Sheaf Theory Through Examples

453 Pages·2022·3.83 MB·English

by  Daniel Rosiak

SheafTheorythroughExamples

SheafTheorythroughExamples
DanielRosiak
TheMITPress
Cambridge,Massachusetts
London,England
©2022MassachusettsInstituteofTechnology
ThisworkissubjecttoaCreativeCommonsCC-BY-NC-NDlicense.
Subjecttosuchlicense,allrightsarereserved.
TheopenaccesseditionofthisbookwasmadepossiblebygenerousfundingfromArcadia—a
charitablefundofLisbetRausingandPeterBaldwin.
The MIT Press would like to thank the anonymous peer reviewers who provided comments on
drafts of this book. The generous work of academic experts is essential for establishing the authority
and quality of our publications. We acknowledge with gratitude the contributions of these otherwise
uncredited readers.
This book was set in LaTeX by the author.
Library of Congress Cataloging-in-Publication Data
Names: Rosiak, Daniel, author.
Title: Sheaf theory through examples / Daniel Rosiak.
Description: Cambridge, Massachusetts : The MIT Press, [2022] | Includes bibliographical
references and index.
Identifiers: LCCN 2021058949 | ISBN 9780262542159 (paperback)
Subjects: LCSH: Sheaf theory.
Classification: LCC QA612.36 .R67 2022 | DDC 514/.224–dc23/eng20220521
LC record available at https://lccn.loc.gov/2021058949
ToFernandoZalamea,whoguidedmyearlyforayintosheaves,andwhose
encouragementandexamplehelpedmyloveofcategorytheoryflourish.

Contents
Acknowledgments ix
Introduction 1
1 Categories 21
1.1 CategoricalPreliminaries 21
1.2 AFewMoreExamples 35
1.3 ReturningtotheDefinitionandDistinctionsofSize 36
1.4 SomeNewCategoriesfromOld 40
1.5 Asideon“NoObjects” 43
2 PreludetoSheaves:Presheaves 45
2.1 Functors 46
2.2 NaturalTransformations 61
2.3 SeeingStructuresasPresheaves 65
2.4 ThePresheafAction 71
2.5 PhilosophicalPass:TheFourActionPerspectives 85
3 UniversalConstructions 89
3.1 LimitsandColimits 89
3.2 PhilosophicalPass:UniversalityandMediation 104
4 Topology:AFirstPassatSpace 109
4.1 Motivation 109
4.2 ADialogueIntroducingtheKeyNotionsofTopology 111
4.3 TopologyandTopologicalSpacesMoreFormally 123
4.4 PhilosophicalPass:OpenQuestions 144
5 FirstLookatSheaves 147
5.1 Sheaves:TheTopologicalDefinition 147
5.2 Examples 151
5.3 PhilosophicalPass:SheafasLocal-GlobalPassage 168
6 There’saYonedaLemmaforThat 171
6.1 First,Enrichment! 171
6.2 DownsetsandYonedaintheMiniature 175
6.3 RepresentabilitySimplified 179
6.4 MoreonRepresentability,FixedPoints,andaParadox 183
viii Contents
6.5 YonedaintheGeneral 187
6.6 PhilosophicalPass:YonedaandRelationality 190
7 Adjunctions 195
7.1 AdjunctionsthroughMorphology 195
7.2 AdjunctionsthroughModalities 210
7.3 SomeAdditionalAdjunctionsandFinalThoughts 237
7.4 PhilosophicalPass:TheIdeaofAdjointness 239
8 SheavesRevisited 241
8.1 ThreeHistoricallySignificantExamples 241
8.2 WhatIsNotaSheaf? 267
8.3 PresheavesandSheavesinOrderTheory 271
9 CellularSheafCohomologythroughExamples 277
9.1 SimplicesandTheirSheaves 279
9.2 SheafCohomology 292
9.3 PhilosophicalPass:SheafCohomology 305
9.4 AGlimpseintoCosheaves 306
10 SheavesonaSite 311
10.1 RevisitingCovers:TowardGeneralSheaves 311
10.2 GrothendieckToposes 314
10.3 AFewMoreExamples 353
10.4 PhilosophicalPass:TheIdeaofToposes 365
11 ElementaryToposes 369
11.1 TheSubobjectClassifier 370
11.2 ExamplesofElementaryToposes 377
11.3 Lawvere-TierneyTopologiesandTheirSheaves 379
11.4 MorphismsofToposes 394
11.5 TowardCohesiveToposes 400
A Appendix(RevisitingTopology) 411
A.1 ConceptualMotivation:TopologyasLogicofFiniteObservations 412
A.2 ExplicitConnectionstoModalLogic 415
A.3 TheIdeaofAllThis 424
A.4 WhyOpens? 426
A.5 WhatIsTopologyReallyAbout? 429
References 431
Index 439
Acknowledgments
I didn’t set out to write a book on sheaves. In 2017, as I was wrapping up my PhD in
philosophy, I received a Fulbright grant to work with Fernando Zalamea in Bogotá, the
ostensible aim of which was to support the completion of a brief supplementary chapter
givingahigh-leveloverviewofcertainphilosophicalelementsofinterestincategorytheory
and sheaf theory. While I had already been doing research in category theory for some
years,asIcametodevotemoreattentiontosheaftheoryinitsownrightthatyearIbecame
entirelyengrossedandfoundmyselfthinkingaboutsheavesalldayeveryday.Bytheend
of that year, I found that I had produced over 200 pages of notes on sheaves with novel
examples and reflections. A number of mathematicians with whom I shared these early
notes found them stimulating and encouraged me to circulate them. While it was never
myintentiontowriteabook,manyreadersofthoseearlynotesencouragedmetodevelop
them into something more. Over the next four years, as my research led me further into
the land of sheaves, and guided by feedback from new readers, I continued to add new
examplesandcontenthereandthere,andatsomepointitbecameevidentthatthesewere
nolongernotes.Ithadgrownintosomethingclosetothepresentbook.
Overtheyears,Ihavetriedtofilloutthestoryinanumberofways.Butonaccountofits
somewhat accidental genesis, the book remains necessarily incomplete and my selection
ofexamplesandpointsoffocusiscertainlypartial.Ontheotherhand,afteryearsofnew
additionsandrevisions,Idohopesomeoftheinitialexcitementofthoseearlynotesstill
shinesthrough.
Therearemanypeoplewhosefeedbackovertheyearshasgreatlyimprovedthiswork,
and many people from whom I have learned an immense amount along the way. Above
all, I would like to express my profound gratitude to Fernando Zalamea, without whose
support and encouragement, from the earliest stages, this book would never have come
to be. Among the colleagues who offered encouragement and/or valuable feedback on
various versions of this manuscript, I would like to thank Colin McLarty, Emily Riehl,
MichaelRobinson,DavidSpivak,WalterTholen,andPhilippZahn.Iamespeciallygrate-
fultoJustinCurryandAlanHylton,whosedetailedcomments,suggestionsforsubstantial
improvements to the organization of the manuscript, stimulating discussions, and ongo-
ing encouragement helped me get the book over the finish line. Let me also express my
indebtednesstoWilliamLawvere,whoseworkhasbeenawellofinspirationformeover
theyears;withouthisparticularwisdom,IamsurethatIwouldneverhavecometoaspire
to the new knowledge of reality that is thinking categorically. I would also like to thank
x Acknowledgments
theanonymousreviewersfromMITPressfortheirhelpfulfeedbackandcorrectionsonan
earlierversion.
IamverygratefultothecontributorstothenLab,forspendingtheirtimeputtingtogether
manyimpressivelyrichreflectionsoncategorytheory,sheaves,andevensomephilosophy.
Inanagewhenitissoeasytodespairofthemountainsofsuperficialcontentonline,and
soeasytofindunrewardingwaysofspendingone’stimelookingatascreen,thankyoufor
creatingacorneroftheinternetwithsomethingprofoundandrewarding.
Finally, I would like to thank the editors at MIT Press, Jermey Matthews, Haley Bier-
mann,andMarcyRoss;aswellasYasuyoIguchiwhodesignedthecover;MichaelDurnin
who copyedited the book; and Jim Mitchell and Theresa Carcaldi who managed produc-
tionofthebook.Thebookismuchimprovedfromwhatitwasmanyversionsago.Thanks
alsototheteamforhelpingwithvariousLaTeXissuesthatcameup,andforsavingmethe
headaches.IamespeciallygratefultoMITPressforhelpingtomakethebookavailablein
anopenaccessversion,allowingittoreachmorereaders.
And of course, my family and friends: for all their patience, love, and support. With-
out Michelle, I am certain that this book wouldn’t have seen the light: thank you for
encouraging me whenever I wanted to give it up and for reminding me of the point of
itall.