category theory lecture notes .pdf

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1、CategoryTheoryLectureNotesDanieleTuriLaboratoryforFoundationsofComputerScienceUniversityofEdinburghSeptember1996{December2001PrologueThesenotes,developedoveraperiodofsixyears,werewrittenforaneighteenlecturescourseincategorytheory.AlthoughheavilybasedonMacLane'sCategoriesfortheWorkingMathem

2、atician,thecoursewasdesignedtobeself-contained,drawingmostoftheexamplesfromcategorytheoryitself.Thecoursewasintendedforpost-graduatestudentsintheoreticalcomputerscienceattheLaboratoryforFoundationsofComputerScience,UniversityofEdinburgh,butwasattendedbyavariedaudience.Mostsectionsareareaso

3、nableaccountofthematerialpresentedduringthelectures,butsome,mostnotablythesectionsonLawveretheories,topoiandKanextensions,arelittlemorethanacollectionofdenitionsandfacts.ContentsIntroduction11UniversalProblems11.1NaturalNumbersinsettheoryandcategorytheory...............11.2Universals.....

4、.................................52BasicNotions72.1Categories......................................72.2Functors.......................................92.3InitialandFinalObjects..............................102.4CommaCategories.................................103Universality114NaturalTransf

5、ormationsandFunctorCategories145Colimits185.1Coproducts.....................................185.2Coequalisers.....................................185.3Pushouts......................................195.4Initialobjectsasuniversalarrows.........................205.5GeneralisedCoproducts......

6、........................225.6FiniteColimits...................................225.7Colimits.......................................23ii6DualityandLimits256.1Universalarrowsfromafunctortoanobject..................256.2Products.......................................266.3Equalisers............

7、..........................266.4Pullbacks......................................276.5Monos........................................286.6Limits........................................297Adjunctions317.1FromUniversalArrowstoAdjunctions......................317.2Adj