayoungmath-ematician.WashingtonhadrespondedtoSputnikwithalotofmoneyforscienceeducationandthescientists,blessthem,saidthattheycouldnotdoanythinguntilstudentsknewmath-ematics.WhatSputnikproved,incrediblyenough,wasthatthecountryneededmoremathematicians.
Publishersgotthemessage.AtannualAMSmeetingsyoucouldspendentireeveningscrawlingpublishers’cocktailparties.Theyweren’tlookingforbookbuyers,theywerelookingforwritersandsomehowtheyhadconcludedthatthebestwaytogetmathematicianstowriteelementarytextswastopublishtheiradvancedtexts.WordhadgoneoutthatIwaswritingatextonsomethingcalled“categorytheory”andwhateveritwas,somebignamesseemedtobeinterested.Ilostcountofthebookmenwhovisitedmyofficebearinggiftcopiesoftheiradvancedtexts.IchoseHarper&Rowbecausetheypromised
Originallypublishedas:AbelianCategories,HarperandRow,1964.Receivedbytheeditors2003-11-10.TransmittedbyM.Barr.Reprintpublishedon2003-12-17.Footnoted
referencesaddedtotheForewordandposted2004-01-20.2000MathematicsSubjectClassification:18-01,18B15.Keywordsandphrases:Abeliancategories,exactembedding.c©PeterJ.Freyd,1964.Permissiontocopyforprivateusegranted.
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course,tobereplacedbytheword“equalizer”.Pages29–30:Exercise1–Dwouldhavebeenmucheasierif
ithadbeendelayeduntilafterthedefinitionsofgeneratorandpushout.Thecategory[→]isbestcharacterizedasageneratorforthecategoryofsmallcategoriesthatappearsasaretractofeveryothergenerator.Thecategory[→→]isapushoutofthetwomapsfrom1to[→]andthischaracterizationalsosimpli-fiesthematerialinsection3:ifafunctorfixesthetwomapsfrom1to[→]thenitwillbeshowntobeequivalenttotheidentityfunctor;if,instead,ittwiststhemitisequivalenttothedual-categoryfunctor.Thesecharacterizationshaveanotherad-vantage:theyarecorrect.Ifonestartswiththethetwo-elementmonoidthatisn’tagroup,viewsitasacategoryandthenfor-mally“splitstheidempotents”(asinExercise2–B,page61)theresultisanothertwo-objectcategorywithexactlythreeendo-functors.Andthesupposedcharacterizationof[→→]iscoun-terexampledbythedisjointunionof[→]andthecyclicgroupoforderthree.
Page35:Theaxiomsforabeliancategoriesareredundant:eitherA1orA1*suffices,thatis,eachinthepresenceoftheotheraxiomsimpliestheother.Theproof,whichisnotstraight-forward,canbefoundonsection1.598ofmybookwithAndreScedrov
1,henceforthtobereferredtoasCats&Alligators.Sec-
tion1.597ofthatbookhasanevenmoreparsimoniousdefinitionofabeliancategory(whichIneededforthematerialdescribedbelowconcerningpage108):itsufficestorequireeitherprod-uctsorsumsandthateverymaphasa“normalfactorization”,towit,amapthatappearsasacokernelfollowedbyamapthatappearsaskernel.
Pages35–36:Oftheexamplesmentionedtoshowthein-
1Categories,Allegories,NorthHolland,1990
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PETER J. FREYD
a low price (≤ $8) and—even better—hundreds of free copies tomathematicians of my choice. (This was to be their first mathpublication.)
On the day I arrived at Harper’s with the finished manuscriptI was introduced, as a matter of courtesy, to the Chief of Pro-duction who asked me, as a matter of courtesy, if I had anypreferences when it came to fonts and I answered, as a matterof courtesy, with the one name I knew, New Times Roman.
It was not a well-known font in the early 60s; in those daysone chose between Pica and Elite when buying a typewriter—notfonts but sizes. The Chief of Production, no longer acting just oncourtesy, told me that no one would choose it for something likemathematics: New Times Roman was believed to be maximallydense for a given level of legibility. Mathematics required a morespacious font. All that was news to me; I had learned its nameonly because it struck me as maximally elegant.
The Chief of Production decided that Harper’s new mathseries could be different. Why not New Times Roman? Thebook might be even cheaper than $8 (indeed, it sold for $7.50).We decided that the title page and headers should be sans serifand settled that day on Helvetica (it ended up as a rather non-standard version). Harper & Row became enamored with thoseparticular choices and kept them for the entire series. (And—coincidently or not—so, eventually, did the world of desktoppublishing.) The heroic copy editor later succeeded in convinc-ing the Chief of Production that I was right in asking for nega-tive page numbering. The title page came in at a glorious –11and—best of all—there was a magnificent page 0.
The book’s sales surprised us all; a second printing was or-dered. (It took us a while to find out who all the extra buyerswere: computer scientists.) I insisted on a number of changes
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(this time Harper’s agreed to make them without deducting frommy royalties; the correction of my left-right errors—scores ofthem—for the first printing had cost me hundreds of dollars).But for reasons I never thought to ask about, Harper’s didn’tmark the second printing as such. The copyright page, –8, is al-most identical, even the date. (When I need to determine whichprinting I’m holding—as, for example, when finding a copy forthis third “reprinting”—I check the last verb on page –3. In thesecond printing it is has instead of have).
A few other page-specific comments:Page 8: Yikes! In the first printing there’s no definition of
natural equivalence. Making room for it required much short-ening of this paragraph from the first printing:
Once the definitions existed it was quickly noticedthat functors and natural transformations had be-come a major tool in modern mathematics. In 1952Eilenberg and Steenrod published their Foundationsof Algebraic Topology [7], an axiomatic approach tohomology theory. A homology theory was definedas a functor from a topological category to an alge-braic category obeying certain axioms. Among themore striking results was their classification of such“theories,” an impossible task without the notion ofnatural equivalence of functors. In a fairly explosivemanner, functors and natural transformations havepermeated a wide variety of subjects. Such monu-mental works as Cartan and Eilenberg’s HomologicalAlgebra [4], and Grothendieck’s Elements of Alge-braic Geometry [1] testify to the fact that functorshave become an established concept in mathematics.
Page 21: The term “difference kernel” in 1.6 was doomed, of
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dependenceofA3andA3*oneisclear,theotherrequireswork:itisnotexactlytrivialthatepimorphismsinthecategoryofgroups(abelianornot)areonto—oneneedsthe“amalgama-tionlemma”.(Giventhesymmetryoftheaxiomseitheroneoftheexampleswould,note,havesufficed.)FortheindependenceofA2(hence,bytakingitsdual,alsoofA2*)letRbearing,commutativeforconvenience.Thefullsubcategory,F,offinitelypresentedR-modulesiseasilyseentobeclosedundertheformationofcokernelsofarbitrarymaps—quiteenoughforA2*andA3.WithalittleworkonecanshowthatthekernelofanyepiinFisfinitelygeneratedwhichguaranteesthatitistheimageofamapinFandthat’senoughforA3*.Thenec-essaryandsufficientconditionthatFsatisfyA2isthatRbe“coherent”,thatis,allofitsfinitelygeneratedidealsbefinitelypresentedasmodules.Forpresentpurposeswedon’tneedthenecessaryandsufficientcondition.So:letKbeafieldandRbetheresultofadjoiningasequenceofelementsXnsubjecttotheconditionthatXiXj=0alli,j.Thenmultiplicationby,say,X1definesanendomorphismonR,thekernelofwhichisnotfinitelygenerated.Moretothepoint,itfailstohaveakernelinF.
Page60:Exercise2–Aonadditivecategorieswasentirelyredoneforthesecondprinting.Amongtheproblemsinthefirstprintingweretheword“monoidal”inplaceof“pre-additive”(clashingwiththemodernsenseofmonoidalcategory)and—wouldyoubelieveit!—theabsenceofthedistributivelaw.
Page72:Areviewermentionedasanexampleofoneofmyprivatejokesthesizeofthefontforthetitleofsection3.6,bifunctors.Goodheavens.Iwasnotreallyawareofhowmanyjokes(privateorotherwise)hadaccumulatedinthetext;Imusthavebeenawareofeachoneoftheminitstimebut
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PETERJ.FREYD
refusedtoengageinthemyriaddiscussionsabouttheissuesdis-cussedinthematerialthatstartsonthebottomofpage85.Itwasagoodrule.Ihad(correctly)predictedthatthecontro-versywouldevaporateandthat,inthemeantime,itwouldbeawasteoftimetoamplifywhatIhadalreadywritten.Ishould,though,havefiguredoutawaytopointoutthattheforgetfulfunctorforthecategory,B,describedonpages131–132hasalltheconditionsneededforthegeneraladjointfunctorexceptforthesolutionsetcondition.Ironicallytherewasalreadyinhandamuchbetterexample:theforgetfulfunctorfromthecategoryofcompletebooleanalgebras(andbi-continuoushomomorphisms)tothecategoryofsetsdoesnothavealeftadjoint(putanotherway,freecompletebooleanalgebrasarenon-existentlylarge).Theproof(albeitforadifferentassertion)wasinHaimGaif-man’s1962dissertation
5.
Page87:Theterm“co-well-powered”should,ofcourse,be“well-co-powered”.
Pages91–93:IlosttrackofthemanyspecialcasesofExercise3–Oonmodeltheorythathaveappearedinprint(mostofteninproofsthataparticularcategory,forexamplethecategoryofsemigroups,iswell-co-poweredandinproofsthataparticularcategory,forexamplethecategoryofsmallskeletalcategories,isco-complete).Inthisexercisethemostconspicuousomissionresultedfrommynottakingthetroubletoallowmany-sortedtheories,whichmeantthatIwasnotabletomentiontheeasytheoremthatBAisacategoryofmodelswheneverAissmallandBisitselfacategoryofmodels.
Page107:CharacteristiczeroisnotneededinthefirsthalfofExercise4–H.Itwouldbebettertosaythatafieldarisingastheringofendomorphismsofanabeliangroupisnecessar-
5InfiniteBooleanPolynomialsI.Fund.Math.541964
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PETER J. FREYD
I kept no track of their number. So now people were seekingthe meaning for the barely visible slight increase in the size ofthe word bifunctors on page 72. If the truth be told, it wasfrom the first sample page the Chief of Production had sent mefor approval. Somewhere between then and when the rest ofthe pages were done the size changed. But bifunctors didn’tchange. At least not in the first printing. Alas, the joke wasremoved in the second printing.
Pages 75–77: Note, first, that a root is defined in Exercise3–B not as an object but as a constant functor. There wasa month or two in my life when I had come up with the no-tion of reflective subcategories but had not heard about adjointfunctors and that was just enough time to write an undergrad-uate honors thesis2. By constructing roots as coreflections intothe categories of constant functors I had been able to prove theequivalence of completeness and co-completeness (modulo, as Ithen wrote, “a set-theoretic condition that arises in the proof”).The term “limit” was doomed, of course, not to be replaced by“root”. Saunders Mac Lane predicted such in his (quite favor-able) review3, thereby guaranteeing it. (The reasons I give onpage 77 do not include the really important one: I could notfor the life of me figure out how A×B results from a limitingprocess applied to A and B. I still can’t.)
Page 81: Again yikes! The definition of representable func-tors in Exercise 4–G appears only parenthetically in the firstprinting. When rewritten to give them their due it was nec-essary to remove the sentence “To find A, simply evaluate theleft-adjoint of S on a set with a single element.” The resulting
2Brown University, 19583The American Mathematical Monthly, Vol. 72, No. 9. (Nov., 1965),
pp. 1043-1044.
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paragraph is a line shorter; hence the extra space in the secondprinting.
Page 84: After I learned about adjoint functors the maintheorems of my honors thesis mutated into a chapter about thegeneral adjoint functor theorems in my Ph.D. dissertation4. Iwas still thinking, though, in terms of reflective subcategoriesand still defined the limit (or, if you insist, the root) of D → Aas its reflection in the subcategory of constant functors. If I hadreally converted to adjoint functors I would have known thatlimits of functors in AD should be defined via the right adjointof the functor A → AD that delivers constant functors. Alas,I had not totally converted and I stuck to my old definition inExercise 4–J. Even if we allow that the category of constantfunctors can be identified with A we’re in trouble when D isempty: no empty limits. Hence the peculiar “condition zero” inthe statement of the general adjoint functor theorem and anynumber of requirements to come about zero objects and such,all of which are redundant when one uses the right definition oflimit.
There is one generalization of the general adjoint functor the-orem worth mentioning here. Let “weak-” be the operator ondefinitions that removes uniqueness conditions. It suffices thatall small diagrams in A have weak limits and that T preservesthem. See section 1.8 of Cats & Alligators. (The weakly com-plete categories of particular interest are in homotopy theory. Amore categorical example is coscanecof, the category of smallcategories and natural equivalence classes of functors.)
Pages 85–86: Only once in my life have I decided to refrainfrom further argument about a non-baroque matter in math-ematics and that was shortly after the book’s publication: I
4Princeton, 1960
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ABELIANCATEGORIES
ilyaprimefield(hencethecategoryofvectorspacesoveranynon-primefieldcannotbefullyembeddedinthecategoryofabeliangroups).TheonlyreasonIcanthinkofforinsistingoncharacteristiczeroisthattheproofsforfiniteandinfinitecharac-teristicsaredifferent—astrangereasongiventhatneitherproofispresent.
Page108:Icameacrossagoodexampleofalocallysmallabeliancategorythatisnotveryabelianshortlyafterthesecondprintingappeared:towit,thetargetoftheuniversalhomol-ogytheoryonthecategoryofconnectedcw-complexes(finitedimensional,ifyouwish).JoelCohencalleditthe“Freydcat-egory”inhisbook
6,butitshouldbenotedthatJoeldidn’t
nameitafterme.(Healwaysinsistedthatitwasmydaugh-ter.)It’ssuchanicecategoryit’sworthdescribinghere.Toconstructit,startwithpairsofcw-complexes〈X′,X〉whereX′
isanon-emptysubcomplexofXandtaketheobviousconditiononmaps,towit,f:〈X′,X〉→〈Y′,Y〉isacontinuousmapf:X→Ysuchthatf(X′)⊆Y′.Nowimposethecongruencethatidentifiesf,g:〈X′,X〉→〈Y′,Y〉whenf|X′andg|X′arehomotopic(asmapstoY).Finally,taketheresultofformallymakingthesuspensionfunctoranautomorphism(whichcan,ofcourse,berestatedastakingareflection).ThiscanallbefoundinJoel’sbookorinmyarticlewiththesametitleasJoel’s
7.
Thefactthatitisnotveryabelianfollowsfromthefactthatthestable-homotopycategoryappearsasasubcategory(towit,thefullsubcategoryofobjectsoftheform〈X,X〉)andthatcategorywasshownnottohaveanyembeddingatallintothe
6StableHomotopyLectureNotesinMathematicsVol.165Springer-
Verlag,Berlin-NewYork19707StableHomotopy,Proc.oftheConferenceofCategoricalAlgebra,
Springer-Verlag,1966
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PETERJ.FREYD
Page159:TheYonedalemmaturnsoutnottobeinYoneda’spaper.When,sometimeafterbothprintingsofthebookap-peared,thiswasbroughttomy(muchchagrined)attention,IbroughtittheattentionofthepersonwhohadtoldmethatitwastheYonedalemma.HeconsultedhisnotesanddiscoveredthatitappearedinalecturethatMacLanegaveonYoneda’streatmentofthehigherExtfunctors.Thename“Yonedalemma”wasnotdoomedtobereplaced.
Pages163–164:AllowsandGeneratingweremissingintheindexofthefirstprintingaswaspage129forMitchell.StillmissinginthesecondprintingareNaturalequivalence,8andPre-additivecategory,60.Notmissing,alas,isMonoidalcate-gory.
FINALLY,acommentonwhatI“hopedtobeageodesiccourse”tothefullembeddingtheorem(mentionedonpage10).Ithinkthehopewasjustifiedforthefullembeddingtheorem,butifonesettlesfortheexactembeddingtheoremthenthegeodesiccourseomittedanimportantdevelopment.Bybroad-eningtheproblemtoregularcategoriesonecanfindachoice-freetheoremwhich—asidefromitswiderapplicabilityinatopos-theoreticsetting—hastheadvantageofnaturality.Theproofrequiresconstructionsinthebroadercontextbutifoneappliesthegeneralconstructiontothespecialcaseofabeliancategories,weobtain:
Thereisaconstructionthatassignstoeachsmallabeliancat-egoryAanexactembeddingintothecategoryofabeliangroupsA→GsuchthatforanyexactfunctorA→Bthereisanat-uralassignmentofanaturaltransformationfromA→GtoA→B→G.WhenA→Bisanembeddingthensoisthetransformation.
TheproofissuggestedinmypamphletOncanonizingcat-
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PETER J. FREYD
category of sets in Homotopy Is Not Concrete8. I was surprised,when reading page 108 for this Foreword, to see how similar inspirit its set-up is to the one I used 5 years later to demonstratethe impossibility of an embedding of the homotopy category.
Page (108): Parenthetically I wrote in Exercise 4–I, “Theonly [non-trivial] embedding theorem for large abelian categoriesthat we know of [requires] both a generator and a cogenerator.”It took close to ten more years to find the right theorem: anabelian category is very abelian iff it is well powered (which itshould be noticed, follows from there being any embedding at allinto the category of sets, indeed, all one needs is a functor thatdistinguishes zero maps from non-zero maps). See my paperConcreteness9. The proof is painful.
Pages 118–119: The material in small print (squeezed inwhen the first printing was ready for bed) was, sad to relate,directly disbelieved. The proofs whose existence are being as-serted are natural extensions of the arguments in Exercise 3–Oon model theory (pages 91–93) as suggested by the “conspicuousomission” mentioned above. One needs to tailor Lowenheim-Skolem to allow first-order theories with infinite sentences. Butit is my experience that anyone who is conversant in both modeltheory and the adjoint-functor theorems will, with minimal prod-ding, come up with the proofs.
Pages 130–131: The Third Proof in the first printing washopelessly inadequate (and Saunders, bless him, noticed thatfact in his review). The proof that replaced it for the secondprinting is ok. Fitting it into the alloted space was, if I may sayso, a masterly example of compression.
8The Steenrod Algebra and its Applications, Lecture Notes in Mathe-matics, Vol. 168 Springer, Berlin 1970
9J. of Pure and Applied Algebra, Vol. 3, 1973
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Pages 131–132: The very large category B (Exercise 6–A)—with a few variations—has been a great source of counterexam-ples over the years. As pointed out above (concerning pages85–86) the forgetful functor is bi-continuous but does not haveeither adjoint. To move into a more general setting, drop thecondition that G be a group and rewrite the “convention” tobecome f(y) = 1G for y /∈ S (and, of course, drop the conditionthat h : G → G′ be a homomorphism—it can be any function).The result is a category that satisfies all the conditions of aGrothendieck topos except for the existence of a generating set.It is not a topos: the subobject classifier, Ω, would need to be thesize of the universe. If we require, instead, that all the values ofall f : S → (G, G) be permutations, it is a topos and a booleanone at that. Indeed, the forgetful functor preserves all the rel-evant structure (in particular, Ω has just two elements). In itscategory of abelian-group objects—just as in B—Ext(A, B) is aproper class iff there’s a non-zero group homomorphism from Ato B (it needn’t respect the actions), hence the only injective ob-ject is the zero object (which settled a once-open problem aboutwhether there are enough injectives in the category of abeliangroups in every elementary topos with natural-numbers object.)
Pages 153–154: I have no idea why in Exercise 7–G I didn’tcite its origins: my paper, Relative Homological Algebra MadeAbsolute10.
Page 158: I must confess that I cringe when I see “A manlearns to think categorically, he works out a few definitions, per-haps a theorem, more likely a lemma, and then he publishes it.”I cringe when I recall that when I got my degree, Princeton hadnever allowed a female student (graduate or undergraduate). Onthe other hand, I don’t cringe at the pronoun “he”.
10Proc. Nat. Acad. Sci., Feb. 1963
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egorytheoryoronfunctorializingmodeltheory11
.Itusesthestrangesubjectofτ-categories.Moreaccessibly,itisexposedinsection1.54ofCats&Alligators.
PhiladelphiaNovember18,2003
∫–
11Mimeographednotes,Univ.Pennsylvania,Philadelphia,Pa.,1974
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F.WilliamLawvere,S.U.N.Y.atBuffalo:[email protected],UniversitedeStrasbourg:[email protected],UniversityofUtrecht:[email protected],UnionCollege:[email protected],DalhousieUniversity:[email protected],UniversityofCambridge:[email protected],MountAllisonUniversity:[email protected],Man-agingEditorJiriRosicky,MasarykUniversity:[email protected],UniversityofNorthCarolina:[email protected],MacquarieUniversity:[email protected],YorkUniversity:[email protected],RutgersUniversity:[email protected],UniversityofInsubria:[email protected],DalhousieUniversity:[email protected]
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