FOUNDATIONS FOR ALMOST RING THEORY May 16, 2014 Release 6.5 Ofer Gabber IHES Le ...

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FOUNDATIONS FOR ALMOST RING THEORY
OFERGABBERANDLORENZORAMERO
January23,2018
Release 7
OferGabber
I.H.E.S.
LeBois-Marie
35,routedeChartres
F-91440Bures-sur-Yvette
e-mailaddress: [email protected]
LorenzoRamero
Universite´ deLilleI
LaboratoiredeMathe´matiques
F-59655Villeneuved’AscqCe´dex
e-mailaddress: [email protected]
webpage: http://math.univ-lille1.fr/˜ramero
Acknowledgements WethankNielsBorne,MichelEmsalem,Pierre-YvesGaillard,LutzGeissler,W.-P.Heidorn,
FabriceOrgogozo,OlafSchnu¨rerandPeterScholzeforpointingoutsomemistakesinearlierdrafts,andforuseful
suggestions.
1
2 OFERGABBERANDLORENZORAMERO
CONTENTS
0. Introduction................................................................. 4
1. Basiccategorytheory........................................................ 7
1.1. Categories,functorsandnaturaltransformations .............................. 8
1.2. Presheavesandlimits....................................................... 21
1.3. AdjunctionsandKanextensions............................................. 33
1.4. Specialpropertiesofthecategoriesofpresheaves ............................. 44
1.5. Finalandcofinalfunctors................................................... 52
1.6. Localizationsofcategories.................................................. 61
2. 2-Categorytheory ........................................................... 69
2.1. 2-Categoriesandpseudo-functors............................................ 69
2.2. Pseudo-naturaltransformationsandtheirmodifications........................ 82
2.3. Theformalismofbasechange............................................... 95
2.4. Adjunctionsin2-categories................................................. 106
2.5. 2-Limitsand2-colimits.....................................................128
2.6. 2-CategoricalKanextensions ............................................... 138
3. Specialcategories ........................................................... 154
3.1. Fibrations................................................................. 154
3.2. 2-Fibrations............................................................... 179
3.3. Fibrationsingroupoids..................................................... 195
3.4. Sievesanddescenttheory...................................................198
3.5. ProfinitegroupsandGaloiscategories ....................................... 213
3.6. Tensorcategoriesandabeliancategories......................................221
4. Sitesandtopoi .............................................................. 239
4.1. Topologiesandsites........................................................239
4.2. Continuousandcocontinuousfunctors....................................... 248
4.3. Morphismsofsites.........................................................256
4.4. Topoi..................................................................... 270
4.5. Fibredsites................................................................281
4.6. Fibredtopoi............................................................... 289
4.7. Localizationandpointsofatopos ........................................... 304
4.8. Algebraonatopos......................................................... 320
4.9. Torsorsonatopos ......................................................... 336
5. Stacks...................................................................... 343
5.1. Prestacksandstacksonasite ............................................... 343
5.2. Coveringmorphismsofprestacks............................................356
5.3. Localcalculusoffractions.................................................. 365
5.4. Functorialpropertiesofthecategoriesofstacks............................... 370
5.5. Sheavesofcategories.......................................................387
5.6. Stacksingroupoidsandind-finitestacks ..................................... 401
5.7. Stacksonfibredsitesandfibredtopoi........................................ 416
6. Monoidsandpolyhedra ...................................................... 434
6.1. Monoids.................................................................. 435
6.2. Integralmonoids...........................................................451
6.3. Polyhedralcones...........................................................461
6.4. Fineandsaturatedmonoids................................................. 475
6.5. Fans...................................................................... 489
6.6. Specialsubdivisions........................................................504
7. Homologicalalgebra.........................................................521
FOUNDATIONSFORALMOSTRINGTHEORY 3
7.1. Complexesinanadditivecategory........................................... 521
7.2. Filteredcomplexesandspectralsequences....................................533
7.3. Derivedcategoriesandderivedfunctors...................................... 541
7.4. Simplicialobjects.......................................................... 561
7.5. Simplicialsets............................................................. 587
7.6. Gradedrings .............................................................. 620
7.7. Differentialgradedalgebras.................................................630
7.8. Koszulalgebrasandregularsequences....................................... 635
7.9. FilteredringsandReesalgebras............................................. 649
7.10. Somehomotopicalalgebra.................................................655
7.11. Injectivemodules,flatmodulesandindecomposablemodules................. 669
8. Complementsoftopologyandtopologicalalgebra.............................. 680
8.1. Spectralspacesandconstructiblesubsets..................................... 680
8.2. Topologicalgroups.........................................................702
8.3. Topologicalrings .......................................................... 712
8.4. Topologicallylocalandtopologicallyhenselianrings.......................... 729
8.5. Gradedstructuresontopologicalrings ....................................... 735
8.6. Homologicalalgebrafortopologicalmodules.................................741
9. Complementsofcommutativealgebra......................................... 754
9.1. Valuationtheory........................................................... 754
9.2. Huber’stheoryofthevaluationspectrum..................................... 769
9.3. Wittvectors............................................................... 787
9.4. Fontainerings............................................................. 811
9.5. Dividedpowermodulesandalgebras ........................................ 821
9.6. Regularrings.............................................................. 831
9.7. Excellentrings ............................................................ 846
10. Cohomologyandlocalcohomologyofsheaves................................ 857
10.1. Cohomologyoftopoiandtopologicalspaces ................................ 857
10.2. Cˇechcohomology ........................................................ 870
10.3. Quasi-coherentmodules................................................... 883
10.4. Depthandcohomologywithsupports....................................... 893
10.5. Depthandassociatedprimes............................................... 906
10.6. Cohomologyofprojectiveschemes......................................... 919
11. Dualitytheory..............................................................937
11.1. Dualityforquasi-coherentmodules.........................................937
11.2. Cousincomplexes ........................................................ 944
11.3. Dualityovercoherentschemes............................................. 955
11.4. Schemesoveravaluationring..............................................976
11.5. Localduality............................................................. 991
11.6. Hochster’stheoremandStanley’stheorem..................................1.007
12. Logarithmicgeometry ......................................................1020
12.1. Logtopoi................................................................1020
12.2. Logschemes.............................................................1031
12.3. Logarithmicdifferentialsandsmoothmorphisms ............................1046
12.4. Logarithmicblowupofacoherentideal ....................................1060
12.5. Regularlogschemes......................................................1076
12.6. Resolutionofsingularitiesofregularlogschemes............................1094
12.7. Localpropertiesofthefibresofasmoothmorphism.........................1.113
13. E´talecoveringsofschemesandlogschemes ..................................1120
13.1. Acyclicmorphismsofschemes............................................1.120
4 OFERGABBERANDLORENZORAMERO
13.2. Localasphericityofsmoothmorphismsofschemes..........................1135
13.3. E´talecoveringsoflogschemes.............................................1147
13.4. Localacyclicityofsmoothmorphismsoflogschemes........................1167
14. Thealmostpuritytoolbox...................................................1180
14.1. Non-flatalmoststructures .................................................1180
14.2. Inversesystemsofalmostmodules .........................................1204
14.3. Almostpurepairs........................................................1.224
14.4. Normalizedlengths.......................................................1242
14.5. Finitegroupactionsonalmostalgebras.....................................1267
14.6. AlmostWittvectors......................................................1.275
14.7. Complements: locallymeasurablealgebras .................................1286
15. Continuousvaluationsandadicspaces.......................................1.303
15.1. Formalschemes..........................................................1303
15.2. Analyticallynoetherianrings ..............................................1319
15.3. Continuousvaluations....................................................1.332
15.4. Affinoidringsandaffinoidschemes........................................1.342
15.5. Adicspaces..............................................................1356
15.6. Speciallociofquasi-affinoidschemes ......................................1378
15.7. Etalecoveringsofquasi-affinoidschemes...................................1383
16. Perfectoidringsandperfectoidspaces........................................1396
16.1. Distinguishedelementsandtransversalpairs................................1.396
16.2. P-rings ..................................................................1401
16.3. Perfectoidrings..........................................................1.410
16.4. Homologicaltheoryofperfectoidrings .....................................1439
16.5. Perfectoidquasi-affinoidrings .............................................1456
16.6. Gradedperfectoidrings ...................................................1484
16.7. Perfectoidspaces.........................................................1504
16.8. Almostpurity ............................................................1522
16.9. PerfectoidTaterings......................................................1541
17. Applications...............................................................1549
17.1. Modelalgebras...........................................................1549
17.2. Almostpurity: thelogregularcase........................................1.564
References.....................................................................1.577
Itisnotincumbentuponyoutocompletethework,
butneitherareyouatlibertytodesistfromit.
(Avot2:21)
0. INTRODUCTION
Both the focus of this monograph and its subject matter have evolved considerably in the
last few years. On the one hand, the insistence on making the text self-contained (aside from a
reduced canon of basic references, which should ideally contain only EGA and some parts of
Bourbaki’sE´le´ments),hasresultedinaratherweightymassofmaterialofindependentinterest,
thatisappliedto,butiscompletelyseparatefromalmostringtheory,andwhoserelationshipto
p-adicHodgetheoryisthusevenmoreindirect. Ratherthanstemmingfromawell-thought-out
plan, this part is the outcome of a haphazard process, lumbering between alternating phases of
accretion and consolidation, with new topics piled up as dictated by need, or occasionally by
whim, when we just branched out from the main flow to pursue a certain line of thought to its
FOUNDATIONSFORALMOSTRINGTHEORY 5
logicaldestination. Nevertheless,afewthemeshavespontaneouslyemerged,aroundwhichthe
originallyamorphousmagmahasbeenabletosettle,tothepointwherebynowadistinctshape
isfinallydiscernible,anditisperhapstimetopauseandtakestockofitsbroadoutlines.
Nowthen,wemaydistinguish:
• Firstofall,aratherthoroughexpositionofthefoundationsoflogarithmicalgebraicgeom-
etry,comprisingchapters6,12and13. Inevitably,ourtreatmentowesalottotheworksofKato
andhisschool: ourcontributionisforemostthatofgatheringandtidyingupthesubject,which
until now was scattered in a disparate number of research articles, many of them still unpub-
lishedandevenunfinished. Acloserscrutinywouldalsorevealafewtechnicalinnovationsthat
we hope will become standard issue of the working algebraic log-geometer : we may mention
the systematic use of pointed monoids and pointed modules, the projective fan associated with
a graded monoid, or a definition of α-flatness for log structures which refines and generalizes
anoldernotionof“toricflatness”. Furthermore,wetooktheoccasiontorepairafewsmall(and
notsosmall)mistakesandinaccuraciesthatwedetectedintheliterature.
• Two other chapters are dedicated to local cohomology and Grothendieck’s duality theory.
Earlyon,theemphasisherewasongeneralizations: especially,wewereinterestedinremoving
from the theory the pervasive noetherian assumptions, to pave the way for our recasting of
Faltings’salmostpurityintheframeworkofvaluationtheory. Applicationsoflocalcohomology
tonon-archimedeananalyticgeometryfurnishedanotherinfluentialmotivation,thoughonethat
hasremained,sofar,hiddenfromview. Morerecently,thenoetherianaspectshavealsobecome
relevant to our project, and this latest release contains a detailed account of the most important
properties of noetherian rings endowed with a dualizing complex. The latter, in turn, could be
dealt with satisfactorily only after a thorough revisitation of the general theory of the dualizing
complex, so that our chapters 10 and 11 can also be regarded as complementary to Conrad’s
book [31] (dedicated to the trace morphism and the deeper aspects of duality) : totaling our
respective efforts, it should eventually become possible to bypass entirely Hartshorne’s notes
[59]which,asiswellknown,arewantinginmanyways.
• Chapters 7 and 9 present (for the time being, anyway) a looser structure : a miscellanea
of self-contained units devoted to more or less independent topics. However, there is at least
one thread running through several sections, and whose stretch can be traced all the way back
to the earliest beginnings of almost ring theory; it connects sections 7.4 and 7.5 – on simplicial
homotopy theory – to a section 7.10 dedicated to homotopical algebra, then on to sections 9.6
and 9.7, which make extensive use of the cotangent complex to derive important characteriza-
tionsofregularandexcellentrings,includinganup-to-datepresentationofclassicalresultsdue
to Andre´, extracted from his monograph [2], and from his paper [3] on localization of formal
smoothness. This homotopical algebraic thread resurfaces again in section 14.1, but there we
arealreadysquarelyintoalmostringtheoryproper.
On the other hand, two recent notable developments are compelling a revision of our un-
derstanding of almost ring theory itself, and of its situation within commutative algebra and
algebraicgeometryatlarge:
• The first is Scholze’s PhD thesis [97] on perfectoid spaces, that contains both a maximal
generalizationofthealmostpuritytheorem,andamajorsimplificationofitsproof,basedonhis
“tilting”technique(andcompletelydifferentfromFaltings’s). However,therangeofScholze’s
theory transcends the domain of p-adic Hodge theory (which was not even his original moti-
vation) : to drive the point home, his thesis concludes with a clever application to the long
standing weight monodromy conjecture, thus affording the unusual spectacle of a tool which
was fashioned out of purely p-adic concerns, and ends up playing a crucial role in the solution
ofapurely(cid:96)-adicproblem.
6 OFERGABBERANDLORENZORAMERO
• The second spectacular development is Yves Andre´’s proof of the direct summand con-
jecture ([4]); the latter is a deceptively simple assertion that has been a central problem in
commutative algebra for the last thirty years : it asserts that every finite injective ring homo-
morphism f : B → A from a regular local ring B, admits a B-linear splitting. The relevance
of almost purity to this question was first surmised by Paul Roberts in 2001 (after a talk by the
second author at the University of Utah), and has been widely advertised by him ever since.
Andre´’s solution uses perfectoid techniques, and builds on earlier work by Bhargav Bhatt, who
in[12]provedtheconjectureinthecasewhereB isessentiallysmoothoveramixedcharacter-
istic discrete valuation ring and f ⊗ Q is e´tale outside a relative normal crossings divisor of
Z
SpecB. Moreover, Bhatt has subsequently simplified some of Andre´’s arguments and shown
how the same method yields a more general “derived version” of the conjecture, for proper
schemesoveranyregularring: see[13].
We see then, that almost ring theory has emancipated itself from its former ancillary role
in the exclusive service of p-adic Hodge theory, and is now elbowing out a niche in the wider
ecosystemofalgebraicgeometry.
Thepresentreleasecompletestheprojectannouncedintheintroductionofthe6threlease:
• Firstweintroduceaclassoftopologicalringsthatgeneralizetheperfectoidringsof[97];it
isveryeasytosaywhata(generalized)perfectoidF -algebrais: namely,itisjustaperfectand
p
completetopologicalF -algebrawhosetopologyislinear,definedbyanidealoffinitetype. The
p
generaldefinitionissomewhatmoreinvolved,butweprovethefollowingcharacterization. For
anyperfectoidF -algebraE,weconsidertheringofWittvectorsW(E),andweendowitwith
p
a natural topology, induced from that of E; then every perfectoid ring is a topological quotient
oftheformA := W(E)/aW(E),forsuchasuitableE,andwherea ∈ W(E)iswhatwecalla
distinguishedelement: seedefinition16.1.6. Moreover,justasinScholze’swork,theperfectoid
F -algebra E can be recovered from A via a tilting functor that establishes, more precisely, an
p
equivalencebetweenthecategoryofallperfectoidringsandthatofpairs(E,I)consistingofa
perfectoidF -algebraE andaprincipalidealI ⊂ W(E)generatedbyadistinguishedelement
p
(as it is well known, this construction is rooted in Fontaine and Winterberger’s theory of the
field of norms). The distinguished ideal I represents an extra parameter that remains hidden
in Scholze’s original approach : the reason is that he fixes from the start a base perfectoid field
K, thereby implicitly fixing as well a distinguished element a in the ring of Witt vectors of the
tilt of K, and then every perfectoid ring in his work is supposed to be a K-algebra, which –
from our viewpoint – amounts to restricting to perfectoid rings whose associated distinguished
ideal is generated by a. Having thus removed the parameter I, he can then also do away with
Wittvectors altogether, andthe inverseto thetilting construction isobtained in[97] viaa more
abstract deformation theoretic argument. This route is precluded to us, so we rely instead on
direct and rather concrete Witt vectors calculations. A similar strategy has been proposed in
[75], and our viewpoint can indeed be described fairly as an interpolation of those of Scholze
andKedlaya-Liu,thoughweonlystudied[97]indetail.
• The first three sections of chapter 16 are devoted to exploring this new class of perfectoid
rings and its manifold remarkable features. The rest of the chapter then merges the theory of
perfectoid rings with Huber’s adic spaces, to forge the perfectoid spaces that are the main tool
forourproofofalmostpurity,whosemostgeneralformisgivenbytheorem16.8.39andapplies
to formal perfectoid rings, i.e. to topological rings whose completion is perfectoid. The proof
proceeds via several preliminary reductions : first, to the case of a perfectoid quasi-affinoid
ring, covered by theorem 16.8.28, then – by exploiting the local geometry of adic spaces – to
thecaseofaperfectoidvaluationring,whichwastreatedalreadyinourmonograph[48]. What
enables here this localization argument is a basic feature of the e´tale topology of arbitrary adic
spaces : the fibred category of finite e´tale coverings of the affinoid subsets of an adic space is a
stack. Thelatterresultisinturnaspecialcaseofourtheorem15.7.6.
FOUNDATIONSFORALMOSTRINGTHEORY 7
• We also include a detailed treatment of the foundations of the theory of adic spaces, that
essentially follows [65], but contains some modest improvement : notably, the systematic use
of analytically noetherian rings (borrowed from [45]) allows us to unify the two classes of
topologicalringsthatHuberdealtwithseparatelyinhiswork(thestronglynoetherianringsand
the f-adic rings with a noetherian ring of definition). We also point out a henselian variant of
the structure sheaf that is available on the adic spectrum of any f-adic ring, with no restriction
whatsoever.
• Thelastchapterproposesfornowacoupleofapplications: insection17.1weintroducea
classofmodelalgebrasoveranyrankonevaluationringK+ ofmixedcharacteristic(0,p),and
we show that when K+ is deeply ramified, such algebras are formal perfectoid rings for their
p-adictopology;hencethetheoryofchapter16immediatelyyieldsanalmostpuritytheoremfor
model algebras. Likewise, section 17.2 proves an almost purity theorem for certain very ram-
ified towers of log-regular rings; again, after some preliminary reductions, the proof amounts
to the observation that the inductive limits of such towers are formal perfectoid for their p-adic
topology. These instances of almost purity were already contained in a previous draft of our
work (Release 6), where they were proven by an extension of Faltings’s method, that relied on
deep results from logarithmic algebraic geometry, and also entailed the construction of certain
normalized lengths for torsion modules over model algebras, and respectively over the rings
occurring in section 17.2. Neither of these two ingredients intervenes any longer in the new
proofs; however, we have found worthwhile to explain how model algebras arise from suitable
veryramifiedtowersoflog-smoothK+-algebras,andwehavealsoretainedtheconstructionof
normalized lengths for model algebras and for limits of towers log-regular rings, since they are
sufficientlyinterestingintheirownright,andmightbeusefulforotherapplications(normalized
lengthsfortorsionK+-modulesareexploitedin[98]).
In the next release of this treatise, we shall complete chapter 17 with our account of Andre´’s
work on the direct summand conjecture, and with some further applications of our theory of
generalizedperfectoidrings.
1. BASIC CATEGORY THEORY
The purpose of this chapter is to fix some notation that shall stand throughout this work, and
to collect, for ease of reference, a few well known generalities on categories and functors that
are frequently used. Our main reference on general nonsense is the treatise [15], and another
goodreferenceisthemorerecent[72].
Sooner or later, any honest discussion of categories and topoi gets tangled up with some
foundational issues revolving around the manipulation of large sets. For this reason, to be
able to move on solid ground, it is essential to select from the outset a definite set-theoretical
framework(amongtheseveralcurrentlyavailable),andsticktoitunwaveringly.
Thus, throughout this work we will accept the so-called Zermelo-Fraenkel system of axioms
for set theory. (In this version of set theory, everything is a set, and there is no primitive notion
ofclass,incontrasttootheraxiomatisations.)
Additionally, following [5, Exp.I, §0], we shall assume that, for every set S, there exists a
universeV suchthatS ∈ V. (Forthenotionofuniverse,thereadermayalsosee[15,§1.1].)
Throughout this chapter, we fix some universe U such that N ∈ U (where N is the set of
natural numbers; the latter condition is required, in order to be able to perform some standard
set-theoretical operations without leaving U). A set S is U-small (resp. essentially U-small), if
S ∈ U(resp. ifS hasthecardinalityofaU-smallset). Ifthecontextisnotambiguous,weshall
justwritesmall,insteadofU-small.
8 OFERGABBERANDLORENZORAMERO
1.1. Categories, functors and natural transformations. A category C is the datum of a set
Ob(C)ofobjectsand,foreveryA,B ∈ Ob(C),asetofmorphismsfromAtoB,denoted:
Hom (A,B)
C
andasusualwewritef : A → B tosignifyf ∈ Hom (A,B). Furthermore,weset
C
Morph(C) := {(A,B,f)|A,B ∈ Ob(C), f ∈ Hom (A,B)}.
C
Foranyf := (A,B,f) ∈ Morph(C),theobjectAiscalledthesourceoff,andB isthetarget
off. Wealsooftenusethenotation
End (A) := Hom (A,A)
C C
and the elements of End (A) are called the endomorphisms of A in C. We say that a pair of
C
elements(f,g)ofMorph(A)iscomposableifthetargetoff equalsthesourceofg. Moreover,
foreveryA,B,C ∈ Ob(C)wehaveacompositionlaw
Hom (A,B)×Hom (B,C) → Hom (A,C) : (f,g) (cid:55)→ g ◦f
C C C
fulfillingthefollowingtwostandardaxioms:
• ForeveryA ∈ Ob(C)thereexistsanidentityendomorphism1 ofA,suchthat
A
1 ◦f = f g ◦1 = g foreveryB,C ∈ Ob(C)andeveryf : B → Aandg : A → C.
A A
• Thecompositionlawisassociative,i.e. wehave
(h◦g)◦f = h◦(g ◦f)
foreveryA,B,C,D ∈ Ob(C)andeveryf : A → B,g : B → C andh : C → D.
Clearly,itfollowsthat(End (A),◦,1 )isamonoid,andwegetagroup:
C A
Aut (A) ⊂ End (A)
C C
ofinvertibleendomorphisms,i.e. theautomorphismsoftheobjectA.
1.1.1. WesaythatthecategoryC isU-small(orjustsmall),ifbothOb(C)andMorph(C)are
smallsets. WesaythatC hassmallHom-setsifHom (A,B) ∈ UforeveryA,B ∈ Ob(C).
C
AsubcategoryofC isacategoryB withOb(B) ⊂ Ob(C)andMorph(B) ⊂ Morph(C).
TheoppositecategoryCo isthecategorywithOb(Co) = Ob(C),andsuchthat:
Hom (A,B) := Hom (B,A) foreveryA,B ∈ Ob(C)
Co C
(with composition law induced by that of C, in the obvious way). Given A ∈ Ob(C), some-
times we denote by Ao the same object, viewed as an element of Ob(Co); likewise, given a
morphismf : A → B inC,wewritefo forthecorrespondingmorphismBo → Ao inCo.
1.1.2. Amorphismf : A → B inC issaidtobeamonomorphismiftheinducedmap
Hom (X,f) : Hom (X,A) → Hom (X,B) g (cid:55)→ f ◦g
C C C
isinjective,foreveryX ∈ Ob(C). Dually,wesaythatf isanepimorphismiffo isamonomor-
phism in Co. Also, f is an isomorphism if there exists a morphism g : B → A such that
g ◦ f = 1 and f ◦ g = 1 . Obviously, an isomorphism is both a monomorphism and an
A B
epimorphism. Theconversedoesnotnecessarilyhold,inanarbitrarycategory.
Two monomorphisms f : A → B and f(cid:48) : A(cid:48) → B are equivalent, if there exists an
isomorphism h : A → A(cid:48) such that f = f(cid:48) ◦h. A subobject of B is defined as an equivalence
classofmonomorphismsA → B. Dually,aquotient ofB isasubobjectofBo inCo.
OnesaysthatC iswell-poweredif,foreveryA ∈ Ob(C),theset:
Sub(A)
ofallsubobjectsofAisessentiallysmall. Dually,C isco-well-powered,ifCo iswell-powered.
FOUNDATIONSFORALMOSTRINGTHEORY 9
1.1.3. LetA andB beanytwocategories;afunctorF : A → B isapairofmaps
Ob(A) → Ob(B) Morph(A) → Morph(B)
bothdenotedalsobyF,suchthat
• F assignstoanymorphismf : A → A(cid:48) inA,amorphismFf : FA → FA(cid:48) inB
• F1 = 1 foreveryA ∈ Ob(A)
A FA
• F(g ◦ f) = Fg ◦ Ff for every A,A(cid:48),A(cid:48)(cid:48) ∈ Ob(A) and every pair of morphisms
f : A → A(cid:48),g : A(cid:48) → A(cid:48)(cid:48) inA.
IfF : A → B andG : B → C areanytwofunctors,wegetacomposition
G◦F : A → C
whichisthefunctorwhosemapsonobjectsandmorphismsarethecompositionsoftherespec-
tivemapsforF andG. Wedenoteby
Fun(A,B)
the set of all functors A → B. Moreover, any such F induces a functor Fo : Ao → Bo with
FoAo := (FA)o andFofo := (Ff)o foreveryA ∈ Ob(A)andeveryf ∈ Morph(A).
Definition1.1.4. LetF : A → B beafunctor.
(i) We say that F is faithful (resp. full, resp. fully faithful), if for every A,A(cid:48) ∈ Ob(A) it
inducesinjective(resp. surjective,resp. bijective)maps:
Hom (A,A(cid:48)) → Hom (FA,FA(cid:48)) : f (cid:55)→ Ff.
A B
(ii) We say that F reflects monomorphisms (resp. reflects epimorphisms, resp. is conserva-
tive) if the following holds. For every morphism f : A → A(cid:48) in A, if the morphism Ff of B
isamonomorphism(resp. epimorphism,resp. isomorphism),thenthesameholdsforf.
(iii) If A is a subcategory of B, and F is the natural inclusion functor, then F is obviously
faithful,andwesaythatA isafullsubcategoryofB,ifF isfullyfaithful.
(iv) The essential image of F is the full subcategory of B whose objects are the objects of
B that are isomorphic to an object of the form FA, for some A ∈ Ob(A). We say that F is
essentiallysurjectiveifitsessentialimageisB.
(v) WesaythatF isanequivalence,ifitisfullyfaithfulandessentiallysurjective.
Remark1.1.5. Forlateruse,itisconvenienttointroducethenotionofn-faithfulfunctor,forall
integers n ≤ 2. Namely : if n < 0, every functor is n-faithful; a functor F : A → B (between
any two categories A and B) is 0-faithful, if it is faithful; F is 1-faithful, if it is fully faithful;
finally,wesaythatF is2-faithful,ifitisanequivalence.
Example 1.1.6. (i) The collection of all small categories, together with the functors between
them,formsacategory
U-Cat.
Unless we have to deal with more than one universe, we shall usually omit the prefix U, and
writejustCat. ItiseasilyseenthatCatisacategorywithsmallHom-sets.
(ii) The category of all small sets shall be denoted U-Set or just Set, if there is no need to
emphasizethechosenuniverse. Thereisanaturalfullyfaithfulembedding:
Set → Cat.
Indeed,toanysetS onemayassignitsdiscretecategoryalsodenotedS,i.e. theuniquecategory
such that Ob(S) = S and Morph(S) = {(s,s,1 ) | s ∈ S}. If S and S(cid:48) are two discrete
s
categories,thedatumofafunctorS → S(cid:48)isclearlythesameasamapofsetsOb(S) → Ob(S(cid:48)).
Noticealsothenaturalfunctor
Ob : Cat → Set C (cid:55)→ Ob(C)
10 OFERGABBERANDLORENZORAMERO
thatassignstoeachfunctorF : C → D theunderlyingmapOb(C) → Ob(D): C (cid:55)→ FC.
(iii) Recall that a preordered set is a pair (I,≤) consisting of a set I and a binary relation ≤
onI whichisreflexiveandtransitive. Inthiscase,wealsosaythat≤isapreorderingonI. We
saythat(I,≤)isapartiallyorderedset,if≤isalsoantisymmetric,i.e. ifwehave
(x ≤ y andy ≤ x) ⇒ x = y foreveryx,y ∈ I.
We say that (I,≤) is a totally ordered set, if it is partially ordered and any two elements are
comparable, i.e. for every x,y ∈ I we have either x ≤ y or y ≤ x. An order-preserving map
f : (I,≤) → (J,≤)betweenpreorderedsetsisamappingf : I → J suchthat
x ≤ y ⇒ f(x) ≤ f(y) foreveryx,y ∈ I.
We denote by Preorder (resp. POSet) the category of small preordered (resp. partially
ordered)sets,withmorphismsgivenbytheorder-preservingmaps. Toanypreorderedset(I,≤)
one assigns a category whose set of objects is I, and whose morphisms are given as follows.
Foreveryi,j ∈ I,thesetofmorphismsi → j containsexactlyoneelementwheni ≤ j,andis
emptyotherwise. Clearly,thisruledefinesafullyfaithfulfunctor
Preorder → Cat.
NoticethatifacategoryC liesintheessentialimageofthisfunctor,thenthesameholdsforCo.
Indeed, if C corresponds to the preordered set (I,≤), then Co corresponds to the preordered
set (Io,≤) with Io := I and x ≤ y in Io if and only if y ≤ x in I, for every x,y ∈ I. Clearly
(I,≤)isapartiallyorderedsetifandonlyifthesameholdsfor(Io,≤).
1.1.7. LetA,B betwocategories,F,G : A → B twofunctors. Anaturaltransformation
(1.1.8) α : F ⇒ G
from F to G is a family of morphisms (α : FA → GA | A ∈ Ob(A)) of B such that, for
A
everymorphismf : A → B inA,thediagram:
FA αA (cid:47)(cid:47) GA
(1.1.9)
Ff Gf
(cid:15)(cid:15) (cid:15)(cid:15)
FB αB (cid:47)(cid:47) GB
commutes. If α is an isomorphism for every A ∈ Ob(A), we say that α is a natural iso-
A
morphismoffunctors. Forinstance,therulethatassignstoanyobjectAtheidentitymorphism
1 : FA → FA, defines a natural isomorphism 1 : F ⇒ F. A natural transformation
FA F
(1.1.8)isalsoindicatedbyadiagramofthetype:
F(cid:31)(cid:31)(cid:31)(cid:31) (cid:41)(cid:41)
A (cid:11)(cid:19) α(cid:53)(cid:53) B.
G
1.1.10. The natural transformations between functors A → B can be composed; namely, if
α : F ⇒ Gandβ : G ⇒ H aretwosuchtransformations,weobtainanaturaltransformation
β (cid:12)α : F ⇒ H bytherule: A (cid:55)→ β ◦α foreveryA ∈ Ob(A).
A A
Withthiscomposition,Fun(A,B)isthesetofobjectsofacategorywhichweshalldenote
Fun(A,B).