From Groups to Categorial Algebra: Introduction to Protomodular and Mal’tsev Categories

113 Pages·2017·1.769 MB·English

by  Dominique Bourn (auth.)

Compact Textbooks in Mathematics
Dominique Bourn
From Groups
to Categorial
Algebra
Introduction to Protomodular
and Mal’tsev Categories
CompactTextbooksinMathematics
Forfurthervolumes:
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DominiqueBourn
From Groups to
Categorial Algebra
IntroductiontoProtomodularandMal’tsevCategories
DominiqueBourn
DépartementdeMathématiques
UniversitéduLittoral
Calais,France
ISSN2296-4568
CompactTextbooksinMathematics
ISBN978-3-319-57218-5 ISBN978-3-319-57219-2(eBook)
DOI10.1007/978-3-319-57219-2
LibraryofCongressControlNumber:2017944916
MathematicsSubjectClassification(2010):08-XX,18-XX
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àIr.M.
Preface
Protomodular categories were introduced more than 25 years ago now [4]. The
book,writtenwithmycolleagueF.Borceux[3],whichgaveafirstoverviewofthe
theory and of its strong relationship with a conceptual non-abelian homological
algebra,isnowmorethan10yearsold.Sothetimeseemedtohavecometopresent
thefieldinarefreshedandrenewedway,drawinguponclarificationsandadvances
madeduringthelastdecade.
This could have remained a wishful thinking without an invitation of Prof.
ZurabJanelidzetogiveaseriesoflecturesformasterstudentsattheDepartment
ofMathematics&AppliedMathematicsoftheUniversityofStellenboschduring
the2ndMathematicalStructuresWorkshopinNovember2015.
It turned the project to the more precise direction of an elementary and syn-
theticintroductionaimingatmasterandpostgraduatestudents,andstartingfrom
verybasicalgebraicobservations.Italsoallowedmetocarryouttheoldprojectof
understandingthemainpropertiesregardingtheequivalencerelationsinthecon-
textofregularcategories withoutresorting toanymetatheoremandonlydealing
withdiagrammaticproofs,seeforinstanceLemma5.2.7andTheorem5.2.8.
I am very grateful for the numerous fruitful discussions on the topic I had
for years with myoldfriends andcolleagues Francis Borceux,Marino Gran and
GeorgeJanelidze.IthankverymuchDr.AmartyaGoswanifromtheUniversityof
LimpopowhotookfaithfulnotesoftheselecturesandmadeafirstLaTeXdraft.
Paris,November17,2017 DominiqueBourn
vii
Introduction
We suppose nobody will contest that the paradigmatic example of an algebraic
structureisthestructureofgroupand,accordingly,thattheparadigmaticexample
of a category in Algebra is the category Gp of groups. This would mean that it
couldcapturetheessenceofthemajoralgebraicstructuresincategoricalterms.
These lecture notes are an attempt to substantiatethis assertion: they will be
mainlydevoted to investigatingtheconceptualproperties of this category andto
showinghowthesepropertiesaresharedwithmanyotherimportantcategoriesin
Algebra. They are written as a self-contained document.Since the lectures were
intended for an audience of master students, only little knowlegde on Category
Theoryisassumedhere.However,formoredetails,wecanmention[28],[2],[31]
and[27].
SincethecategoryGpispartofalargertableofinclusions:
CoM Mon
Ab Gp
whereMonisthecategoryofmonoids,AbthecategoryofabeliangroupsandCoM
thecategoryofcommutativemonoids,severalobservationsaboutthesethreeother
basicalgebraicstructureswillfindtheirplacesintheselecturenotesnaturally.
ThereisnosuchparadigmaticcategoryinAnalysis,howeverthecategoryTop
oftopologicalspacesiscertainlyaveryimportantoneandwillbeoftenusefulas
aninterestingcounterpointtothealgebraiccontexts.
Certainly one aspect of the art of doing mathematics is to extract from very
basic facts the most extreme consequences, and it is probably one of the major
aspectsofcategorytheory.Itiswhatweconstantlytriedtodohere,andwhywe
constantlytriedtobringtheproofstothehighestdegreeofclarityandconciseness.
Here are the four very basic facts in Algebra which will be used as starting
points:
(A)inMon
givenanypairofmonoids.M;(cid:2)/,.N;(cid:2)/,then8.x;y/intheproductM (cid:3)N we
get:
.x;y/D.x;1/(cid:2).1;y/D.1;y/(cid:2).x;1/
(B)inMon
(B )givenacommutativemonoid.M;C/,thebinaryoperationCisahomomor-
1
phismofmonoids:
C
.M;C/(cid:3).M;C/(cid:4)!.M;C/
viii Introduction
(B )thereisakindofconverseofthepreviousone,knownastheEckmann-Hilton
2
argument[19]:givenanymonoid.M;(cid:2)/andanybinaryoperation(cid:5)withunitein-
sidethecategoryMon,namelywherebothfunctionsaremonoidhomomorphisms:
(cid:5) e
.M;(cid:2)/(cid:3).M;(cid:2)/(cid:4)!.M;(cid:2)/ (cid:4).1;(cid:2)/
thenthetwobinaryoperation(cid:2)and(cid:5)coincideandthemonoid.M;(cid:2)/isnecessarily
commutative.
Proof
SincetheuniteW1!M isamonoidhomomorphism,ecoincidesnecessarilywiththeunit1of
.M;(cid:2)/.Ontheotherhandsayingthat(cid:5)isamonoidhomomorphismissayingthat8.x;y;x0;y0/2
M4weget:
.x(cid:2)x0/(cid:5).y(cid:2)y0/D.x(cid:5)y/(cid:2).x0(cid:5)y0/
settingx0D1Dy,wegetx(cid:5)y0Dx(cid:2)y0
settingxD1Dy0,wegetx0(cid:5)yDy(cid:2)x0,sothatx(cid:2)yDy(cid:2)x. (cid:2)
(C)inAb
givenanysplitepimorphism.f;s/WA(cid:2)B (f:s D1 )inAb,theidentity:
B
aDsf.a/C.a(cid:4)sf.a//
produces the uniquedecompositiona D s.b/Ck with k 2 Kerf; in classical
terms,thissaysthatwehaveadirectsumADs.B/˚Kerf.
(D)inGp
givenanysplitepimorphism.f;s/WX (cid:2)Y (f:s D1 )inGp,theidentity:
Y
x Dsf.x/:.sf.x(cid:4)1/:x/
showsthatanysubgroupX0 ofX whichcontainsthesubgroupss.Y/andKerf
isnecessarilyX or,inotherwords,WthatX isthesupremumofthesubgroupss.Y/
andKerf,namelythatX Ds.Y/ Kerf.
Asemphasizedabove,weshallconcentrateourattentionongroups,namelyon
theobservation(D),fromwhichwillbeextractedourmainnotionofprotomodular
category.Weshallshowthatanypointedprotomodularcategorysharesfivestriking
featuresofthecategoryGpofgroups,namely:
(a) aninjectivehomomorphismf WX !Y ischaracterizedbythetrivialityof
itskernel:Kerf D1
(b) asurjectivehomomorphismisnecessarilythecokernelofitskernel
(c) thereisaveryspecificclassofsubobjects,namelythenormalsubobjects
(d) beingcommutativeisaproperty
(e) anyreflexiverelationisanequivalencerelation.
ix
Introduction
Andalsoasannouncedabove,weshallseethattherearemanyclassesofalgebraic
structures which share with the category Gp the property of being protomodu-
lar:thecategoriesRgandURgofnon-unitaryringsandofunitaryrings;givena
ringR,thecategoriesofanykindofnon-unitaryR-algebras(associative,commu-
tative, Lie, Jordan, Poisson, Leibniz, etc) and of any kind of unitary R-algebras
(associative,commutative,Jordan).
This is the case of any category Gp(E) of internal groups in E as well, and
inparticularofthecategoryGp(Top)oftopologicalgroups.Therearemoreunex-
pectedexamplesasthedualcategoriesSetop andSetopofthecategoriesofsetsand
(cid:5)
ofpointedsets.
In a way, the observations (C) and (D) show that the distance between the
categoriesGp ofgroupsand Abofabelian groupscan bepreciselymeasured,in
termsofsplitepimorphisms,bythedistancebetweenthenotionofsupremumand
thenotionofdirect sum. It is exactly thesamesituation concerningthe distance
betweenpointedprotomodularcategoriesandadditivecategories.Accordinglywe
haveakindofpreciseratio:
Ab ˚ direct sum additivecategory
D W D D
Gp supremum protomodularcategory
ToconcludewewouldliketoemphasizethatCategoryTheoryisnotonlyacon-
ceptualcontextallowingtodescribemathematicalsituationsthankstothenotion
ofobjects,arrowsandcompositionofarrows:
B
f g
A C
g:f
but also how it produces a highly effective way of apprehending thoroughly an
object,throughthenotionofuniversalproperty,which isdeveloppedindetailin
7Sect.1.5.Wemustconfessthatittookussometime(wemeanyears)torealize
thatitisakindofheuristicmini-revolution:besidestheusualwayofidentifying
orcharacterizinganobjectorastructure
4 bytheknowledgeofitsinside(=itselementsandtheirrelationships)
itgivesustheabilitytodoitexclusively
4 bytheknowledgeofitsrelationshipwithitsneighbours,withitsfellowenti-
ties(=theuniversalproperty)
Wehopethatthesenoteswill allowthereader to appreciate howproductivethis
mini-revolutionis.