Involutive Category Theory

250 Pages·2020·13.474 MB·English

by  Donald Yau

Lecture Notes in Mathematics 2279
Donald Yau
Involutive
Category
Theory
Lecture Notes in Mathematics
Volume 2279
Editors-in-Chief
Jean-MichelMorel,CMLA,ENS,Cachan,France
BernardTeissier,IMJ-PRG,Paris,France
SeriesEditors
KarinBaur,UniversityofLeeds,Leeds,UK
MichelBrion,UGA,Grenoble,France
CamilloDeLellis,IAS,Princeton,NJ,USA
AlessioFigalli,ETHZurich,Zurich,Switzerland
AnnetteHuber,AlbertLudwigUniversity,Freiburg,Germany
DavarKhoshnevisan,TheUniversityofUtah,SaltLakeCity,UT,USA
IoannisKontoyiannis,UniversityofCambridge,Cambridge,UK
AngelaKunoth,UniversityofCologne,Cologne,Germany
ArianeMézard,IMJ-PRG,Paris,France
MarkPodolskij,UniversityofLuxembourg,Esch-sur-Alzette,Luxembourg
SylviaSerfaty,NYUCourant,NewYork,NY,USA
GabrieleVezzosi,UniFI,Florence,Italy
AnnaWienhard,RuprechtKarlUniversity,Heidelberg,Germany
This series reports on new developments in all areas of mathematics and their
applications-quickly,informallyandatahighlevel.Mathematicaltextsanalysing
newdevelopmentsinmodellingandnumericalsimulationarewelcome.Thetypeof
materialconsideredforpublicationincludes:
1. Researchmonographs
2. Lecturesonanewfieldorpresentationsofanewangleinaclassicalfield
3. Summerschoolsandintensivecoursesontopicsofcurrentresearch.
Textswhichareoutofprintbutstillindemandmayalsobeconsiderediftheyfall
withinthesecategories.Thetimelinessofamanuscriptissometimesmoreimportant
thanitsform,whichmaybepreliminaryortentative.
Moreinformationaboutthisseriesathttp://www.springer.com/series/304
Donald Yau
Involutive Category Theory
123
DonaldYau
DepartmentofMathematics
OhioStateUniversityatNewark
Newark,OH,USA
ISSN0075-8434 ISSN1617-9692 (electronic)
LectureNotesinMathematics
ISBN978-3-030-61202-3 ISBN978-3-030-61203-0 (eBook)
https://doi.org/10.1007/978-3-030-61203-0
MathematicsSubjectClassification:16W10,18-01,18D10,18D50,46C50,18C15,46L05
©TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusivelicensetoSpringerNatureSwitzerland
AG2020
Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher,whether
thewholeorpartofthematerialisconcerned,specificallytherightsoftranslation,reprinting,reuse
ofillustrations,recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,and
transmissionorinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilar
ordissimilarmethodologynowknownorhereafterdeveloped.
Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication
doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant
protectivelawsandregulationsandthereforefreeforgeneraluse.
Thepublisher,theauthors,andtheeditorsaresafetoassumethattheadviceandinformationinthisbook
arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor
theeditorsgiveawarranty,expressedorimplied,withrespecttothematerialcontainedhereinorforany
errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional
claimsinpublishedmapsandinstitutionalaffiliations.
ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG
Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland
ToEunSooandJacqueline
Preface
For an object X in a category, such as a set or a vector space, an involution is a
degree2self-map,i.e.,amapj :X (cid:2)(cid:2) Xsuchthatj ◦j =Id .Aninvolutive
X
category is a categorification of an object with an involution. More precisely, an
involutivecategoryisacategoryCequippedwith
(cid:129) afunctorI :C (cid:2)(cid:2) C,calledtheinvolutionfunctor,and
(cid:129) anaturalisomorphismι:Id ∼= (cid:2)(cid:2) I2,calledtheunitofI,
C
suchthatthecoherencecondition:
ιI =Iι:I (cid:2)(cid:2) I3,
calledthetriangleidentity,issatisfied.Involutivecategoriesareimportantbecause
they are the natural setting to study objects with involutive structures. A basic
example is the involutive category Vect(C) of complex vector spaces, with the
involutionfunctorI givenbytakingconjugatevectorspaces.Star-algebras,which
satisfy
(ab)∗ =b∗a∗,
and their differential graded analogues both arise naturally as involutive objects
in some involutive categories. These ∗-algebras are important in applications to
physics,inparticularasalgebrasofobservablesinquantumtheory.
Thisbookisaboutinvolutivecategoriesandinvolutiveoperads,withapplications
totheGNSconstructionandalgebraicquantumfieldtheory.Whiletherearemany
books on category theory, this book is the first one that focuses on involutive
categories.Someofthematerials—includingpartsofChapters2,4,7,and8—are
basedontheworkofJacobs[Jac12]andBenini,Schenkel,andWoike[BSW19b].
Other parts—including Chapters 3, 5, and 6 on coherence and strictification of
involutive categories and involutive (symmetric) monoidal categories—are new
materialsthathavenotappearedanywhereelsebefore.
vii
viii Preface
This book is written at the first- to second-year graduate level. With detailed
explanationandexercisesattheendofeverychapter,thisbookissuitableforinde-
pendentstudyandgraduatecourses.Itisalsoausefulreferenceformathematicians
andphysicistswhoseworkinvolvesinvolutiveobjects.
Abriefdescriptionofeachchapterfollows.
Chapter1.CategoryTheory
To make this book as self-contained as possible, in this chapter we recall some
basic category theory, including monoidal categories and monads. Since we will
discusscolimitsrelatedtoinvolutivecategories,inthischapterweprovideasimple
and constructive proof that the category Cat of small categories has all small
colimits. This is a well-known and often used fact, but a direct proof is not easily
accessibleintheliterature.
Chapter2.InvolutiveCategories
Involutive analogues of basic categorical concepts, including functors, natural
transformations, and adjunctions, are defined in this chapter. In involutive cate-
gories,thereisanaturalconceptofaninvolutiveobject,whichconsistsofanobject
Xandamorphismj :X (cid:2)(cid:2) IX,suchthatthediagram:
is commutative. For a given involutive category (C,I,ι), the category Inv(C) of
involutive objects inherits from C an involutive structure and other categorical
properties. One main result of this chapter is Theorem 2.3.8. It states that for an
adjunction between involutive categories, if one of the functors is an involutive
functor, then so is the other one, and the adjunction is an involutive adjunction.
As a consequence, if an involutive functor is an equivalence of categories in the
usualsense,thenitisaninvolutiveequivalence.
Chapter3.CoherenceofInvolutiveCategories
Witheachtypeofcategoriescomesasetofcoherencequestions.Inthischapter,
weproveseveralcoherencepropertiesofinvolutivecategories.Aninvolutivecate-
goryisstrictiftheunitistheidentitynaturaltransformationId : Id (cid:2)(cid:2) I2 =
C
Id . We construct the free strict involutive category, the cofree strict involutive
C
category,andthefreeinvolutivecategoryofacategory.Then,weobservethatthere
isauniquestrictinvolutiveequivalencefromthefreeinvolutivecategorytothefree
strictinvolutivecategoryofacategory.Next,weshowthatinvolutivecategoriesare
algebrasoveramonad.Wealsoshowthatthecategoryofinvolutivecategoriesand
strictinvolutivefunctorsislocallyfinitelypresentableandthatitsforgetfulfunctor
toCatadmitsarightadjoint.
Preface ix
Chapter4.InvolutiveMonoidalCategories
This chapter is about the involutive analogues of monoidal categories. For an
involutive monoidal category C, its category Inv(C) of involutive objects inherits
fromCthestructureofaninvolutivemonoidalcategory.MonoidsinInv(C)arethe
sameasinvolutiveobjectsinMon(C),whichisthecategoryofmonoidsinCwith
aninheritedinvolutivestructure.IfCisaninvolutivesymmetricmonoidalcategory,
then Mon(C) has another involutive structure that incorporates the symmetry iso-
morphisminC.Thecorrespondinginvolutiveobjectsarecalledreversinginvolutive
monoids. Complex unital ∗-algebras and their differential graded analogues are
both examples of reversing involutive monoids. This chapter ends with a section
oninvolutivemonads,whichyieldmoregeneralobjectswithinvolutivestructures.
Chapter5.CoherenceofInvolutiveMonoidalCategories
In this chapter, we prove several coherence properties of involutive monoidal
categories. First, we construct the free involutive monoidal category and the free
involutivestrictmonoidalcategoryofacategoryandshowthattheyareequivalent
via a strict involutive strict monoidal equivalence. Then, we prove that all formal
diagrams in an involutive monoidal category are commutative. Next, we prove a
strictificationresultthatsaysthateveryinvolutivemonoidalcategoryisequivalent
toaninvolutivestrictmonoidalcategoryviaaninvolutiveadjointequivalence.This
chapterendswiththeconstructionsofthefreeinvolutivemonoidalcategoryandthe
freeinvolutivestrictmonoidalcategoryofaninvolutivecategory.
Chapter6.CoherenceofInvolutiveSymmetricMonoidalCategories
This chapter is the symmetric analogue of the previous chapter. We prove a
numberofcoherenceresultsregardinginvolutivesymmetricmonoidalcategories.In
particular,weproveastrictificationresultthatsaysthateveryinvolutivesymmetric
monoidalcategoryisequivalenttoaninvolutivestrictsymmetricmonoidalcategory
via an involutive adjoint equivalence. This strictification result allows us to safely
ignore parentheses in multiple monoidal products in an involutive symmetric
monoidal category. This is particularly important when we discuss involutive
operadsinChapter8.
Chapter7.CategoricalGelfand-Naimark-SegalConstruction
The GNS construction is an important result in functional analysis that has
consequencesinquantummechanics.Thischaptercontainsacategoricalanalogue
oftheGNSconstructionasanisomorphismbetweentwocategories.Thisresultis
due to Jacobs [Jac12], and we provide detailed proofs of some technical Lemmas
thatarenotincludedinthatpaper.
Chapter8.InvolutiveOperads
Anoperadisageneralizationofacategorythatencodesalgebraicstructureswith
multipleinputsandoneoutput.Withoutassuminganypriorknowledgeofoperads,
inthischapterwefirstrecallthedefinitionsofanoperadandofanalgebraoveran
operad and provide a list of examples. Involutive operads are involutive monoids
in an involutive monoidal category of colored symmetric sequences. We observe
thateveryinvolutiveoperadyieldsaninvolutivemonad,whoseinvolutivealgebras
x Preface
are defined as those of the involutive operad. For example, reversing involutive
monoidsareinvolutivealgebrasovertheinvolutiveassociativeoperad,andsimilarly
forthediagramversion.Next,weprovethateveryinvolutivesymmetricmonoidal
functorinducesaninvolutivefunctorbetweentherespectiveinvolutivecategoriesof
involutiveoperads.Asexamples,weobservethattheinvolutiveassociativeoperad
and the algebraic quantum field theory involutive operad of [BSW19b] both arise
fromthischange-of-categoryconstruction.
Newark,OH,USA DonaldYau