Higher Categories and Homotopical Algebra

ThisbookprovidesanintroductiontomodernhomotopytheorythroughthelensofhighercategoriesafterJoyalandLurie,givingaccesstomethodsusedattheforefrontofresearchinalgebraictopologyandalgebraicgeometryinthetwenty-firstcentury.Thetextstartsfromscratch-revisitingresultsfromclassicalhomotopytheorysuchasSerre'slongexactsequence,Quillen'stheoremsAandB,Grothendieck'ssmooth/properbasechangeformulas,andtheconstructionoftheKan–Quillenmodelstructureonsimplicialsets-anddevelopsanalternativetoasignificantpartofLurie'sdefinitivereferenceHigherToposTheory,withnewconstructionsandproofs,inparticular,theYonedaLemmaandKanextensions.Thestrongemphasisonhomotopicalalgebraprovidesclearinsightsintoclassicalconstructionssuchascalculusoffractions,homotopylimitsandderivedfunctors.Forgraduatestudentsandresearchersfromneighbouringfields,thisbookisauser-friendlyguidetoadvancedtoolsthatthetheoryprovidesforapplication.