Higher-Dimensional Categories an Illustrated Guidebook

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1、Higher-DimensionalCategories:anillustratedguidebookEugeniaChengandAaronLaudaUniversityofCambridgeDraftVersionPreparedforIMAWorkshop1June2004Revised19August2004Higher-dimensionalcategoriesarelikeavastmountainthatmanypeoplearetryingtoconquer.Someintrepidexplorers

2、havemadetheascent,eachtakingadierentrouteandeachencounteringdierenthazards.Eachhasmadeamapofhisroute,butdoweknowhowallthesemapsttogether?Doweknowthattheyttogetheratall?Infact,areweevenclimbingthesamemountain?Thisworkisanillustratedguidebooktotheworldofhighe

3、r-dimensionalcategories.Amapwouldbemoredetailedandprecise.Anencyclopediawouldbemorecomprehensive.Ouraimisneitherrigournorcompleteness.Ouraimistoprovidewould-bevisitorswithasenseofwhattheymightndonarrival;togivethemanideaofwhatlandmarkstolookoutfor;towarnthemof

4、thehazardsoftheterritory;tointroducethemtothelanguageoftheplace;towhettheirappetiteforexploringbythemselves.Tothisend,weadoptaninformalandfriendlytone.Wein-cludeenoughdetailtogivevisitorstheirbearingsbutnotsomuchthattheyneedamagnifyingglasstondwhatthey'relooki

5、ngfor.And,mostimportantly,weprovidecopiouspicturestoillustrateourdescriptions.Inasmuchasthisisaguidebook,wewillassumethatthereaderisalreadyinterestedinvisiting.Thisisnotanadvertisingcampaign.However,wehopethat,withtheappearanceofaguidebook,morepeoplewillconside

6、rvisiting:::iiContentsPrefacevi1Introduction11.1Startingpointsforgeneralisation..................11.1.1CategoriesI:graphswithstructure.............21.1.2CategoriesII:objectsandmorphisms...........31.1.3CategoriesIII:nerves....................51.2Keypointsofdie

7、rence.......................51.2.1TheData-Structure-Properties(DSP)trichotomy.....51.2.2Algebraicvsnon-algebraic..................61.2.3Cellshape...........................71.2.4Weakn-categoriesorn-weakcategories?..........71.2.5Beheadingvsheadshrinking........

8、.........81.2.6Conclusion..........................82Penon92.1Introduction..............................92.2Intuition................................102.2.1Magmas...........