Locally constant n-operads as higher braided operads

Transcription

1 J. Noncommut. Geom. 4 (21), DOI /JNCG/54 Journal of Noncommutatve Geometry European Mathematcal Socety Locally constant n-operads as hgher braded operads Mchael A. Batann Abstract. We ntroduce a category of locally constant n-operads whch can be consdered as the category of hgher braded operads. For n D 1; 2; 1 the homotopy category of locally constant n-operads s equvalent to the homotopy category of classcal nonsymmetrc, braded and symmetrc operads, respectvely. Mathematcs Subject Classfcaton (21). 18D2, 18D5, 55P48. Keywords. Brad groups, operads, homotopy category, recognton prncple. 1. Introducton It s well nown that contractble nonsymmetrc operads detect 1-fold loop spaces, contractble braded operads detect 2-fold loop spaces and that contractble symmetrc operads detect 1-fold loop spaces. A natural queston arses: s there a sequence of groups G.n/ DfG.n/ g together wth a noton of G.n/ -operad, whch we would call n-braded operad, such that the algebras of a contractble such operad are n-fold loop spaces? Wth some natural mnor assumptons one can prove that the answer to the above queston s negatve. Ths s because for such an operad A the quotent A =G.n/ s a K.G.n/ ;1/-space. One can show, however, that such a quotent must have a homotopy type of the space of unordered confguratons of ponts n R n, whch s a K.; 1/-space only for n D 1; 2; 1. In ths paper we show that there s a category of operads whch we can thn of as a correct replacement for the nonexstent category of G.n/ -operads n all dmensons. We call them locally constant n-operads. For n D 1; 2; 1 the homotopy category of locally constant n-operads s equvalent to the homotopy category of classcal nonsymmetrc, braded and symmetrc operads, respectvely. Here s a bref overvew of the paper. In Secton 2 we recall the defntons of symmetrc and braded operads. In Secton 3 we ntroduce the category of n-ordnals as hgher dmensonal analogue of the category of fnte ordnals. Usng ths category and ts subcategory of quas-bjectons we defne n-operads and quassymmetrc n- operads n Secton 4. In Secton 5 we show that the category of quas-bjectons s The author holds the Scott Russell Johnson Fellowshp n the Centre of Australan Category Theory at Macquare Unversty.

2 238 M.A. Batann closely related to the classcal Fox Neuwrth stratfcaton of confguraton spaces. As a corollary we observe that the nerve of ths category has homotopy type of unordered confguratons of ponts n R n. We also prove two techncal lemmas that we use n Secton 6 to relate dfferent operadc notons. Fnally n Secton 7 we ntroduce locally constant operads and compare them wth symmetrc, braded and quassymmetrc operads. We also state our recognton prncple for n-fold loop spaces. 2. Symmetrc and braded operads For a natural number n we wll denote by Œn the ordnal <1<<n: We denote an empty ordnal by Œ1. A morphsm from Œn! Œ s any functon between underlyng sets. It can be order-preservng or not. It s clear that we then have a category. We denote ths category by s. Of course, s s equvalent to the category of fnte sets. In partcular, the symmetrc group S nc1 s the group of automorphsms of Œn. Let W Œn! Œ be a morphsm n s and let. Then the premage 1./ has a lnear order nduced from Œn. Hence, there exsts a unque object Œn 2 s and a unque order-preservng bjecton Œn! 1./. We wll call Œn the fber of over and wll denote t 1./ or Œn. Analogously, gven a composte of morphsms n s,! Œn! Œl! Œ; (1) we wll denote the -th fber of ;.e., the pullbac 1.! 1.//! 1./ Œ1 Œn Œl! Œ. Let be the subcategory of bjectons n s. Ths s a strct monodal groupod wth tensor product gven by ordnal sum and wth Œ1 as ts untal object. A rght symmetrc collecton n a symmetrc monodal category V s a functor AW op! V. The value of A on an object Œn wll be denoted A n. Notce, that ths s not a standard operadc notaton. Classcally, the notaton for A Œn s A nc1 to stress the fact that A nc1 s the space of operatons of arty n C 1. The followng defnton s classcal May defnton [7] of symmetrc operad. Defnton 2.1. A (rght) symmetrc operad n V s a rght symmetrc collecton A equpped wth the followng addtonal structure:

3 Locally constant n-operads as hgher braded operads 239 a morphsm e W I! A, for every order-preservng map W Œn! Œ n s a morphsm W A.A n A n /! A n ; where Œn D 1./. They must satsfy the followng denttes: (1) For any composte of order-preservng morphsms n s the dagram Œn!! Œl! Œ; A A l A n A n A n ' A A l1 A n A l A n A l A n A l A n A n A n A A n A n commutes. Here A l D A l A l ; A n D A n A m n and A n D A n A n : (2) For an dentty D d W Œn! Œn the dagram A n A A A n d commutes. (3) For the unque morphsm Œn! Œ the dagram commutes. A A n A n d A n I I I A n

4 24 M.A. Batann The followng equvarance condtons are also requred: (1) For any order-preservng W Œn! Œ and any bjecton W Œ! Œ the followng dagram commutes: A.A n./ A n./ / A././ A n A./ A.A n A n / A n, where./ s the symmetry n V whch corresponds to permutaton and D S.I 1;:::;1/s the permutaton, whch permutes the fbers Œn ;:::;Œn accordng to and whose restrcton on each fber s an dentty. (2) For any order-preservng W Œn! Œ and any set of bjectons W Œn! Œn,, the followng dagram commutes: A.A n A n / d A. / A. / A n A. / A.A n A n / A n. We can gve an alternatve defnton of symmetrc operad [2]. Defnton 2.2. A (rght) symmetrc operad n V s a rght symmetrc collecton A equpped wth the followng addtonal structure: a morphsm e W I! A, for every order-preservng map W Œn! Œ n s a morphsm W A.A n A n /! A n ; where Œn D 1./. They must satsfy the same condtons as n the defnton 2.1 wth respect to order-preservng maps and denttes but the equvarance condtons are replaced by the followng: (1) For every commutatve dagram n s, Œn Œn Œ Œ,

5 Locally constant n-operads as hgher braded operads 241 whose vertcal maps are bjectons and whose horzontal maps are order-preservng the followng dagram commutes: A.A n./ A n./ / A././ A n A./ A.A n A n / A n, where./ s the symmetry n V whch corresponds to permutaton. (2) For every commutatve dagram n s, Œn Œn Œn Œ, where, are bjectons and, are order-preservng maps, the followng dagram commutes: A.A n A n / A n 1 A. / A. / A.A n A n / A n 1 A. / A. / A. / A./ A.A n A n / A n. Proposton 2.1. Defntons 2.1 and 2.2 are equvalent. We leave ths proposton as an exercse for the reader. Let Br be the groupod of brad groups. We wll regard the objects of Br as ordnals. There s a monodal structure on Br gven by ordnal sum on objects and concatenaton of brads on morphsm. The ordnal Œ1 s the untal object. The followng s the defnton of braded operad from [4]. A rght braded collecton n a symmetrc monodal category V s a functor AW Br op! V. The value of A on an object Œn wll be denoted A n. Defnton 2.3. A rght braded operad n V s a rght braded collecton A equpped wth the followng addtonal structure: a morphsm e W I! A,

6 242 M.A. Batann for every order-preservng map W Œn! Œ n s a morphsm W A.A n A n /! A n ; where Œn D 1./. They must satsfy the denttes (1) (3) from the defnton 2.1 and the followng two equvarancy condtons: (1) For any order-preservng W Œn! Œ and any brad W Œ! Œ the followng dagram commutes: A.A n./ A n./ / A././ A n A./ A.A n A n / A n, where./ s the symmetry n V whch corresponds to the brad and D B.I 1;:::;1/s a brad obtaned from by replacng the -th strand of by n parallel strands for each. (2) For any order-preservng W Œn! Œ and any set of brads W Œn! Œn,, the followng dagram commutes: A.A n A n / d A. / A. / A n A. / A.A n A n / A n. 3. n-ordnals and quasbjectons Defnton 3.1. An n-ordnal conssts of a fnte set T equpped wth n bnary relatons < ;:::;< n1 satsfyng the followng axoms: (1) < p s nonreflexve; (2) for every par a, b of dstnct elements of T there exsts exactly one p such that a< p b or b< p ai (3) f a< p b and b< q c then a< mn.p;q/ c. Every n-ordnal can be represented as a pruned planar tree wth n levels. For example, the 2-ordnal < 1; < 2; < 3; 1 < 1 2; 2 < 1 3; 2 < 1 3 (2)

7 Locally constant n-operads as hgher braded operads 243 s represented by the pruned tree See [1] for a more detaled dscusson.. Defnton 3.2. A map of n-ordnals W T! S s a map W T! S of underlyng sets such that mples that (1)./ < r.j/ for some r p,or (2)./ D.j/,or (3).j/ < r./ for r>p. < p j n T For every 2 S the premage 1./ (the fber of over ) has a natural structure of an n-ordnal. We denote by Ord.n/ the seletal category of n-ordnals. The category Ord.n/ s monodal. The monodal structure s defned as follows. For two n-ordnals S and T the n-ordnal S T has as an underlyng set the unon of underlyng sets of S and T. The orders < restrcted to the elements of S and T concde wth respectve orders on S and T, and a< b f a 2 S and b 2 T. The untal object for ths monodal structure s empty n-ordnal. An n-ordnal structure on T determnes a lnear order (called total order) on the elements of T as follows: a<b ff a< r b for some r n 1: We denote by ŒT the set T wth ts total lnear order. In ths way we have a monodal functor ŒW Ord.n/! s : Ths functor s fathful but not full. For example, no morphsm from the 2-ordnal (2) to the 2-ordnal < 1 1 can reverse the order of 1, 2 and 3 We also ntroduce the category of 1-ordnals Ord.1/. Defnton 3.3. An 1-ordnal conssts of a fnte set T equpped wth a sequence of bnary relatons <, < 1, < 2, satsfyng the followng axoms: (1) < p s nonreflexve;

8 244 M.A. Batann (2) for every par a, b of dstnct elements of T there exsts exactly one p such that a< p b or b< p ai (3) f a< p b and b< q c then a< mn.p;q/ c. The defnton of morphsm between 1-ordnals concdes wth the Defnton 3.2. The category Ord.1/ s the seletal category of 1-ordnals. As for Ord.n/ we have a functor of total order ŒW Ord.1/! s : For a -ordnal R, n we consder ts.n /-th vertcal suspenson S n R, whch s an n-ordnal wth the underlyng set R, and the order < m s equal to the order < m on R (so < m s empty for m<n). We also can consder the horzontal.n 1/-suspenson T n R, whch s a n-ordnal wth the underlyng set R, and the order < m s equal to the order on R (so < m s empty for 1<m n 1). The vertcal suspenson provdes us wth a functor S W Ord.n/! Ord.nC1/. We also defne an 1-suspenson functor Ord.n/! Ord.1/ as follows. For an n-ordnal T ts 1-suspenson s an 1-ordnal S 1 T whose underlyng set s the same as the underlyng set of T, and a< p b n S 1 T f a< ncp1 b n T. It s not hard to see that the sequence Ord./ S! Ord.1/ S! Ord.2/ S! exhbts Ord.1/ as a colmt of Ord.n/. S! Ord.n/! S 1! Ord.1/ Defnton 3.4. A map of n-ordnals s called a quasbjecton f t s a bjecton of the underlyng sets. Let Q n, 1 n 1,bethe subcategory of quasbjectons of Ord.n/. The total order functor nduces then a functor whch we wll denote by the same symbol: ŒW Q n! : Defnton 3.5. A map of n-ordnals 1 n 1s called order-preservng f t preserves the total orders n the usual sense, or, equvalently, only condtons 1 and 2 from the Defnton 3.2 hold for. Lemma 3.1. For every morphsm W T! S n Ord.n/, 1 n 1, there exsts a factorsaton T! T! S; where s a quasbjecton, s order-preservng and preserves total order on fbers of.

9 Locally constant n-operads as hgher braded operads 245 Proof. For n D 1 ths factorsaton s trval, snce all maps of 1-ordnals are orderpreservng. Let n D 2. Let W T! S be a map of 2-ordnals and let S D SŒ be a suspenson of the 1-ordnal Œ. Let T be the 2-ordnal whose underlyng set s the same as that of T, whose only nonempty order s < 1 and whose total order concdes wth ŒT. SoT tself s a vertcally suspended 1-ordnal. Now one can factorse the map ŒW ŒT! ŒS n s : ŒT! ŒT! ŒS; wth beng total order-preservng and a bjecton whch preserves the order on the fbers of [2]. Obvously, can be consdered as a map of 2-ordnals and t s order-preservng. Let us chec that s also a map of 2-ordnals. Indeed, f, j are from the same fber of then preserves ther order. If < j n T and they are from dfferent fbers, then there s no restrcton on snce T s a suspended 1-ordnal. Fnally, f < 1 j n T and they are from dfferent fbers then./ < 1.j/; so./ < 1.j/ because s order-preservng. Fnally, f S s an arbtrary 2-ordnal, then S D S 1 S for some suspended 1-ordnals S 1 ;:::;S and moreover, D 1 W T D T 1 T! S 1 S : By applyng the prevous result to each we obtan a requred factorsaton of. The factorsaton for n>2can be obtaned smlarly. 4. Quassymmetrc n-operads We now recall the defnton of pruned.n 1/-termnal n-operad [1]. Snce we do not need other types of n-operads n ths paper we wll call them smply n-operads. The notaton U n means the termnal n-ordnal. Let V be a symmetrc monodal category. For a morphsm of n-ordnals W T! S the n-ordnal T s the fber 1./. Defnton 4.1. An n-operad n V s a collecton A T, T 2 Ord.n/, of objects of V equpped wth the followng structure: a morphsm e W I! A Un (the unt), for every morphsm W T! S n Ord.n/ a morphsm m W A S A T A T! A T (the multplcaton/: They must satsfy the followng denttes:

10 246 M.A. Batann for any composte T! S!! R, the assocatvty dagram A R A S A T A T A T ' A R A S A T 1 A S A T A S A T A S A T 1 A T commutes, where A T A R A T A T and A S D A S A S ; A T D A T A m T ; A T D A T A T I for an dentty D d W T! T the dagram A T A Un A Un A T d commutes; for the unque morphsm T! U n the dagram commutes. A Un A T I A T d A T A T I I Let W T! S be a quasbjecton and A be a pruned n-operad. Snce a fber of s the termnal n-ordnal U n, the multplcaton n composton wth the morphsm W A S.A Un A Un /! A T A S! A S.I I/! A S.A Un A Un / nduces a morphsm A./W A S! A T : It s not hard to see that n ths way A becomes a contravarant functor on Q n.

11 Locally constant n-operads as hgher braded operads 247 Defnton 4.2. We call a pruned n-operad A quassymmetrc f for every quasbjecton W T! S the morphsm s an somorphsm. A./W A S! A T The desymmetrsaton functor from symmetrc to n-operads for fnte n was defned n [2] usng pullng bac along the functor ΠW Ord.n/! s. It was shown that ths functor has a left adjont whch we call symmetrsaton. We can obvously extend these defntons to n D1. By constructon the desymmetrsaton of a symmetrc operad s a quassymmetrc n-operad for any n. Let Q n be the fundamental groupod of Q n. A quassymmetrc operad provdes, therefore, a contravarant functor on Q n. Defnton 4.3. A Q n -collecton s a contravarant functor on Q n.a Q n -collecton s a contravarant functor on Q n. Defnton 4.4. A Q n -operad s a Q n -collecton A together wth the followng structure: for every order-preservng map W T! S the usual operadc map W A S.A T A T /! A T : Ths collecton of maps must satsfy the usual assocatvty and untarty condtons plus two equvarancy condtons: For every commutatve dagram T S T S, where vertcal maps are quas-bjectons and horzontal maps are order-preservng, the dagram A S.A T A T / A T A S.A T A T / A T commutes.

12 248 M.A. Batann For every commutatve dagram T T T S, where, are quas-bjectons and, are order-preservng, the dagram A S.A T A T / A T A S.A T A T / A T A S.A T A T / A T commutes. Theorem 4.1. The category of Q n -operads s equvalent to the category of quassymmetrc n-operads. Proof. Obvously, every quassymmetrc n-operad s a Q n -operad. Let us construct an nverse functor. Gven a Q n -operad C we defne a quassymmetrc operad A on an n-ordnal T to be equal to C T. We have to defne A on an arbtrary map of n-ordnals W T! S. Let us choose a factorsaton of accordng to Lemma 3.1. Now we can defne operadc multplcaton by the followng commutatve dagram: A S.A T A T / / A S.A T A T / A T A T. The second equvarancy axom mples that ths defnton does not depend on a chosen factorsaton. Suppose now that we have a composte T! S!! R:

13 Locally constant n-operads as hgher braded operads 249 It generates the followng factorzaton dagram T T S T T R whch n ts turn generates the followng huge dagram: S A R A S? A T? :::A T? A R A S? A T? :::A T? A R A S? A T? :::A T? A R A S A T? :::A S A T? ARAS AT? :::A S A T? A R A S? A T? :::A T? equvarancy 1 A R A S? A T? :::A T?! ARAS A 1 T? :::A S A T? assocatvty A S A T? :::A T? A R A T :::A T A T A R A T :::A T equvarancy 1 equvarancy 2 A T A T A S A T? :::A T? A S A T? :::A T? A S A T? :::A T? A S A T? :::A T? A T A R A T :::A T A R A T :::A T

14 25 M.A. Batann In ths dagram we omt the symbol to shorten the notatons. Then the central regon of the dagram commutes because of assocatvty of A wth respect to orderpreservng maps of n-ordnals. Other regons commute ether by one of equvarancy condtons ether by naturalty ether by functoralty. The commutatvty of ths dagram means the assocatvty of A wth respect to composton of maps of n- ordnals. 5. The category of quas-bjectons and confguraton spaces It s clear that the category Q n s the unon of connected components Q n./ where s the cardnalty of the n-ordnals. Theorem 5.1. (1) For a fnte n the space N.Q n.// has homotopy type of unordered confguraton spaces of -ponts n R n. (2) The localsaton functor l 2 W Q 2! Q 2 nduces a wea equvalence of the nerves. (3) The groupod Q 2 s equvalent to the groupod of brads. (4) The localsaton functor l 1 W Q 1! Q 1 nduces a wea equvalence of the nerves. (5) The groupods Q n, 3 n 1, are equvalent to the symmetrc groups groupod. Proof. We gve a setch of the proof. A detaled dscusson can be found n [1], [3]. Consder the confguraton space of ordered -ponts n R n : Conf.R n / Df.x 1 ;:::;x / 2.R n / j x x j f j g: It admts a so-called Fox Neuwrth stratfcaton. Let o S np1 C denote the open.n p 1/-hemsphere n R n, p n 1: o S np1 C Dfx 2 R n j x1 2 CCx2 n D 1; x pc1 >;x D f 1 pg: Smlarly, o S np1 Dfx 2 R n j x1 2 CCx2 n D 1; x pc1 <;x D f 1 pg: Let u j W Conf.R n /! S n1 be the functon u j.x 1 ;:::;x / D x j x x j x :

15 Locally constant n-operads as hgher braded operads 251 The Fox Neuwrth cell correspondng to an n-ordnal T wth ŒT D Œ 1 s a subspace of Conf.R n /, FN T Dfx 2 Conf.R n / j u j.x/ 2 S o np1 C f < p j n T; u j.x/ 2 S o np1 f j< p n T g: Each Fox Neuwrth cell s an open convex subspace of.r n /. We also have Conf.R n S / D FN T : ŒT DŒ1; 2S Here FN T means a space obtaned from FN T by renumberng ponts accordng to the permutaton. Let J n./ be the Mlgram poset of all possble n-ordnal structures on the set f;:::;1g [1]. The group S acts on J n./ and the quotent J n./=s s somorphc to Q n./. One can thn of an element from J n./ as a par.t; / where T s an n-ordnal and s a permutaton from S and.t;/>.s;/n J n./ when there exsts a quasbjecton W T! S and D. We also can assocate a convex subspace of the confguraton space FN.T; / D FN T wth every element of J n./. Moreover, f.t;/>.s;/then FN.S; / s on the boundary of the closure of FN.T; /. Let us defne S FN.T; / D FN.S; /:.S;/.T;/ The spaces FN.T; / are contractble and, moreover, we have a functor FN W Jn op./! Top: We then have the followng zg-zag of wea equvalences N.J op n.// hocolm FN! colm FN ' Conf.R n /: The frst statement of the theorem follows then from the quotent of the zg-zag above by the acton of the symmetrc group. The second and the thrd statements are the consequences of the fact that the space Conf.R 2 / s the K.Br ;1/-space. The ffth statement follows from the fact that the fundamental group of Conf.R n / s trval for n>3. Fnally the fourth statement can be obtaned usng the formula Q 1 D colm n Q n. We shall now, n Lemmas 5.1 and 5.2, mae the equvalence between Q 2 and Br more explct. These results wll then be used n Secton 6 to relate dfferent operadc notons. The total order functor ŒW Q 2! nduces by the unversal property a functor s 2 W Q 2!.

16 252 M.A. Batann Let p W Br! be the canoncal functor. The map p admts a secton q, whch s not a homomorphsm. For 2 S n we construct a brad q./ whch, for <jsuch that./ >.j/, has a strand from to./ whch goes over the strand from j to.j/ and there s no crossng f preserves the order of and j. Lemma 5.1. The composte Q 2 Œ!! qbr s a functor. The functor nduced by the unversal property of Q 2, s an equvalence of groupods. b W Q 2! Br; Proof. To prove that qœ s a functor we have to prove that t preserves composton. We observe that n a composte of quas-bjectons of 2-ordnals T! S! R f reverses the total order of two elements ;j 2 T then cannot reverse the order of./ and.j/. So, the resultng overcrossngs n the composte qœqœ are the same as n qœ. To prove the second clam t s suffcent to chec that the nduced morphsm of groups b W Q 2.SŒn 1; SŒn 1/! Br n s an somorphsm. It s obvously an epmorphsm. So we have to prove that t s also a monomorphsm. For ths t wll be enough to prove that f a zg-zag z W SŒn 1 TŒn 1! SŒn 1!SŒn 1; where each arrow s gven by a permutaton of two consecutve elements or an dentty permutaton, s such that the correspondng brad b.z/ s trval, then z s trval n Q 2. Ths can be done f we prove that the morphsms n Q 2.SŒn 1; SŒn 1/, N W SŒn 1 1 TŒn 1! SŒn 1; where the left arrow s gven by an dentty and the rght arrow s gven by permutaton whch changes the order of and C 1, satsfy the classcal Artn brad relatons. Then we can prove trvalty of z usng the same rewrtng process as for b.z/.

17 Locally constant n-operads as hgher braded operads 253 Let j>c1 and choose m, l such that Œm1 Œl 1 D Œn1 and 2 Œm1, j D m C 1. The followng commutatve dagram n Q 2 proves that N N j DN j N : SŒn 1 TŒn 1 TŒn 1 j. SŒn 1 ; j / SŒm 1 SŒl 1 SŒn 1 j TŒn 1 j TŒn 1 j SŒn 1 In ths dagram all unnamed morphsms are denttes on the underlyng sets. The morphsm. ; j / acts as on Œm 1 and as on Œl 1. For the proof of the Yang Baxter relatons N N C1 N D N C1 N N C1 we should consder the followng commutatve dagram n Q 2, whch expresses the morphsm N C1 N N C1 : TŒn 1 C1 SŒn 1 TŒn 1 SŒn 1 SŒn 1 C1 SŒ SŒn 2 TŒn 1 C1 SŒ C 1 SŒn 3 C1 SŒ C 1 SŒn 3 C1 C1 C1 T Œn 1 SŒn 1 C1 C1 T Œn 1 SŒn 1 An analogous dagram (the mrror mage of the above dagram) can be wrtten for N N C1 N. The relaton follows from t mmedately.

18 254 M.A. Batann So we have a commutatve dagram of categores and functors Q 2 s 2 Œ b Q 2 a Br p, where c s an adjont equvalence to b. Notce that all functors n ths dagram are strct monodal functors. Lemma 5.2. Let z W S T! R be a zg-zag of quas-bjectons of n-ordnals such that s 2.z/ D 1 : Then there exst brads b, 1, such that p.b / D, 1, and b.z/ D b 1 b : Proof. Our am s to prove that there exst quas-bjectons L W T! S D SŒn, W T! R D SŒn, 1, W S L! S, W R! R, and W L T! T such that the dagram L S L L T L L R S T R commutes and b./ D b./ D B.I 1;:::;1/for a brad on strands. Then the result wll follow from an elementary observaton that the brad b.s/ b./1! L b.s / L b. / 1! L b.t / L b. /! L b.r / b./! b.r/ s equal to L L b.s / b.1 /! L L b.t / b.! / L b.r /: It s enough to proof the lemma for D 2. The rest follows by nducton. Also wthout loss of generalty we can assume that S D SŒn and T D T Œn. Now p.s/ s the ordnal sum Œl Œm, n D m C 1 C 1, and the mage of the restrcton of the map 1 on f;:::;lg s f;:::;lg, whle the mage of the restrcton on fl C 1;:::;mC l C 1g s fl C 1;:::;mC l C 1g.

19 Locally constant n-operads as hgher braded operads 255 We put S 1 D SŒl, T 1 D TŒland S 2 D SŒm, T 1 D T Œm. We have to construct quas-bjectons ; W T! S ; D 1; 2; and also quas-bjectons whch mae the dagram ;W S 1 S 2! S; W T 1 T 2! T; S 1 S 2 S 1 2 T 1 T 2 T 1 2 S 1 S 2 (3) S commutatve. The quasbjecton s smply the dentty. Let us descrbe 1. Let.Œl/ be the mage of the set f;:::;lg n the ordnal Œn. Ths mage gets an nduced order from Œn whch maes t somorphc to Œl. Let 1 W.Œl/! Œl be ths unque somorphsm. We defne 1 as the composte Smlarly, we defne 2 as the composte Œl!.Œl/ 1! Œl: Œm!.Œm/ 2! Œm; where.œm/ s the mage of fl C1;:::;mClC1g, and we gve analogous defntons for 1 and 2. Fnally, we defne by the formula.x/ D x/ f x 2¹;:::;lº;.x/ f x 2¹l C 1;:::;mC l C 1º: We use a smlar argument to defne. The commutatvty of the dagram (3) follows from the defnton. 6. Quassymmetrc n-operads vs symmetrc and braded operads Theorem 6.1. The category of quassymmetrc 2-operads and the category of braded operads are equvalent. Proof. We frst prove that the category of quassymmetrc 2-operads s equvalent to the category whose objects are mxed 2-operads n the sense of the defnton below and whose morphsms are multplcatons and unts preservng morphsms of the underlyng braded collectons.

20 256 M.A. Batann Defnton 6.1. A mxed 2-operad n V s a rght braded collecton A equpped wth the followng addtonal structure: a morphsm e W I! A, for every order-preservng map W Œn! Œ n s a morphsm W A.A n A n /! A n ; where Œn D 1./. They must satsfy the denttes (1-3) from the defnton of symmetrc operad and the followng two equvarance condtons: (1) For any two quas-bjectons of 2-ordnals, and two order-preservng maps ; 2 s such that the dagram ŒT ŒS Œ T S commutes n s the followng nduced dagram commutes: Œ A.A n./ A n./ / A.b.//./ A n A.b.// A.A n A n / A n, where./ s the symmetry n V whch corresponds to the permutaton Œ. (2) For any two quas-bjectons, and two order-preservng maps ; 2 s such that the dagram ŒT Œ Œ ŒT Œ T S commutes n s the followng dagram commutes: A.A n A n / A n 1 A.b. // A.b. // A.A n A n / A n 1 A.b. // A.b. // A.b. // A.b.// A.A n A n / A n.

21 Locally constant n-operads as hgher braded operads 257 For a quassymmetrc 2-operad A we defne a mxed 2-operad B by pullng bac along the equvalence c W Br! Q 2. And vce versa, we produce a quassymmetrc 2-operad from a mxed 2-operad by pullng bac along b W Q 2! Br. Itsnot hard to chec that ths ndeed gves the necessary equvalence of the correspondng operadc categores. Now we wll prove that the category of mxed 2-operads s equvalent to the category of braded operads. Let A be an operad n the sense of 6.1. We have to chec that A also satsfes the Fedorowcz equvarance condtons. Let us start from the second condton. For each let us choose a zgzag of 2L morphsms n Q 2 such that D b.t 1 R 1 2! R 2 R 2L 2! S /: Obvously, such a zg-zag exsts and L can be chosen ndependently on. Then the followng square commutes for each odd j : Œn Œ L j Œn Œ L j C1 Œn Œ. Hence, the applcaton of the second equvarance condton of defnton 6.1 L tmes gves the second Fedorowcz equvarance condton. For the frst equvarance condton we do an analogous constructon by choosng a presentaton of the brad as an mage of a zgzag. Let A be an operad n the sense of 2.3. We construct an operad B n the sense of 6.1 as follows. As a braded collecton B concdes wth A. Its multplcaton s the same as n A as well. The only nontrval statement to chec s that B satsfes the equvarance condtons from Defnton 6.1. To prove the second condton we use Lemma 5.2. It s obvous also that the frst equvarance condton s satsfed n the followng specal case. Let W T! S be an order-preservng map and let W S! S be a quasbjecton. Apply Lemma 3.1 to produce a quasbjecton.; /W T! T and order-preservng map.; /W T! S such that D.; /.; /. Then b..; // D B.b./I 1;:::;1/and we can apply the frst equvarance Fedorowcz condton. Then the frst equvarance condton s satsfed n general because of the second equvarance condton of the Defnton 6.1 appled to the commutatve dagram ŒT Œ ŒT Œ.; // Œ ŒT Œ ŒS.

22 258 M.A. Batann Theorem 6.2. The category of Q n -operads 3 n 1and the category of symmetrc operads are equvalent. Proof. The proof s a repetton of the above proof wth the smplfcaton that s n W Q n! for 3 n 1s an equvalence. 7. Locally constant n-operads The quassymmetrc n-operads are defned n any symmetrc monodal category V. But accordng to Theorems 6.1 and 6.2 they are dfferent from symmetrc operads only when n D 1; 2. As we have seen before the man reason why quassymmetrc operads collapse to symmetrc operads for n>2s that the confguraton space Conf.R n / s smply connected and so localsng wth respect to quas-bjectons can only produce a groupod equvalent to. The correct procedure, therefore, should be to tae the wea!-groupod 1 Q n and consder presheaves on t wth values n V as the category of collectons. There are, however, consderable techncal dffcultes wth ths approach. Fortunately, the results of Csns [5] show a way around ths problem by consderng as the category of collectons the category of locally constant functors from Q op n to V. Pursung ths dea we gve the followng defnton. Defnton 7.1. Let V be a symmetrc monodal category and W (wea equvalences) be a subclass of ts morphsms. A locally constant n-operad n.v; W/ s an n-operad A n V such that for every quasbjecton W T! S the morphsm A./W A S! A T s a wea equvalence. Remar. We have chosen the name locally constant n-operads (whch some people prefer to call homotopcally locally constant n-operads) for two reasons. Frst, we would le our termnology to agree wth the termnology of [5]. But a more mportant reason s about phlosophy. The noton of locally constant n-operad (and locally constant functor) depends only on the class of wea equvalences but not on the choce of homotopy theory n V. For example, f V s a symmetrc monodal category and Iso s the class of all somorphsms, a locally constant n-operad n.v; Iso/ s the same as a quassymmetrc n-operad n V. So, the word homotopcal s a lttle bt msleadng. Compare ths stuaton wth the theory of homotopy lmts developed n [6]. We beleve that a true reason for ths phenomenon s that homotopy lmt and locally constant functors are hgher categorcal rather than homotopcal notons. But the homotopy theory s helpful n computatons. As far as we now a smlar argument s behnd Csns s choce of termnology. An example of an nterestng locally constant n-operad n the model category of topologcal spaces whch s not a quassymmetrc n-operad s the Getzler Jones

23 Locally constant n-operads as hgher braded operads 259 n-operad GJ n constructed n [1] for all n<1. One can also construct an 1-verson GJ 1 by the formula GJ 1 T D GJn xt, where < n s the mnmal nonempty relaton n the 1-ordnal T, the n-ordnal xt has the same underlyng set as T and the relaton < np1 n xt concdes wth the relaton < p n T. Let V be a symmetrc monodal category equpped wth a class of wea equvalences W. We ntroduce the followng notatons: SO s the category of symmetrc operads n V ; BO s the category of braded operads n V ; O n s the category of n-operads n V ; QO n s the full subcategory of O n of quassymmetrc n-operads n V ; LCO n s the full subcategory of O n of locally constant n-operads n.v; W/. Defnton 7.2. A morphsm of operads (n any of the categores above) s a wea equvalence f t s a termwse wea equvalence of the collectons. The homotopy category of operads s the category of operads localsed wth respect to the class of wea equvalences. Let us descrbe the relatons between the dfferent categores of operads we deal wth n ths paper. We have already done t for the case W D Iso n Secton 6. Let us fx a base symmetrc monodal model category V and let W be ts class of wea equvalences n the model category theoretc sense. Moreover, we wll assume that V satsfes the condtons from Secton 5 of [1], whch means that there s a model structure on the category of collectons transferable to the category of operads (see [1] for the detals). For n D 1 the relatonshps between operadc categores above s smple. The followng categores are somorphc to the category of nonsymmetrc operads O 1 ' LCO 1 ' QO 1 ; and we have a classcal adjuncton between nonsymmetrc operads and symmetrc operads. All ths s true on the level of homotopy categores. For n D 2 we have the followng dagram of categores and rght and left adjont functors: Sym 2 O 2 SO Des 2 L 2 I 2 F 2 U 2 K 2 B LCO 2 2 QO2 BO. J 2 In ths dagram the functor Des 2 s rght adjont to Sym 2 (see [1], [2] for the constructon). The functors I 2 and J 2 are natural nclusons. The functor K 2 s left adjont to J 2 and L 2 s left adjont to the composte J 2 I 2. Usng the theory of A 2

24 26 M.A. Batann nternal operads from [2] one can show that L 2 on the level of collectons s gven by the left Kan extenson along the localsaton functor l 2 W Q 2! Q 2 : L 2.A/ D Lan l2.a/: (4) We have also the same formula for K 2. The functor A 2 s a rght adjont and B 2 s a left adjont part of the equvalence constructed n Secton 6. Fnally, U 2 s the functor whch produces a braded operad from a symmetrc operad by pullng bac along the functor p W Br! and F 2 s ts left adjont gven by quotentng wth respect to the acton of the pure brad groups. Theorem 7.1. The homotopy category of locally constant 2-operads and the homotopy category of quassymmetrc 2-operads and the homotopy category of braded operads are equvalent. The functor of symmetrsaton Sym 2 can be factorsed as L 2 B 2 F 2. A base space X s a 2-fold loop space (up to group completon) f and only f t s an algebra of a contractble 2-operad, f and only f t s an algebra of a contractble braded operad (Fedorowcz s recognton prncple [4]). Proof. Snce Q 2 s a groupod, the localsaton functor l 2 s locally constant n the sense of [5], By the formal Serre spectral sequence [5], Prop. 1.15, we get that the homotopy left Kan extenson along l 2 s a left adjont to the restrcton functor between homotopy categores of collectons. The functor l 2 nduces a wea equvalence of the nerves and so, by Qullen s Theorem B, t s also asphercal n the sense of [5], 1.4. So, by [5], Prop. 1.16, the homotopy left Kan extenson along l 2 s an equvalence of homotopy categores of collectons. Tang nto account the formula (4) we see that to prove the equvalence of homotopy categores of operads t s enough to show that for an n-operad A (1 n 1) there exsts a cofbrant replacement B.A/ such that the underlyng Q n -collecton of B.A/ s cofbrant n the projectve model structure. Recall [1] that ph n s the categorcal symmetrc operad representng the 2-functor of nternal pruned n-operads. In partcular an n-operad A s represented by an operadc functor AW Q ph n! V. If we forget about operadc structures then for any we wll have a functor AQ W ph n! V. Tae the bar-resoluton B.L; L; C. A//, Q where.l;;/s the monad on the functor category Œd.ph n /; V generated by restrcton and left Kan extenson along the ncluson of dscretsaton d.ph n / of phn to phn and C.A/ s the termwse cofbrant replacement of the underlyng n-collecton of A. These functors for all form an operadc functor B.A/W ph n! V and, hence, determne an n-operad B.A/ whch s a cofbrant replacement for A [2]. Snce B.A/ s a bar-constructon on cofbrant collecton t s cofbrant n the projectve model category of functors. Recall also that there s a symmetrc categorcal operad rh n representng the 2-functor of nternal reduced n-operads [1] and a projecton p W ph n! rh n. A typcal fber (n a strct sense) of ths projecton over an object w 2 rh n s a category wth a termnal object s.w/. The map s assembles to

25 Locally constant n-operads as hgher braded operads 261 the (nonoperadc) functor s W rh n! ph n, whch s by defnton a secton of p and s also a rght adjont to p. The count of ths adjuncton s the dentty and the unt s the unque map to the termnal object s.w/. The smple calculatons wth ths adjuncton show that the restrcton functor s preserves the cofbrant objects for projectve model structures and so s.b.a// s cofbrant. There s also an ncluson j W Jn op! rh n [1]. It s not hard to see also that the categores Jn op./ and rh n are Reedy categores. Recall that the objects of rhn are planar trees decorated by pruned n-trees (.e., n-ordnals). One can choose the total number of edges of n-trees n a decorated planar tree as a degree functon and see that each morphsm decreases strctly ths functon. 1 It follows from these consderatons that the functor s.b.a// satsfes the followng property charactersng cofbrant objects n the projectve model categores for functor categores over Reedy categores: colm.s.b.a//.w//! s.b.a//.t / (5) s a cofbraton. Here the colmt s taen over the category of all w! T, w T n rh n op. It was proved n [1] that Jn./ s cofnal n rh n. Exactly the same argument shows that n the colmt (5) one can replace w 2 rh n op wth the objects from J n./. And, therefore the restrcton j s.b.a// s cofbrant as well. The quotent functor q W Jn op./! Qn op./ nduces the restrcton functor q on functor categores whch s fully fathful. It follows from ths that q reflects cofbratons. We observe that q.u.b.a// D j s.b.a//; and so u.b.a// s cofbrant. Hence the frst statement of the theorem s proved. The statement about symmetrsaton s obvous snce Des 2 D U 2 A 2 J 2 I 2. Fnally, a contractble operad s locally constant, so the thrd statement follows from the frst statement, Theorem 8.6 from [1] and the fact that the functors U 2, A 2, J 2, I 2 preserve endomorphsm operads. For 3 n 1the correspondng dagram s Sym n O n SO Des 2 I n LCO n B n L n A n J n QOn. K n 1 In fact, rh n s a poset, but we dd not provde a proof of ths fact n [1].

26 262 M.A. Batann Theorem 7.2. For 3 n 1the category of symmetrc operads s equvalent to the category of quassymmetrc n-operads. For 3 n<1 a base space X s an n-fold loop space (up to group completon) f and only f t s an algebra of a contractble n-operad. The homotopy category of locally constant 1-operads, the homotopy category of quassymmetrc 1-operads and the homotopy category of symmetrc operads are equvalent. A base space X s an nfnte loop space (up to group completon) f and only f t s an algebra of a contractble 1-operad f and only f t s an algebra of a contractble symmetrc operad (May s recognton prncple [7]). Proof. The proof s analogous to the proof of Theorem 7.1. An nterestng queston whch we do not consder here s the exstence of model structures on the varous categores of operads. The results of [5] ndcate that ths mght be possble. But t s a subject for a future paper. Acnowledgements. I would le to than Dens-Charles Csns for hs nce answers [5] to my sometmes nave questons. I wsh to express my grattude to C. Berger, I. Galvez, E. Getzler, V. Gorbunov, A. Davydov, R. Street, A. Tons, M. Weber for many useful dscussons and to the anonymous referee for useful comments concernng the presentaton of the paper. I also gratefully acnowledge the fnancal support of Scott Russel Johnson Memoral Foundaton, Max Plan Insttut für Mathemat and Australan Research Councl (grant No. DP558372). References [1] M. A. Batann, Symmetrsaton of n-operads and compactfcaton of real confguraton spaces. Adv. Math. 211 (27), Zbl MR , 245, 25, 251, 259, 26, 261 [2] M. A. Batann, The Ecmann Hlton argument and hgher operads. Adv. Math. 217 (28), Zbl MR , 245, 247, 259, 26 [3] C. Berger, Combnatoral models for real confguraton spaces and E n -operads. In Operads: Proceedngs of Renassance Conferences (Hartford, CT/Lumny, 1995), Contemp. Math. 22, Amer. Math. Soc., Provdence, RI, 1997, Zbl MR [4] Z. Fedorowcz, The symmetrc bar constructon. Preprnt , 26 [5] D.-D. Csns, Locally constant functors. Math. Proc. Cambrdge Phlos. Soc. 147 (29), Zbl MR , 26, 262 [6] W. G. Dwyer, P. S. Hrschhorn, D. M. Kan, and J. H. Smth, Homotopy lmt functors on model categores and homotopcal categores. Math. Surveys Monogr. 113, Amer. Math. Soc., Provdence, RI, 24. Zbl MR

27 Locally constant n-operads as hgher braded operads 263 [7] J. P. May, The geometry of terated loop spaces. Lectures Notes n Math. 271, Sprnger- Verlag, Berln Zbl MR , 262 Receved May 29, 28; revsed January 17, 29 M.A. Batann, Department of Mathematcs, Macquare Unversty, NSW 219, Australa E-mal: mbatann@cs.mq.edu.au