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Symmetric Monoidal Categories Model All Theory and Applications of CategoriesVol No pp SYMMETRIC MONOIDAL CATEGORIES MODEL ALL CONNECTIVE SPECTRA R W THOMASON Transmitted by Myles Tierney Abstract The classical innite lo opspace machines in fact induce an equivalence of categories b etween a lo calization of the category of symmetric monoidal categories and the stable homotopy category of connective sp ectra Intro duction Since the early seventies it has b een known that the classifying spaces of small symmetric monoidal categories are innite lo op spaces the zeroth space in a sp ectrum a sequence of spaces X i with given homotopyequivalences to the lo ops on the succeeding i space X X Indeed many of the classical examples of innite lo op spaces i i were found as such classifying spaces eg Ma Se These innite lo op spaces and sp ectra are of great interest to top ologists The homotopy category formed byinverting the weak equivalences of sp ectra is the stable homotopy categorymuch b etter b ehaved than but still closely related to the usual homotopy category of spaces eg Ad I I I One has in fact classically a functor Spt from the category of small symmetric mo noidal categories to the category of connective sp ectra those sp ectra X for which i X when ki Ma Se Th Moreover anytwosuch functors satisfying k i the condition that the zeroth space of SptS is the group completion of the classifying space B S are naturally homotopy equivalent Ma Th The aim of this article is to prove the new result Thm that in fact Spt induces an equivalence of categories b etween the stable homotopy category of connective sp ectra and the lo calization of the category of small symmetric monoidal categories by inverting those morphisms that Spt sends to weak homotopyequivalences In particular each connectivespectrumisweak equivalenttoSptS for some symmetric monoidal category S Received by the Editors April Published on July Mathematics Sub ject Classication Primary P Secondary C D D D P Key words and phrases club connective sp ectrum E space op erad lax algebra sp ectrum stable homotopy symmetric monoidal c R W THOMASON Permission to copy for private use granted Theory and Applications of Categories Vol No Thus the category of symmetric monoidal categories provides an alternate mo del for the connective stable homotopy category one which lo oks rather dierent from the classical mo del category of sp ectra It is really co ordinatefree in that there are no susp ension co ordinates at all in contrast to Mays co ordinatefree sp ectra Ma I I which use all nite subspaces of an innite vector space as co ordinates and thus are just free of choices of co ordinates The category of E spaces spaces with an action of an E op erad Ma is similarly a mo del for connective sp ectra which is really co ordinate free But a symmetric monoidal category structure is much more rigid than a general E space structure and as a consequence can b e sp ecied and manipulated much more readily As convincing evidence for this claim I refer to my talk at the Collo que en lhonneur de Michel Zisman at lUniversiteParis VI I in June There I used this alternate mo del of stable homotopytogive the rst known construction of a smash pro duct which is asso ciative and commutative up to coherent natural isomorphism in the mo del category This will b e the sub ject of an article to app ear The pro of that the functor Spt induces an equivalence b etween a lo calization of the category of symmetric monoidal categories and the connective stable homotopy category b egins by constructing an inverse functor The construction is made in several steps First there are known equivalences of homotopy categories induced by functors between the category of connective sp ectra and the category of E spaces Thus one reduces to nding an appropriate functor to the category of symmetric monoidal categories from that of E spaces AnyspaceX is weak homotopy equivalenttothe classifying space of the category NullX of weakly contractible spaces over X When X is an E space the op erad action on X induces op erations on this category For example for eachinteger n there is an nary functor sending the ob jects C Q n X C X C X to E n C C C E n X X where the n n Q n last arrow E n X X is given by the op erad action The internal comp osition functions of the op erad induce certain natural transformations b etween comp osites of these op erations on NullX Using Kellys theory of clubs Ke Ke one nds that NullX has b een given the structure of a lax algebra for the club of symmetric monoidal categories The next step is to replace this lax symmetric monoidal category by a symmetric monoidal one There is a Go dement resolution of a lax algebra byfreelax algebras yielding a simplicial lax algebra By coherence theory the latter is degreewise stably homotopy equivalent to a simplicial free symmetric monoidal categoryTaking a sort of homotopy colimit of the last simplicial ob ject Th yields the desired symmetric monoidal category The layout of the article is as follows The rst section is a review starting with the denitions of nonunital symmetric monoidal categories and of lax strong and strict symmetric monoidal functors I recall in the basic prop erties of the classical functor from symmetric monoidal categories to sp ectra Next in comes a review of the denitions of oplax functors into a category and left oplax natural transformations between such The homotopy colimit or oplax colimit of a diagram of symmetric Theory and Applications of Categories Vol No monoidal functors is recalled in and its go o d homotopy theoretic prop erties stated The rst section closes with a pro of in that all the variant categories of symmetric monoidal categories considered have equivalent homotopy categories Section b egins with a review of lax algebras over a do ctrine A Go dementtyp e simplicial resolution of lax algebras by free lax algebras in given in This resolution is shown to b e left oplax natural with resp ect to lax morphisms of lax algebras In I consider the sp ecial case of lax symmetric monoidal categories The functor to Sp ectra is extended to these in In I use the Go dement resolution and the homotopy colimit to show how to replace a lax symmetric monoidal category by a symmetric monoidal category Section reviews the notion of an E space and the equivalence of homotopy categories between E spaces and Sp ectra Section contains the construction of a lax symmetric monoidal category asso ciated to an E space Finally Section nishes the pro of by showing various roundtrip functors made by joining the ab ove pieces are linked to the identitybya chain of stable homotopy equivalences Symmetric monoidal categories and homotopy colimits I will need to use the homotopy colimit of diagrams of symmetric monoidal categories The based version of the homotopy colimit do es not have go o d homotopy b ehavior except under stringent nondegenerate basep oint conditions essentially one would have to ask that the symmetric monoidal unit have no nonidentity automorphisms and that every morphism to or from the unit b e an isomorphismSuch a symmetric monoidal category is equivalent to the disjoint union of with a p ossibly nonunital symmetric monoidal category and nally all homotopy colimit results are easier to state if one works with variant categories of nonunital symmetric monoidal categories from the b eginning Thus Definition By unital symmetric monoidal category I mean a symmetric monoidal category in the classic sense a category S provided with an ob ject and a functor SS S together with natural isomorphisms of asso ciativity commutativity and unitaricity for which certain simple diagrams are to commute For the details see for example McL VI I x x Definition A symmetric monoidal category is a category S together with a functor SSS and natural isomorphisms A B C A B C A B B A which are such that the following twopentagon and hexagon diagrams commute Theory and Applications of Categories Vol No A B C D A B C D A B C D O A B C D A B C D A B C C A B A B C A C B A C B C A B The usual results of coherence theory that every diagram commutes continue to hold for these nonunital symmetric monoidal categories indeed the precise statements and the demonstrations b ecome easier without the additional structure of the unitaricity natural isomorphisms Ep Ke To compare with the existing literature it will b e useful to recall Ma and McLa VI I x that a permutative category is a unital symmetric monoidal category where the natural asso ciativity isomorphism is the identity It follows from coherence theory that every unital symmetric monoidal category is equivalenttoapermutative category Indeed the equivalence is realized by strong unital symmetric monoidal func tors and unital symmetric monoidal natural isomorphisms See Ma or Ke As to morphisms b etween symmetric monoidal categories one needs to consider three kinds the lax the strong and the strict symmetric monoidal functors The usual denitions adapt to the the nonunital case easily Definition For S and T symmetric monoidal categories a lax symmetric mo noidal functor from S to T consists of a functor F ST together with a natural transformation of functors from SS f FA FB F A B such that the following two diagrams commute f f FA F B C F A B C FA FB FC F FA FB FC F A B FC F A B C f f Theory and Applications of Categories Vol No f F A B FA
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