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1、MichaelBarrCharlesWellsToposes,TriplesandTheoriesVersion1.110September2000Copyright2000byMichaelBarrandCharlesFrederickWells.Thisversionmaybedownloadedandprintedinunmodi edformforprivateuseonly.Itisavailableathttp://www.cwru.edu/artsci/math/wells/pub/t
2、tt.htmlandftp.math.mcgill.ca/pub/barrasanyofthe lesttt.dvi,ttt.ps,ttt.ps.zip,ttt.pdf,ttt.pdf.zip.MichaelBarrPeterRedpathProfessorEmeritusofMathematics,McGillUniversitybarr@barrs.orgCharlesWellsProfessorEmeritusofMathematics,CaseWesternReserveUniversity
3、AliateScholar,OberlinCollegecharles@freude.comToMarciaandJaneContentsPrefacevi1.Categories11.1De nitionofcategory11.2Functors111.3Naturaltransformations161.4ElementsandSubobjects201.5TheYonedaLemma261.6Pullbacks291.7Limits351.8Colimits481.9Adjointfunc
4、tors541.10Filteredcolimits671.11NotestoChapterI712.Toposes742.1BasicIdeasaboutToposes742.2SheavesonaSpace782.3PropertiesofToposes862.4TheBeckConditions922.5NotestoChapter2953.Triples973.1De nitionandExamples973.2TheKleisliandEilenberg-MooreCategories10
5、33.3Tripleability1093.4PropertiesofTripleableFunctors1223.5SucientConditionsforTripleability1283.6MorphismsofTriples1303.7AdjointTriples1353.8HistoricalNotesonTriples1424.Theories1444.1Sketches1454.2TheEhresmann-KennisonTheorem1494.3Finite-ProductTheo
6、ries1524.4LeftExactTheories1584.5NotesonTheories170iv5.PropertiesofToposes1735.1TripleabilityofP1735.2SlicesofToposes1755.3LogicalFunctors1785.4ToposesareCartesianClosed1835.5ExactnessPropertiesofToposes1865.6TheHeytingAlgebraStructureon1936.Permanence
7、PropertiesofToposes1986.1Topologies1986.2SheavesforaTopology2036.3Sheavesformatopos2096.4Leftexactcotriples2116.5Leftexacttriples2156.6CategoriesinaTopos2206.7GrothendieckTopologies2266.8Giraud'sTheorem2317.RepresentationTheorems2407.1Freyd'sRepresenta
8、tionTheorems2407.2TheAxiomofChoice2457.3MorphismsofSites2497.4Deligne'sTheorem2567.5NaturalNumberObjects2577.6CountableToposesandSeparableToposes2657.7Barr'sTheorem2727.8NotestoChapter72748.CoconeTheories2778.1RegularTheories2778.2Finit
