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FREE MONOID IN MONOIDAL ABELIAN CATEGORIES

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FREE MONOID IN MONOIDAL ABELIAN CATEGORIES BRUNO VALLETTE Abstract. We give an explicit construction of the free monoid in monoidal abelian categories when the monoidal product does not necessarily preserve coproducts. Then we apply it to several new monoidal categories that appeared recently in the theory of Koszul duality for operads and props. This gives a conceptual explanation of the form of the free operad, free dioperad and free properad. Contents Introduction 1 1. Conventions 2 2. Reflexive coequalizers 4 3. Construction of the free monoid 6 4. Split analytic functors 10 5. Applications 11 5.1. Free properad 11 5.2. Free 12 -prop 14 5.3. Free dioperad 15 5.4. Free special prop 16 5.5. Free colored operad 16 References 16 Introduction The construction of the free monoid in monoidal categories is a general problem that appears in many fields of mathematics. In a monoidal category with denumer- able coproducts, when the monoidal product preserves coproducts, the free monoid on an object V is well understood and is given by the words with letters in V (see [MacL1] Chapter VII Section 3 Theorem 2). In general, the existence of the free monoid has been established, under some hypotheses, by M. Barr in [B]. When the monoidal product preserves colimits over the simplicial category, E. Dubuc de- scribed in [D] a construction for the free monoid.

  • preserves reflexive

  • biadditive monoidal

  • associated free monoids

  • free properad

  • sub-objects a1

  • multiplication functors

  • monoidal abelian

  • categories

  • preserve coproducts

  • reflexive coequalizers


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FREEMONOIDINMONOIDALABELIANCATEGORIESBRUNOVALLETTEAbstract.Wegiveanexplicitconstructionofthefreemonoidinmonoidalabeliancategorieswhenthemonoidalproductdoesnotnecessarilypreservecoproducts.ThenweapplyittoseveralnewmonoidalcategoriesthatappearedrecentlyinthetheoryofKoszuldualityforoperadsandprops.Thisgivesaconceptualexplanationoftheformofthefreeoperad,freedioperadandfreeproperad.ContentsIntroduction1.Conventions2.Reexivecoequalizers3.Constructionofthefreemonoid4.Splitanalyticfunctors5.Applications5.1.Freeproperad15.2.Free2-prop5.3.Freedioperad5.4.Freespecialprop5.5.FreecoloredoperadReferences12460111114151616161IntroductionTheconstructionofthefreemonoidinmonoidalcategoriesisageneralproblemthatappearsinmanyfieldsofmathematics.Inamonoidalcategorywithdenumer-ablecoproducts,whenthemonoidalproductpreservescoproducts,thefreemonoidonanobjectViswellunderstoodandisgivenbythewordswithlettersinV(see[MacL1]ChapterVIISection3Theorem2).Ingeneral,theexistenceofthefreemonoidhasbeenestablished,undersomehypotheses,byM.Barrin[B].Whenthemonoidalproductpreservescolimitsoverthesimplicialcategory,E.Dubucde-scribedin[D]aconstructionforthefreemonoid.AgeneralcategoricalanswerwasgivenbyG.M.Kellyin[K]whenthemonoidalproductpreservescolimitononeside.Onceagain,itsconstructionrequiresthetensorproducttopreservecolimits.Theproblemisthatthemonoidalproductsthatappearedrecentlyinvariousdo-mainsdonotsharethisgeneralproperty.Inordertostudythedeformationtheoryofalgebraicstructureslikealgebras(e.g.associative,commutative,Liealgebras)andbialgebras(e.g.associativebialgebras,1