Locally constant n-operads as higher braided operads

Lo cally constan t n -op erads as higher brai ded op era ds M.A. Batanin ∗ Macquarie Univ ersit y , NSW 2109, Australia e-mail: m batanin@ics.mq.edu.au No v em b er 20, 2018 Abstract W e introduce a category of loc al ly c onstant n -op er ads whic h can b e considered as the category of h igher braided operads. F or n = 1 , 2 , ∞ the homotopy category of lo cally constant n -op erads is equiv alen t to the homotopy category of classica l nonsymmetric, braided and symmetric op- erads corresp ondingly . 1991 Math. Sub j. Class. 18D20 , 18D50, 55P48 1 In tro duction It is well kno wn that contractible nonsymmetric op era ds detect 1-fo ld lo op spaces, contractible braided op er ads detect 2- fold lo op spa ces and that con- tractible symmetric op era ds detect ∞ -fo ld lo op spaces. A natural question arises : is there a sequence of g roups G ( n ) = { G ( n ) k } k ≥ 0 together with a no tion of G ( n ) -op erad, which we would ca ll n -br aide d op er ad , such that the alg e bras of a co nt ra ctible such oper ad are n - fold lo op spaces? With so me natural minor assumptions one ca n prove that the ans wer to the ab ov e question is negative. This is beca us e for such an op erad A the quotient A k /G ( n ) k is a K ( G ( n ) k , 1)-spac e . One can sho w, how ever, that such a quotien t m ust hav e a ho motopy type o f the space of unorder ed configur ations of k p oints in ℜ n , which is a K ( π, 1)-space only for n = 1 , 2 , ∞ . In this pap er w e show that there is a c a tegory of op era ds whic h we ca n think of as a cor rect replacement for the no nexistent categor y of G ( n ) -op erads in all dimensions. W e call them lo c al ly c onstant n -op er ads. F o r n = 1 , 2 , ∞ the homotopy category of lo cally consta nt n -o pe rads is equiv alent to the homotopy ∗ The author holds the Sc ott Russell Johnson F ellowship in the Cen tre of Australian Cate- gory Theory at Macquarie Universit y 1 category o f clas sical nonsy mmetric, braided and symmetric op er ads cor resp ond- ingly . Here is a brief ov erview of the pa p e r. In section 2 we recall the definitions of symmetric and braided op erads. In Section 3 we in tro duce the categor y o f n - ordinals as higher dimensio nal analogue of th e catego ry of finite or dinals. Using this c a tegory and its sub categ ory of q ua sibijections we define n - o p erads a nd quasisymmetric n -o pe rads in Section 4. In Section 5 we sho w that the categ ory of qua sibijections is closely related to the classical F ox-Newirth stratificatio n of configuratio n spaces. As a co r ollary we observe that the nerve of this catego ry has homoto py type of unordere d configura tions of po int s in ℜ n . W e also prov e t wo techn ical le mmas which we use in Section 6 to relate different op era dic notions. Finally in Section 7 we introduce lo cally consta nt op erads and co mpa re them with symmetric, br a ided and q ua sisymmetric op era ds. W e also state o ur recognition principle for n - fold lo op spaces. 2 Symmetric and braided op erads F or a na tural num b er n we will deno te b y [ n ] the o rdinal 0 < 1 < . . . < n. W e denote an empt y ordinal by [ − 1] . A morphism fr om [ n ] → [ k ] is any function betw een underlying sets. It ca n b e or der pre s erving or not. It is clear that we then hav e a categor y . W e denote this categor y by Ω s . Of c o urse, Ω s is equiv alent to the categ o ry of finite sets . In par ticular, the symmetric gro up S n +1 is the group of automor phisms o f [ n ]. Let σ : [ n ] → [ k ] be a mo rphism in Ω s and let 0 ≤ i ≤ k . Then the preima ge σ − 1 ( i ) has a linea r order induce d from [ n ] . Hence, there exists a unique o b ject [ n i ] ∈ Ω s and a unique order pr eserving bijection [ n i ] → σ − 1 ( i ) . W e will call [ n i ] the fib er of σ over i and will denote it σ − 1 ( i ) or [ n i ] . Analogously , given a comp osite of morphisms in Ω s : [ n ] σ − → [ l ] ω − → [ k ] (1) we will denote σ i the i -th fib er of σ ; i.e. the pullback σ − 1 ( ω − 1 ( i )) ❄ ✲ σ i ω − 1 ( i ) ❄ [1] ❄ ✲ ξ i [ n ] ✲ σ [ l ] [ k ] ✲ . ω Let S be the sub c a tegory of bijections in Ω s . This is a strict monoidal group oid with tenso r pro duct ⊕ giv en b y ordinal sum a nd with [ − 1] as its unital ob ject. 2 A right symmetric c ol le ction in a sy mmetric monoidal ca tegory V is a functor A : S op → V . The v alue of A on an ob ject [ n ] will b e denoted A n . Notice, that this is no t a s ta ndard op eradic no tation. Classically , the no ta tion fo r A [ n ] is A n +1 to stress the fact that A n +1 is the space of op er ations of arity n + 1 . The following definition is classical May definition [7] of symmetric op er ad. Definition 2.1 A (right) symmetric op er ad in V is a ri ght symmetric c ol le ction A e quipp e d with t he fol lowing additional s t ructur e: - a morphism e : I → A 0 - for every or der pr eserving map σ : [ n ] → [ k ] in Ω s a morphism : µ σ : A k ⊗ ( A n 0 ⊗ ... ⊗ A n k ) − → A n , wher e [ n i ] = σ − 1 ( i ) . They must satisfy the fol lowing identities: 1. for any c omp osite of or der pr eserving morphisms in Ω s [ n ] σ − → [ l ] ω − → [ k ] , the fol lowing diagr am c ommutes A k ⊗ A l • ⊗ A n • 0 ⊗ ... ⊗ A n • i ⊗ ... ⊗ A n • k ❄ ❄ A k ⊗ A l 1 ⊗ A n • 0 ⊗ ... ⊗ A l i ⊗ A n • i ⊗ ... ⊗ A l k ⊗ A n • k ≃ ✲ A l ⊗ A n • 0 ⊗ ... ⊗ A n • i ⊗ ... ⊗ A n • k A k ⊗ A n • A n ❳ ❳ ❳ ❳ ❳ ❳ ③ ✘ ✘ ✘ ✘ ✘ ✘ ✾ Her e A l • = A l 0 ⊗ ... ⊗ A l k , A n • i = A n 0 i ⊗ ... ⊗ A n m i i and A n • = A n 0 ⊗ ... ⊗ A n k ; 2. for an identity σ = i d : [ n ] → [ n ] the diagr am ✛ ❄ A n ⊗ A 0 ⊗ ... ⊗ A 0 A n ⊗ I ⊗ ... ⊗ I A n ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✾ id c ommu t es; 3 3. for the unique morphism [ n ] → [0 ] the diagr am ✛ ❄ A 0 ⊗ A n I ⊗ A n A n ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✮ id c ommu t es. The fol lowing e quivarianc e c onditions ar e also r e quir e d: 1. F or any or der pr eserving σ : [ n ] → [ k ] and any bije ction ρ : [ k ] → [ k ] the fol lowing diagr am c ommutes: A k ⊗ ( A n ρ (0) ⊗ ... ⊗ A n ρ ( k ) ) ✻ A ( ρ ) ⊗ τ ( ρ ) ✲ µ σ A n ✻ A k ⊗ ( A n 0 ⊗ ... ⊗ A n k ) ✲ µ σ A n A ( π ) , wher e τ ( ρ ) is t he symmetr y in V which c orr esp onds t o p ermutation ρ and π = Γ S ( ρ ; 1 , . . . , 1) is the p ermutation , which p ermu tes the fib ers [ n 0 ] , . . . , [ n k ] ac c or ding to ρ and whose r est riction on e ach fib er is an iden- tity. 2. F or any or der pr eserving σ : [ n ] → [ k ] and any set of bije ctions ρ i : [ n i ] → [ n i ] , 0 ≤ i ≤ k, the fol lowing diagr am c ommut es A k ⊗ ( A n 0 ⊗ ... ⊗ A n k ) ✻ id ⊗ A ( ρ 0 ) ⊗ ... ⊗ A ( ρ k ) ✲ µ σ A n ✻ A k ⊗ ( A n 0 ⊗ ... ⊗ A n k ) ✲ µ σ A n A ( ρ 0 ⊕ ... ⊕ ρ k ) , W e ca n give an alterna tive definition of symmetric op erad [2]. Definition 2.2 A (right) symmetric op er ad in V is a ri ght symmetric c ol le ction A e quipp e d with t he fol lowing additional s t ructur e: - a morphism e : I → A 0 - for every or der pr eserving map σ : [ n ] → [ k ] in Ω s a morphism: µ σ : A k ⊗ ( A n 0 ⊗ ... ⊗ A n k ) − → A n , 4 wher e [ n i ] = σ − 1 ( i ) . They must satisfy the same c onditions as in the definition 2.1 with r esp e ct to or der pr eserving maps and id entities but the e quivarianc e c onditions a r e r eplac e d by the fol lowing: 1. F or every c ommutative diagr am in Ω s [ n ′ ] ❄ π ρ ✲ σ ′ [ k ′ ] ❄ [ n ] ✲ σ [ k ] whose vertic al maps ar e bije ct ions and whose horizontal maps ar e or der pr eserving the fol lowing diagr am c ommu t es: A k ′ ⊗ ( A n ′ ρ (0) ⊗ ... ⊗ A n ′ ρ ( k ) ) ✻ A ( ρ ) ⊗ τ ( ρ ) ✲ µ σ ′ A n ′ ✻ A k ⊗ ( A n 0 ⊗ ... ⊗ A n k ) ✲ µ σ A n A ( π ) , wher e τ ( ρ ) is the symmetry in V whi ch c orr esp onds t o p ermut ation ρ. 2. F or every c ommutative diagr am in Ω s [ n ′′ ] ❄ σ η ′ ✲ σ ′ [ n ′ ] ❄ [ n ] ✲ η [ k ] wher e σ, σ ′ ar e bije ctions and η , η ′ ar e or der pr eserving maps, the fol lowing diagr am c ommut es 5 A k ⊗ ( A n ′′ 0 ⊗ ... ⊗ A n ′′ k ) ✻ 1 ⊗ A ( σ 0 ) ⊗ ... ⊗ A ( σ k ) 1 ⊗ A ( σ ′ 0 ) ⊗ ... ⊗ A ( σ ′ k ) A ( σ ′ ) A ( σ ) µ η ′ µ η A n ′′ ✻ A k ⊗ ( A n 0 ⊗ ... ⊗ A n k ) ✲ A n A k ⊗ ( A n ′ 0 ⊗ ... ⊗ A n ′ k ) A n ′ ✲ ❄ ❄ Prop ositi on 2.1 The definition 2.1 and 2.2 ar e e quivalent. W e leave this prop os ition as an exercise for the r eader. Let B r be the gr oup oid of br aid gr o ups. W e will r egard the ob jects of B r as ordinals. There is a monoidal structure on B r given by ordinal sum on o b jects and concatenation of braids on morphism. The ordinal [ − 1] is the unital ob ject. The following is the definition o f br aided op erad from [4]. A right br aide d c ol le ct ion in a symmetric monoidal ca tegory V is a functor A : B r op → V . The v alue of A on an ob ject [ n ] will b e deno ted A n . Definition 2.3 A right br aide d op er ad in V is a right br aide d c ol le ction A e quipp e d with t he fol lowing additional s t ructur e: - a morphism e : I → A 0 - for every or der pr eserving map σ : [ n ] → [ k ] in Ω s a morphism : µ σ : A k ⊗ ( A n 0 ⊗ ... ⊗ A n k ) − → A n , wher e [ n i ] = σ − 1 ( i ) . They must satisfy the identities (1-3) fr om the definition 2.1 and the fol low- ing two e qu ivariancy c onditions: 1. F or any or der pr eserving σ : [ n ] → [ k ] and any br aid ρ : [ k ] → [ k ] the fol lowing diagr am c ommutes: A k ⊗ ( A n ρ (0) ⊗ ... ⊗ A n ρ ( k ) ) ✻ A ( ρ ) ⊗ τ ( ρ ) ✲ µ σ A n ✻ A k ⊗ ( A n 0 ⊗ ... ⊗ A n k ) ✲ µ σ A n A ( π ) , wher e τ ( ρ ) is t he symmetry in V which c orr esp onds to the br aid ρ and π = Γ B ( ρ ; 1 , . . . , 1 ) is a br aid obtaine d fr om ρ by r eplacing the i -th stra nd of ρ by n i p ar al lel str ands for e ach i. 6 2. F or any or der pr eserving σ : [ n ] → [ k ] and any set of br aids ρ i : [ n i ] → [ n i ] , 0 ≤ i ≤ k, the fol lowing diagr am c ommut es A k ⊗ ( A n 0 ⊗ ... ⊗ A n k ) ✻ id ⊗ A ( ρ 0 ) ⊗ ... ⊗ A ( ρ k ) ✲ µ σ A n ✻ A k ⊗ ( A n 0 ⊗ ... ⊗ A n k ) ✲ µ σ A n A ( ρ 0 ⊕ ... ⊕ ρ k ) , 3 n -ordinals and quasibijections Definition 3.1 An n -or dinal c onsists of a finite set T e quipp e d with n binary r elations < 0 , . . . , < n − 1 satisfying the fol lowing axioms 1. < p is nonr eflexive; 2. for every p air a, b of distinct elements of T ther e ex ists exactly one p such that a < p b or b < p a ; 3. if a < p b and b < q c then a < min ( p,q ) c. Every n -ordinal can b e represented as a pr uned pla na r tree with n levels. F or ex ample, the 2-or dinal 0 < 0 1 , 0 < 0 2 , 0 < 0 3 , 1 < 1 2 , 2 < 1 3 , 2 < 1 3 (2) is represented by the following pruned tree ❅ ❅   0   1 ❅ ❅ 2 3 See [1] for a more detaile d discuss ion. Definition 3.2 A map of n -or dinals σ : T → S is a map σ : T → S of un derlying sets such that i < p j in T implies that 1. σ ( i ) < r σ ( j ) for some r ≥ p or 2. σ ( i ) = σ ( j ) or 7 3. σ ( j ) < r σ ( i ) for r > p. F or every i ∈ S the preima ge σ − 1 ( i ) ( the fib er of σ over i ) has a na tur al structure of an n -o rdinal. W e denote by Or d ( n ) the skeletal c ate gory of n -or dinals . The category Or d ( n ) is monoida l. The monoidal s tructure ⊕ is defined as follows. F or t wo n -ordinals S and T the n -ordinal S ⊕ T has as an underly ing set the union of underlying sets of S and T . The order s < k restricted to the elements of S and T coincide with resp ective o r ders on S and T . a nd a < 0 b if a ∈ S and b ∈ T . The unital ob ject for this mo noidal structure is empty n -ordinal. An n -ordinal structure on T determines a linea r order (called total or der ) on the elements of T as follows: a < b iff a < r b for so me 0 ≤ r ≤ n − 1 . W e will denote b y [ T ] the se t T with its total linear order. In this wa y we hav e a monoidal functor [ − ] : O r d ( n ) → Ω s . This functor is faithful but not full. F or example, no mor phism from the 2- ordinal (2) to the 2 -ordinal 0 < 1 1 can reverse the o rder of 1 , 2 and 3 W e a lso introduce the categ ory of ∞ -ordinals Or d ( ∞ ) . Definition 3.3 An ∞ -or dinal c onsist s of a finite set T e quipp e d with a se quenc e of binary r elations < 0 , < − 1 , < − 2 , . . . satisfying the fol lowing axioms 1. < p is nonr eflexive; 2. for every p air a, b of distinct elements of T ther e ex ists exactly one p such that a < p b or b < p a ; 3. if a < p b and b < q c then a < min ( p,q ) c. The definitio n of morphism b etw een ∞ -o rdinals coincides with the Definition 3.2. The categor y Or d ( ∞ ) is t he skeletal c ate gory of ∞ -or dinals . As for O r d ( n ) we hav e a functor of total or der [ − ] : O r d ( ∞ ) → Ω s . F or a k -or dinal R , k ≤ n we consider its ( n − k )-th vertic al su sp en s ion S n − k R whic h is an n - ordinal with the under lying set R, a nd the order < m equal the order < m − k on R (so < m are empt y for 0 ≤ m < n − k . ) W e also can consider t he horizontal ( n − 1 ) -susp ension T n − k R whic h is a n -ordinal with the underlying set R, and the order < m equal the or der on R (so < m are empty for k − 1 < m ≤ n − 1 . ) The vertical susp ensio n pr ovides us with a functor S : O r d ( n ) → O rd ( n + 1) . W e also define an ∞ -susp ension functor O rd ( n ) → O r d ( ∞ ) as follows. F or an n -ordinal T its ∞ -s usp ension is an ∞ -ordinal S ∞ T whose underlying s e t is the 8 same as the underlying set o f T a nd a < p b in S ∞ T if a < n + p − 1 b in T . It is not hard to see that the sequenc e Or d (0) S − → Ord (1) S − → O r d (2) − → . . . S − → Ord ( n ) − → . . . S ∞ − → Ord ( ∞ ) , exhibits O rd ( ∞ ) as a c olimit of O rd ( n ) . Definition 3.4 A map of n -or dinals is c al le d a quasibije ction if it is a bije ction of t he u nderlying sets. Let Q n , 1 ≤ n ≤ ∞ b e t he sub c ate gory of quasibije ct ions of Or d ( n ) . The total o rder functor induces then a functor which we will denote by the same symbol: [ − ] : Q n → S . Definition 3.5 A map σ of n -or dinals 1 ≤ n ≤ ∞ is c al le d or der pr eserving if it pr eserves the t otal or ders in the usual sense or e quivalently only c onditions 1 and 2 fr om the Definition 3.2 hold for σ. Lemma 3. 1 F or every morphism σ : T → S in O r d ( n ) 1 ≤ n ≤ ∞ t her e exists a factorisatio n T π − → T ′ ν − → S wher e π is a qu asibije ction, ν is or der pr eserving and π pr eserves total or der on fib ers of ν . Pro of. F or n = 1 this factorisa tion is trivial, since all maps of 1 - ordinals are order preserving . Let n = 2 . Let σ : T → S b e a map o f 2-ordina ls a nd let S = S [ k ] b e a susp ension o f the 1- ordinal [ k ] . Let T ′ be the 2 -ordinal whos e under lying set is the same as tha t of T , whose o nly nonempty order is < 1 and whose total order coincides with [ T ] . So T ′ itself is a vertically suspended 1-o rdinal. Now, o ne can factorise the map [ σ ] : [ T ] → [ S ] in Ω s [ T ] π − → [ T ′ ] ν − → [ S ] with ν b eing tota l order preserving and π a bijection which preser ves the order on the fibers of σ [2]. Obviously , ν ca n be considered as a map of 2 -ordinals and it is o rder pr eserving . Let us chec k that π is also a ma p of 2 -ordina ls . Indeed, if i, j are from the same fiber of σ then π preser ves their order. If i < 0 j in T and they are from different fib er s then there is no res tr iction on π since T ′ is a susp ended 1-ordinal. Finally , if i < 1 j in T and they are fro m differen t fib er s then σ ( i ) < 1 σ ( j ) , so π ( i ) < 1 π ( j ) b eca use ν is or der preserving. Finally , if S is an arbitra ry 2-ordinal then S = S 1 ⊕ . . . ⊕ S k for some susp ended 1-o rdinals S 1 , . . . , S k and moreover, σ = σ 1 ⊕ . . . ⊕ σ k : T = T 1 ⊕ . . . ⊕ T k → S 1 ⊕ . . . ⊕ S k . By applying the pr e vious result to each σ k we obtain a required factorisa tion of σ . The factorisa tion for n > 2 can b e o btained similarly . 9 4 Quasisymmetric n -op erads. W e now reca ll the definition of pruned ( n − 1)- ter minal n -op era d [1]. Since we do not need other types of n - op erads in this pap er we will call them simply n -op erads . The notation U n means the terminal n - ordinal. Let V b e a symmetric monoidal catego r y . F or a morphism of n -ordina ls σ : T → S the n -or dinal T i is the fib e r σ − 1 ( i ) . Definition 4.1 An n -op er ad in V is a c ol le ction A T , T ∈ Ord ( n ) of obje cts of V e quipp e d with the fol lowing st ructur e : - a morphism e : I → A U n (the unit); - for every morphism σ : T → S in O rd ( n ) , a morphism m σ : A S ⊗ A T 0 ⊗ ... ⊗ A T k → A T (the multiplic ation ) . They must satisfy the fol lowing identities: - for any c omp osite T σ → S ω → R, t he asso ciativity diagr am A R ⊗ A S • ⊗ A T • 0 ⊗ ... ⊗ A T • i ⊗ ... ⊗ A T • k ❄ ❄ A R ⊗ A S 0 ⊗ A T • 1 ⊗ ... ⊗ A S i ⊗ A T • i ⊗ ... ⊗ A S k ⊗ A T • k ≃ A S ⊗ A T • 1 ⊗ ... ⊗ A T • i ⊗ ... ⊗ A T • k A R ⊗ A T • A T ❳ ❳ ❳ ❳ ❳ ❳ ③ ✘ ✘ ✘ ✘ ✘ ✘ ✾ c ommu t es, wher e A S • = A S 0 ⊗ ... ⊗ A S k , A T • i = A T 0 i ⊗ ... ⊗ A T m i i and A T • = A T 0 ⊗ ... ⊗ A T k ; - for an identity σ = id : T → T the diagr am ✛ ❄ A T ⊗ A U n ⊗ ... ⊗ A U n A T ⊗ I ⊗ ... ⊗ I A T ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✾ id c ommu t es; - for the unique morphism T → U n the diagr am 10 ✛ ❄ A U n ⊗ A T I ⊗ A T A T ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✮ id c ommu t es. Let σ : T → S be a q uasibijection and A b e a pruned n -op er ad. Since a fiber o f σ is the terminal n -or dinal U n , the m ultiplication µ σ : A S ⊗ ( A U n ⊗ ... ⊗ A U n ) − → A T in comp osition with the mo rphism A S → A S ⊗ ( I ⊗ ... ⊗ I ) → A S ⊗ ( A U n ⊗ ... ⊗ A U n ) induces a morphism A ( σ ) : A S → A T . It is not hard to see that in this way A beco mes a contra v a riant functor on Q n . Definition 4.2 We c al l a prune d n -op er ad A quasisymmetric if for every qua- sibije ction σ : T → S the morphism A ( σ ) : A S → A T is an isomorphism. The desy mmetr isation functor fro m symmetric to n -op erads for finite n was defined in [2] using pulling back a long the functor [ − ] : Or d ( n ) → Ω s . It w as shown tha t this functor has a le ft adjo int which we ca ll symmetrisa tion. W e can o bviously extend these definitions to n = ∞ . By cons truction the desym- metrisation of a symmetric o p erad is a quasisy mmetr ic n - op erad for any n. Let Π Q n be the fundamental gr o up oid of Q n . A quasis y mmetric op er ad provides, therefor e , a contra v ariant f unctor on Π Q n . Definition 4.3 A Q n -c ol le ction is a c ontr avariant fun ctor on Q n . A Π Q n - c ol le ct ion is a c ontra variant functor on Π Q n . Definition 4.4 A Q n -op er ad is a Π Q n -c ol le ction A to gether with the fol lowing structur e • for every or der pr eserving map σ : T → S the usual op er adic map: µ σ : A S ⊗ ( A T 0 ⊗ ... ⊗ A T k ) − → A T . This c ol le ction of maps mus t satisfy the usu al asso ciativity and un it arity c onditions plus t wo e quivariancy c onditions: 11 • F or every c ommutative diagr am T ′ ❄ ✲ σ ′ S ′ ❄ T ✲ σ S wher e vertic al maps ar e quasibije ct ions and horizontal maps ar e or der pr e- serving the diagr am A S ⊗ ( A T 0 ⊗ ... ⊗ A T k ) ❄ ✲ A T ❄ A S ′ ⊗ ( A T ′ 0 ⊗ ... ⊗ A T ′ k ) ✲ A T ′ c ommu t es • F or every c ommutative diagr am T ❄ ✲ σ ′ T ′ ❄ σ η ′ T ′′ ✲ η S wher e σ, σ ′ ar e quasibije ctions and η , η ′ ar e or der pr eserving, the diagr am A S ⊗ ( A T 0 ⊗ ... ⊗ A T k ) ✻ A T ✻ A S ⊗ ( A T ′′ 0 ⊗ ... ⊗ A T ′′ k ) ✲ A T ′′ A S ⊗ ( A T ′ 0 ⊗ ... ⊗ A T ′ k ) A T ′ ✲ ❄ ❄ c ommu t es. Theorem 4 . 1 The c ate gory of Q n -op er ads is e quivalent to the c ate gory of qua- sisymmetric n -op er ads. Pro of. O bviously , every quasisymmetric n -o p erad is a Q n -op erad. Let us con- struct an inv erse functor. Given a Q n -op erad C we define a quas isymmetric 12 op erad A on an n - ordinal T to b e equal to C T . W e have to define A on a n arbitrar y map o f n -ordinals σ : T → S . Let us choo se a factorisation of σ ac c ording to Lemma 3.1. Now we ca n define oper adic multiplication by the following commutativ e diagram A S ⊗ ( A T 0 ⊗ ... ⊗ A T k ) ❄ 1 ⊗ ( α − 1 π 1 ⊗ ... ⊗ α − 1 π k ) ✲ µ σ A T ✻ α π A S ⊗ ( A T ′ 0 ⊗ ... ⊗ A T ′ k ) ✲ µ ν A T ′ The sec o nd eq uiv ariancy a xiom implies that this definition do es no t dep end on a chosen factor isation. Supp ose now we have a comp osite T σ − → S ω − → R. It generates the following fac to rization diagram T ✲     ✒ ❅ ❅ ❅ ❅ ❘ ❅ ❅ ❘ ❅ ❅ ❘ ❅ ❅ ❘   ✒   ✒   ✒ S T ′′′ T ′′ T ′ S ′ ✲ R which in its turn generates the following huge diagram 13 A R A S ⋆ A T ⋆ 0 . . . A T ⋆ k    ✠ ✲ ❅ ❅ ❅ ❘ A R A S 0 A T ⋆ 0 . . . A S k A T ⋆ k A R A S ′ ⋆ A T ⋆ 0 . . . A T ⋆ k A R A S 0 A T ′ ⋆ 0 . . . A S k A T ′ ⋆ k ✁ ✁ ✁ ✁ ☛ ❆ ❆ ❆ ❆ ❯ A S ′ A T ⋆ 0 . . . A T ⋆ k A R A T ′ 0 . . . A T ′ k A S A T ⋆ 0 . . . A T ⋆ k ❄ ❄ A R A T 0 . . . A T k A S A T ′ ⋆ 0 . . . A T ′ ⋆ k ❆ ❆ ❆ ❆ ❯ ✁ ✁ ✁ ✁ ☛ A R A T ′′′ 0 . . . A T ′′′ k A T ′′ A T ′ A T ′′′ A T P P P P q ✏ ✏ ✏ ✏ ✮ ❍ ❍ ❍ ❍ ❥ ✟ ✟ ✟ ✟ ✙    ✠ ❄ associativ ity A R A S ′ ⋆ A T ′′ ⋆ 0 . . . A T ′′ ⋆ k ✄ ✄ ✄ ✄ ✄ ✄ ✎ ✲ ❈ ❈ ❈ ❈ ❈ ❈ ❲ A R A S ′ 1 A T ′′ ⋆ 0 . . . A S ′ k A T ′′ ⋆ k A S ′ A T ′′ ⋆ 0 . . . A T ′′ ⋆ k ❍ ❍ ❍ ❥ ✟ ✟ ✟ ✙ A R A T ′′ 0 . . . A T ′′ k A S ′ A T ′ ⋆ 0 . . . A T ′ ⋆ k    ✠ ❅ ❅ ❅ ❘ ❄ ❅ ❅ ❅ ❅ ❘ A R A S ′ ⋆ A T ′ ⋆ 0 . . . A T ′ ⋆ k P P P q ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✌ ❆ ❆ ❆ ❆ ❆ ❯ eq uiv ar iancy 1 eq uiv ar iancy 2 eq uiv ar iancy 1 A R A S ⋆ A T ′ ⋆ 0 . . . A T ′ ⋆ k ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✌ ❅ ❅ ❅ ❘ ✏ ✏ ✏ ✮ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✌ ✲ ✁ ✁ ✁ ✁ ✁ ✁ ☛ ✟ ✟ ✟ ✙ In this diagram we omit the symbol ⊗ to sho rten the no tations. Then the central r e g ion of the diagr am commut es be c ause of asso ciativity of A with r e - sp ect to order preserving ma ps of n -or dinals. Other r egions commute either by one of eq uiv ar iancy conditions either by naturality either by functor iality . The commu tativity o f this diagram means the asso ciativity of A with resp ect to comp osition of maps of n -ordinals . 5 The category of quasibijections and con fi gu- ration spaces. It is clear that the ca tegory Q n is the union of connected comp onents Q n ( k ) where k is the ca rdinality of the n - ordinals. Theorem 5 . 1 • F or a finite n the sp ac e N ( Q n ( k )) has homotopy typ e of unor der e d c onfigu r ation sp ac es of k -p oints in ℜ n ; 14 • The lo c alisation functors l 2 : Q 2 → Π Q 2 , induc es a we ak e quivalenc e of t he nerves; • The gr oup oid Π Q 2 is e quivalent t o t he gr oup oid of br aids; • The lo c alisation functors l ∞ : Q ∞ → Π Q ∞ , induc es a we ak e quivalenc e of t he nerves; • the gr oup oids Π Q n , 3 ≤ n ≤ ∞ ar e e quivalent to the symmetr ic gr oups gr oup oid. Pro of. W e g ive a sketc h of the pro o f. A detailed disc us sion can b e found in [1, 3]. Consider the configuratio n spa ce of order ed k -po ints in ℜ n : Conf k ( ℜ n ) = { ( x 1 , . . . , x k ) ∈ ( ℜ n ) k | x i 6 = x j if i 6 = j } It admits a so called F ox-Neu wirth stratification. Let o S n − p − 1 + denote the op en ( n − p − 1)-hemisphere in ℜ n , 0 ≤ p ≤ n − 1: o S n − p − 1 + =  x ∈ ℜ n     x 2 1 + . . . + x 2 n = 1 x p +1 > 0 a nd x i = 0 if 1 ≤ i ≤ p  Similarly , o S n − p − 1 − =  x ∈ ℜ n     x 2 1 + . . . + x 2 n = 1 x p +1 < 0 and x i = 0 if 1 ≤ i ≤ p  . Let u ij : Co nf k ( ℜ n ) → S n − 1 be the function u ij ( x 1 , . . . , x k ) = x j − x i || x j − x i || The F ox-Neu wirth cell corres po nding to a n n - o rdinal T with [ T ] = [ k − 1] is a subspace of Conf k ( ℜ n ) F N T =        x ∈ Conf k ( ℜ n )         u ij ( x ) ∈ o S n − p − 1 + if i < p j in T u ij ( x ) ∈ o S n − p − 1 − if j < p i in T        . Each F ox-Neu wirth cell is an op en conv ex subspace of ( ℜ n ) k . W e also hav e Conf k ( ℜ n ) = [ [ T ]=[ k − 1] , π ∈ S k π F N T . 15 Here π F N T means a space o btained from F N T by renum ber ing p oints ac c ording to the p e rmutation π . Let J n ( k ) b e the Milgram pose t of a ll p oss ible n -ordinal s tructures on the set { 0 , . . . , k − 1 } [1]. The group S k acts on J n ( k ) a nd the quotient J n ( k ) /S k is isomorphic to Q n ( k ) . One can think of a n elemen t from J n ( k ) as a pa ir ( T , π ) where T is an n - ordinal and π is a p ermutation from S k and ( T , π ) > ( S, ξ ) in J n ( k ) when there exists a quasibijection σ : T → S and ξ · π = σ . W e a lso can asso cia te a conv ex subs pa ce of the config ur ation space F N ( T , π ) = π F N T with every element of J n ( k ) . Moreover, if ( T , π ) > ( S, ξ ) then F N ( S, ξ ) is on the b oundar y o f the closur e o f F N ( T , π ) . Let us define F N ( T , π ) = [ ( S,ξ ) ≤ ( T ,π ) F N ( S, ξ ) . The spaces F N ( T , π ) a r e c ontractible a nd, moreov er, we ha ve a functor F N : J op n ( k ) → T op. W e then hav e the following zig-zag of weak equiv alences N ( J op n ( k )) ← h ocol i m F N → colim F N ≃ Conf k ( ℜ n ) . The firs t statement of the theorem fo llows then from the quo tient o f the zig- zag above by the ac tion of the symmetric group. The second and the third statements are the consequences of the fact that the spa ce Conf k ( ℜ 2 ) is the K ( B r k , 1)-spac e . The fifth statement follows from the fact that the fundamental group of Conf k ( ℜ n ) is trivia l for n > 3 . Finally the fourth sta tement can b e obtained using the formula Q ∞ = colim n Q n . W e shall now, in Lemmas 5.1 a nd 5.2, make the equiv alence betw een Π Q 2 and B r more explicit. These results will then b e used in sectio n 6 to relate different op eradic notio ns. The total order functor [ − ] : Q 2 → S induces b y the universal pr op erty a functor s 2 : Π Q 2 → S . Let p : B r → S be the c a nonical functor. The ma p p admits a section q , which is not a homomor phism. F or σ ∈ S n we construct a br a id q ( σ ) which for i < j such that σ ( i ) > σ ( j ) has a stra nd from i to σ ( i ) which go es ov er the strand fro m j to σ ( j ) and ther e is no cro s sing if σ preser ves the o rder of i and j. Lemma 5. 1 • The c omp osite Q 2 [ − ] − → S q − → B r is a functor; 16 • The functor induc e d by the u niversal pr op erty of Π Q 2 b : Π Q 2 → B r is an e qu ivalenc e of gr oup oids. Pro of. T o prov e that q [ − ] is a functor we have to pr ov e that it pres erves comp osition. W e obs erve that in a comp osite of quas ibijections o f 2-ordina ls T σ → S ξ → R if σ re verses the total or der of tw o element s i , j ∈ T then ξ ca n not reverse the o rder of σ ( i ) and σ ( j ) . So, the resulting overcrossings in the comp osite q [ σ ] q [ ξ ] are the same as in q [ σ · ξ ] . T o prov e the second claim it is sufficien t t o c heck that the induced morphism of groups b : Π Q 2 ( S [ n − 1] , S [ n − 1]) → B r n is an isomor phism. It is obviously an epimorphism. So we hav e to prov e that it is als o a monomorphism. F or this it will b e enoug h to prov e that if a zig- zag z : S [ n − 1 ] ← T [ n − 1] → S [ n − 1 ] ← . . . → S [ n − 1 ] , where each arrow is g iven by a p er mut ation o f tw o consecutive elements or an ident ity p ermutation, is suc h that the corr esp onding braid b ( z ) is trivial then z is trivial in Π Q 2 . This ca n b e done if we prov e that the morphisms in Π Q 2 ( S [ n − 1] , S [ n − 1 ]) ¯ σ i : S [ n − 1] 1 ← − T [ n − 1 ] σ i − → S [ n − 1 ] , where the left arr ow is given b y an identit y and the r ight a rrow is given by per mutation σ i which c hange the order of i and i + 1 , satisfy the cla ssical Artin braid rela tions. Then we can prove triviality of z using the same r ewriting pro cess as for b ( z ) . Let j > i + 1 and c ho ose m , l such that [ m − 1] ⊕ [ l − 1] = [ n − 1] and i ∈ [ m − 1 ] , j = m + 1 . The following commutativ e diagra m in Q 2 prov es that ¯ σ i ¯ σ j = ¯ σ j ¯ σ i : 17 S [ n − 1] ✛   ✒ ❅ ❅ ❘   ✠ ❅ ❅ ■ ❅ ❅ ❘   ✠ ❅ ❅ ■   ✒   ✒ ❅ ❅ ■ ❅ ❅ ❘   ✠ ✻ ❄ S [ m − 1] ⊕ S [ l − 1] S [ n − 1] S [ n − 1] T [ n − 1] T [ n − 1] T [ n − 1] T [ n − 1] ✲ S [ n − 1] σ i σ i σ i σ i σ j σ j σ j σ j ( σ i , σ j ) In this dia gram all unna med morphisms ar e identities o n the under lying sets. The morphism ( σ i , σ j ) acts as σ i on [ m − 1] and as σ 0 on [ l − 1] . F or the pr o of o f Y ang -Baxter r elations ¯ σ i ¯ σ i +1 ¯ σ i = ¯ σ i +1 ¯ σ i ¯ σ i +1 we should con- sider the following c o mmut ative diag ram in Q 2 which express e s the morphism ¯ σ i +1 ¯ σ i ¯ σ i +1 T [ n − 1] ✲ ❅ ❅ ■   ✒ ❅ ❅ ❘ ❅ ❅ ❘   ✠   ✠ ❅ ❅ ❘ ❅ ❅ ■ ✻ ✡ ✡ ✡ ✡ ✡ ✡ ✣ ✟ ✟ ✟ ✟ ✙ ✟ ✟ ✟ ✟ ✙ ✻ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❯ ❄ ❄ ❄ ❄ S [ i ] ⊕ S [ n − i − 2] T [ n − 1] S [ n − 1] T [ n − 1] T [ n − 1] S [ n − 1] S [ n − 1] S [ i +1] ⊕ S [ n − i − 3] S [ i +1] ⊕ S [ n − i − 3] S [ n − 1] S [ n − 1] ✻ ✲ ✲ ✛ ✛ T [ n − 1] σ i +1 σ i +1 σ i +1 σ i +1 σ i σ i +1 σ i +1 σ i σ i +1 σ i σ i +1 σ i σ i +1 An analogo us diagr am (t he mirr or image of the ab ove diagram) can be writ- ten for ¯ σ i ¯ σ i +1 ¯ σ i . The relation follows from it immediately . So, we have a commutativ e diagram of categor ies and functors Q 2 ✲ ❍ ❍ ❍ ❍ ❍ ❥ ✟ ✟ ✟ ✟ ✟ ✙ Π Q 2 ❄ S ✲ ✛ c [ − ] s 2 p B r b 18 where c is an adjoint equiv alence to b. Notice that all functors in this diagra m are strict monoidal functors . Lemma 5. 2 L et z : S σ ← − T η − → R b e a zig-zag of qu asibije ct ions of n -or dinals such that s 2 ( z ) = τ 1 ⊕ . . . ⊕ τ k . Then t her e exist br aids b i , 1 ≤ i ≤ k su ch that p ( b i ) = τ i , 1 ≤ i ≤ k and b ( z ) = b 1 ⊕ . . . ⊕ b k . Pro of. W e will prov e that there exist qua sibijections σ i : T i → S i = S [ n i ] , η i : T i → R i = S [ n i ] 1 ≤ i ≤ k , tw o qua sibijections ξ : ⊕ i S i → S , ζ : ⊕ i R i → R and a quasibijection κ : ⊕ i T i → T , such that the following diag ram commut es ⊕ i R i ⊕ i S i S ✛ ❄ ✛ ❄ ✲ ⊕ i T i ❄ κ T ξ ζ ✲ R σ η ⊕ i σ i ⊕ i η i and b ( ξ ) = b ( ζ ) = Γ B ( π ; 1 , . . . , 1) for a bra id π on k str ands. Then the result will follow from a n elemen tary o bserv ation that the braid b ( S ) b ( ξ ) − 1 − → ⊕ i b ( S i ) ⊕ i b ( σ i ) − 1 − → ⊕ i b ( T i ) ⊕ i b ( η i ) − → ⊕ i b ( R i ) b ( ξ ) − → b ( R ) is equal to ⊕ i b ( S i ) ⊕ i b ( σ − 1 i ) − → ⊕ i b ( T i ) ⊕ i b ( η i ) − → ⊕ i b ( R i ) . It is e no ugh to pr o of the lemma for k = 2 . The res t will follow by induction. Also without los s o f generality we can assume tha t S = S [ n ] a nd T = T [ n ] . Now, p ( S ) is the o rdinal sum [ l ] ⊕ [ m ] , n = m + 1 + 1 and the image of the restriction of the map σ − 1 η on { 0 , . . . , l } is { 0 , . . . , l } and the ima ge of the restriction on { l + 1 , . . . , m + l + 1 } is { l + 1 , . . . , m + l + 1 } . W e put S 1 = S [ l ] , T 1 = T [ l ] and S 2 = S [ m ] , T 1 = T [ m ] . W e hav e to construct quasibijections σ i , η i : T i → S i i = 1 , 2 , and also quasibijections ξ , ζ : S 1 ⊕ S 2 → S , κ : T 1 ⊕ T 2 → T , which make the dia gram 19 S 1 ⊕ S 2 σ 1 ⊕ σ 2 ← − T 1 ⊕ T 2 η 1 ⊕ η 2 − → S 1 ⊕ S 2 ↓ ↓ ↓ (3) S σ ← − T η − → S commutativ e. The qua sibijection κ is simply the identit y . Let us describ e σ 1 . Let σ ([ l ]) b e the imag e of the set { 0 , . . . , l } in the o rdinal [ n ] . This imag e gets an induced order from [ n ] which ma kes it isomo rphic to [ l ] . Let φ 1 : σ ([ l ]) → [ l ] b e this unique isomor phism. W e define σ 1 as the comp osite [ l ] → σ ([ l ]) φ 1 → [ l ] . Similarly , w e define σ 2 as the comp osite [ m ] → σ ([ m ]) φ 2 → [ m ] , where σ ([ m ]) is the image of { l + 1 , . . . , m + l + 1 } and we g ive analogo us definitions for η 1 and η 2 . Finally , we define ξ by the for mula ξ ( x ) =  φ − 1 1 ( x ) if x ∈ { 0 , . . . , l } φ − 1 2 ( x ) if x ∈ { l + 1 , . . . , m + l + 1 } W e use a similar ar gument to define ζ . The co mmut ativity of the diagram (3) follows from the definitio n. 6 Quasisymmetric n -op erads vs symmetric and braided op erads. Theorem 6 . 1 The c ate gory of quasisymmetric 2 -op er ads and the c ate gory of br aide d op er ads ar e e qu ivalent. Pro of. W e first prove that the ca tegory of q uasisymmetric 2-op er a ds is eq uiv- alent to the catego ry whose ob jects ar e mixe d 2 -op er ads in the sense o f the definition b elow and whose morphisms ar e multiplications and units preserving morphisms of the underlying bra ided collections. Definition 6.1 A mixe d 2 - op er ad in V is a right br aide d c ol le ction A e quipp e d with t he fol lowing addi tional structur e: - a morphism e : I → A 0 - for every or der pr eserving map σ : [ n ] → [ k ] in Ω s a morphism : µ σ : A k ⊗ ( A n 0 ⊗ ... ⊗ A n k ) − → A n , wher e [ n i ] = σ − 1 ( i ) . They must satisfy the identities (1-3) fr om the definition o f symmetric op er ad and t he fol lowing two e quivarianc e c onditions: 20 1. F or any two quasibije ctions of 2 - or dinals π, ρ and two or der pr eserving maps σ, σ ′ ∈ Ω s such that t he fol lowing diagr am c ommutes in Ω s [ T ′ ] ❄ [ π ] [ ρ ] ✲ σ ′ [ S ′ ] ❄ [ T ] ✲ σ [ S ] the fol lowing induc e d diagr am c ommut es: A k ′ ⊗ ( A n ′ ρ (0) ⊗ ... ⊗ A n ′ ρ ( k ) ) ✻ A ( b ( ρ )) ⊗ τ ( ρ ) ✲ µ σ ′ A n ′ ✻ A k ⊗ ( A n 0 ⊗ ... ⊗ A n k ) ✲ µ σ A n A ( b ( π )) , wher e τ ( ρ ) is the symmetry in V which c orr esp onds to t he p ermutation [ ρ ] . 2. F or any two quasibije ctions σ, σ ′ and two or der pr eserving maps η , η ′ ∈ Ω s such that the fol lowing diagr am c ommutes in Ω s [ T ′′ ] ❄ [ σ ] η ′ ✲ [ σ ′ ] [ T ′ ] ❄ [ T ] ✲ η [ S ] the fol lowing diagr am c ommutes A k ⊗ ( A n ′′ 0 ⊗ ... ⊗ A n ′′ k ) ✻ 1 ⊗ A ( b ( σ 0 )) ⊗ ... ⊗ A ( b ( σ k )) 1 ⊗ A ( b ( σ ′ 0 )) ⊗ ... ⊗ A ( b ( σ ′ k )) A ( b ( σ ′ )) A ( b ( σ )) µ η ′ µ η A n ′′ ✻ A k ⊗ ( A n 0 ⊗ ... ⊗ A n k ) ✲ A n A k ⊗ ( A n ′ 0 ⊗ ... ⊗ A n ′ k ) A n ′ ✲ ❄ ❄ F or a quasisymmetric 2-op e r ad A we define a mixed 2-op erad B by pulling ba ck along the equiv ale nce c : B r → Π Q 2 . And vice v ersa , we pro duce a quasisym- metric 2-op era d from a mixed 2-op era d by pulling back along b : Π Q 2 → B r. 21 It is not hard to c heck that this indeed gives the necessa ry equiv alence of the corres p o nding op era dic categorie s . Now, w e will prov e that the catego ry of mixed 2 -op erads is equiv alent to the category o f bra ided o p e rads. Let A b e an op era d in the sense of 6.1. W e hav e to c heck tha t A als o satisfies the Fiedorowicz equiv ariance conditions. Let us start from the second conditio n. F or ea ch ρ i let us choo se a z igzag of 2 L mor phisms in Q 2 , such that ρ i = b ( T i τ 1 ← − R 1 i τ 2 − → R 2 i ← . . . ← − R 2 L i τ 2 k − → S i ) . Obviously , s uch a zig-zag exists and L can b e chosen indep endently on i. Then the following sq ua re comm utes for e ach o dd j : [ n ] ❄ [ ⊕ i τ j i ] σ ✲ [ ⊕ i τ j +1 i ] [ n ] ❄ [ n ] ✲ σ [ k ] Hence, the application of the seco nd eq uiv ar iance condition of definition 6.1 L times gives the sec ond Fiedorowicz equiv ariance condition. F or the first equiv ariance condition we do an analo gous construction by choosing a presentation of the braid ρ as an imag e of a zigzag . Let A b e a n o p erad in the sense of 2.3. W e c onstruct a n op era d B in the sense of 6.1 as follows. As a braide d collection B coincides with A. Its multiplication is the same as in A also. The only nont rivia l statement to chec k is that B satisfies the equiv ariance conditions from Definition 6.1. T o prove the sec o nd condition we use Lemma 5 .2. It is ob vious also that the firs t equiv ariance condition is satisfied in the following sp ecia l case. Let σ ′ : T → S ′ be an o rder pres erving map a nd let ρ : S ′ → S b e a quasibijection. Apply Lemma 3.1 to pro duce a quas ibijection π ( ρ, σ ′ ) : T ′ → T and order pr eserving map σ ( ρ, σ ′ ) : T → S such that σ ′ · ρ = π ( ρ, σ ′ ) · σ ( ρ, σ ′ ) . Then b ( π ( ρ, σ ′ )) = Γ B ( b ( ρ ); 1 , . . . , 1 ) and we can a pply the first equiv ar iance Fiedorowicz condition. Then the first equiv ariance condition is satisfied in genera l beca use of the second equiv aria nce c o ndition of the Definition 6.1 applied to the commutativ e diagram [ T ′ ] ❄ [ π ] [ σ ] ✲ [ π ( ρ, σ ′ ))] [ T ] ❄ [ T ] ✲ [ σ ] [ S ] Theorem 6 . 2 The c ate gory of Q n -op er ads 3 ≤ n ≤ ∞ and the c ate gory of symmetric op er ads ar e e quivalent. 22 Pro of. The pr o of is a rep etition of the ab ove pro of with a simplification that s n : Π Q n → S for 3 ≤ n ≤ ∞ is a n equiv a lence. 7 Lo c ally constan t n - op erads. The quasisymmetric n -op erads are defined in an y symmetric monoidal category V . But acco rding to Theorems 6.1 and 6.2 they are different fr om sy mmet- ric op erads only when n = 1 , 2 . As we have seen b efore the main reason why quasisymmetric oper ads collapse to symmetric ope r ads for n > 2 is that config- uration space C onf k ( ℜ n ) is simply connected a nd so lo calising with r esp ect to quasibijections ca n only pro duce a gro upo id equiv a lent to S . The cor rect pro - cedure, ther efore, should b e to take the weak ω -group oid Π ∞ Q n and co nsider presheav es on it with v alues in V as the categor y of collectio ns. There are , how ever, considerable tec hnical difficulties with this appr oach. F or tunately , the results of Cisinsk i [5] show a wa y around this problem by considering as the catego r y of co llections the categ ory of lo c al ly c onstant functor s from Q op n to V . Pur suing this idea we give the following definition. Definition 7.1 L et V b e a symmetric monoidal c ate gory and W (we ak e quiva- lenc es) b e a sub class of its morphisms. A lo c al ly c onstant n -op er ad in ( V , W ) is an n -op er ad A in V such that for every quasibije ction σ : T → S the morphism A ( σ ) : A S → A T is a we ak e quivalenc e. Remark. W e hav e chosen the name lo cally constant n -op er a ds (which so me peo ple pr efer to call homotopically lo cally constant n -op erads) for tw o reasons. First, we would like o ur ter minology to a gree with the terminolo gy of [5]. But more imp orta nt rea son is a b out philos ophy . The no tion o f lo cally co nstant n - op erad (and lo cally constant functor ) dep ends only on the class of weak equiv- alences but not o n the choice o f homotopy theory in V . F or exa mple, if V is a symmetric monoida l catego ry and I so is the class of all is omorphisms, a lo cally constant n -op er ad in ( V , I so ) is the sa me as a qua sisymmetric n -op er ad in V . So, the w ord ‘ho motopical’ is a little bit mislea ding. Compare this situa tion with the theory of homotopy limits develop ed in [6]. W e b elieve that a ‘true’ reas on for this phenomenon is that homotopy limit and lo cally constant functors are higher categ orical rather then homotopical no tions. But the homotopy theor y is helpful in computations . As far as we know a similar arg ument is b ehind Cisinski’s choice o f terminology . An example o f a n interesting lo ca lly consta nt n -op erad in the mo del catego ry of top o logical space s, which is not a quasis y mmetric n -op er ad is the Getzler - Jones n -op erad GJ n constructed in [1] for all n < ∞ . One can also construct an ∞ -version GJ ∞ by the for mu la GJ ∞ T = GJ n T , where < − n is the minimal nonempty r elation in the ∞ -ordinal T , the n - ordinal T has the same underlying set as T a nd the relation < n − p − 1 in T coincides with the relation < − p in T . Let V b e a symmetric mono idal catego ry equipp ed with a clas s of weak equiv ale nce s W . W e introduce the following notations: 23 • S O is the category of symmetric op era ds in V ; • B O is the categ o ry of braided op erads in V ; • O n is the categ ory of n -op erads in V ; • QO n is the full sub categ ory of O n of quasisymmetric n -op er ads in V ; • LC O n is the full sub catego ry of O n of lo cally constant n -op er ads in ( V , W ). Definition 7.2 A morphism of op er ads (in any of the c ate gories ab ove) is a we ak e quivalenc e if it is a termvise we ak e quivalenc e of the c ol le ctions. Th e homotopy c ate gory of op er ads is the c ate gory of op er ads lo c alise d with r esp e ct to the class of we ak e quivalenc es. Let us descr ib e the r elations b etw een the different categor ies of op era ds we deal with in this paper. W e have already done it for the case W = I so in Section 6. Let us fix a ba se symmetric mono ida l mo del catego r y V and let W b e its class of weak eq uiv alences in the mo del categ ory theo retic sense . Moreover, w e will as s ume that V sa tis fie s the c o nditions fr om Sectio n 5 o f [1], which mea ns that there is a model structure on the catego r y of collections transferable to the category of op erads (see [1] for the details). F or n = 1 the re lationships b etw een op eradic categories a b ove is simple. The following catego r ies are is omorphic to the categ ory of nonsymmetric op erads O 1 ≃ LC O 1 ≃ QO 1 and we have a classical adjunction betw een nonsymmetric operads and symmet- ric op erads. All this is true o n the level of homotopy categories. F or n = 2 w e hav e the following dia gram of categories and right and left adjoint functors: O 2 ❅ ❅ ❅ ❘ ✻ ✲ ✛ Des 2 S y m 2 S O ❄ ✻ LC O 2 ✛ ✲ ✛ ✲ B O QO 2 I 2 L 2 U 2 F 2 K 2 J 2 B 2 A 2 In this dia gram the functor D e s 2 is right adjoin t to S ym 2 (see [1 , 2 ] for the construction). The functor s I 2 and J 2 are natural inclusions. The functor K 2 is left adjoin t to J 2 and L 2 is left adjoin t to the comp osite J 2 · I 2 . Using the theor y of internal op era ds from [2] one can show tha t L 2 on the lev el of collections is given by the left Kan extension along the lo ca lisation functor l 2 : Q 2 → Π Q 2 : L 2 ( A ) = L an l 2 ( A ) . (4) 24 W e hav e a lso the same fo rmula for K 2 . The functor A 2 is a right a djoint and B 2 is a left a djoint par t of the equiv alence constructed in the section 6. Finally , U 2 is the functor which pro duces a braide d oper ad from a symmetric op er ad by pulling back a long the functor p : B r → S and F 2 is its left adjoint given b y quotienting with re s p ect to the action of the pure braid gro ups . Theorem 7 . 1 • The homotopy c ate gory of lo c al ly c onstant 2 -op er ads and the homotopy c ate gory of quasisymmetric 2 -op er ads and t he homotopy c at- e gory of br aide d op er ads ar e e quivalent. • The functor of symmetrisation S ym 2 c an b e factorise d as L 2 · B 2 · F 2 . • A b ase sp ac e X is a 2 -fold lo op sp ac e (u p to gr oup c ompletion) if and only if it is an algebr a of a c ont ra ctible 2 -op er ad , if and only if it is an algebr a of a c ont r actible br aide d op er ad (Fie dor owicz’s r e c o gnition principle [4]). Pro of. Since Π Q 2 is a g r oup oid, the lo calisa tion functor l 2 is lo cally constant in the sense of [5, 1 .14]. By F o rmal Serre sp e ctral s equence [5, P rop. 1.15] we ge t that the homoto py left K an extension along l 2 is a left adjoint to the restriction functor b etw een homotopy categorie s of collections. T he functor l 2 induces a weak eq uiv alence of the ner ves and so, by Quillen theorem B, it is also aspher ic al in the sense o f [5, 1.4]. So, by [5, Pro p. 1.1 6] the homotopy left Kan extension along l 2 is an equiv alence of homotopy categ ories of collections. T ak ing in to acco unt the formula (4) we see that to prov e the equiv alence of homotopy categ ories of op erads it is enoug h to show that for an n -op era d A (1 ≤ n ≤ ∞ ) there ex ists a co fibrant replace ment B ( A ) suc h that the underlying Q n -collection of B ( A ) is cofibra nt in the pro jective mo de l str ucture. Recall [1], that ph n is the categorical symmetric o pe rad representing the 2- functor of internal pruned n - o p erads. In particular an n -opera d A is repr esented by a n op er adic functor ˜ A : ph n → V • . If we forg e t ab out op eradic structures then for any k ≥ 0 w e will hav e a functor ˜ A k : ph n k → V . T ake the ba r - resolution B ( L , L, C ( ˜ A )) , where ( L, µ, ǫ ) is the mo nad on the functor category [ d ( ph n ) , V ] genera ted by r estriction and left Ka n extensio n along the inclusion of discr etisation d ( ph n k ) of ph n k to ph n k and C ( A ) is the ter mwise cofibrant replacement of the underlying n -collection of A. These functors for a ll k ≥ 0 form an op eradic functor B ( A ) : ph n → V • and, hence, deter mine an n -o p erad B ( A ) which is a cofibr ant replacement for A [2]. Since B k ( A ) is a bar- construction on c ofibrant collection it is cofibra nt in the pro jective mo del categor y of functors . Recall also that there is a symmetric categoric al o p erad rh n representing the 2-functor of in ternal reduced n -op er ads [1] and a pro jectio n p : ph n → rh n . A typical fib er (in a strict sense) of this pro jection ov er a n ob ject w ∈ rh n is a catego ry with a terminal ob ject s ( w ) . The map s asse mbles to the (no nop eradic) functor s : rh n → ph n , which is by definition a sec tio n of p and it is also a right adjoint to p. The co unit of this adjunction is the identit y and the unit is the unique map to the ter minal ob ject s ( w ) . 25 The simple calcula tions with this adjunction sho ws that the restriction f unc- tor s ∗ preserves the cofibr ant o b jects for pro jective mo del str uctures a nd so s ∗ ( B k ( A )) is cofibrant. There is also an inclusion j : J op n → rh n [1]. It is not ha r d to see als o that the categ ories J op n ( k ) a nd rh n k are Reedy c a tegories . Recall, that the ob jects of rh n k are pla na r trees decora ted by pruned n -trees (i.e. n - ordinals). One ca n choose the total num b e r of edges of n -trees in a decorated planar tr ee as a degre e function and see that each morphism decreases strictly this function. 1 . It follows from these co nsiderations that the functor s ∗ ( B k ( A )) satisfies the following pro p erty characterising cofibrant ob jects in the pro jective mo del cat- egories for functor catego ries ov er Reedy ca tegories : col im ( s ∗ ( B k ( A ))( w )) → s ∗ ( B k ( A ))( T ) (5) is a cofibration. Here the colimit is taken ov er the category of all w → T , w 6 = T in rh n k . It w as pr ov ed in [1] that J op n ( k ) is cofinal in rh n k . E xactly the same argument shows tha t in the colimit (5) o ne can replace w ∈ rh n k by the o b jects from J op n ( k ) . And, ther efore the restriction j ∗ s ∗ ( B k ( A )) is cofibrant as w ell. The quotient functor q : J op n ( k ) → Q op n ( k ) induces the restriction functor q ∗ on functor catego ries which is fully faithful. It follows from this that q ∗ reflects cofibrations. W e obse r ve that q ∗ ( u ( B ( A )) = j ∗ s ∗ ( B k ( A )) and so u ( B ( A )) is cofibrant. Hence the firs t statement o f the theor em is pr oved. The statement ab out symmetrisation is ob vious since Des 2 = U 2 · A 2 · J 2 · I 2 . Finally , a contractible o p erad is lo c ally constant so the third statement follows from the first statement, Theorem 8.6 from [1] and the fact that the functors U 2 , A 2 , J 2 , I 2 preserve endomorphism op er ads. F or 3 ≤ n ≤ ∞ the cor resp onding diagram is: A n B n K n J n I n L n O n ❍ ❍ ❍ ❍ ❍ ❍ ❥ ✻ ✲ ✛ Des n S y m n S O ❄ ✻ LC O n ✲ ✛ QO n Theorem 7 . 2 • F or 3 ≤ n ≤ ∞ the c ate gory of symmetric op er ads is e quiv- alent to the c ate gory of quasisymmetric n -op er ads; • F or 3 ≤ n < ∞ a b ase s p ac e X is an n -fold lo op sp ac e (up to gr oup c ompletion) if and only if it is an algebr a of a c ontr actible n -op er ad; 1 In fact, rh n k is a p oset but w e did not provide a pr oof of this fact in [1]. 26 • The homotopy c ate gory of lo c al ly c onstant ∞ -op er ads, t he homotopy c ate- gory of quasisymmetric ∞ -op er ads and the homotopy c ate gory of s ymm et - ric op er ads ar e e quivalent. • A b ase s p ac e X is an infinite lo op sp ac e (up to gr oup c ompletion) if and only if it is an algebr a of a c ont r actible ∞ -op er ad if and only if it is an algebr a of a c ontr actible symmetric op er ad (May’s r e c o gnition principle [7]). Pro of. The pro of is ana logous to the pro of of Theor e m 7.1. An interesting question which we do not consider here is the existence o f mo del str uctures on the v arious categ ories of op e rads. The results o f [5] indicate that this might b e p o ssible. But it is a sub ject for a future pap er. Ac kno wledgements. I would lik e to thank Denis-Char les Cisinski for his nice answers [5] to my so me time naive questions. I wish to expre ss my gr atitude to C.B e r ger, I.Galvez, E.Getzle r , V.Gorbunov, A.Davydov, R.Street, A.T o nks, M.W eb er for many useful discussions and to my anonymous r eferee for useful comments concer ning the presentation o f the pap er. I als o g ratefully ackno wledge the financial supp or t of Sco tt Russel John- son Memor ial F ounda tio n, Max P lank Institut f¨ ur Mathematik and Austr a lian Research Council (gra nt No. DP0558 372). Bibliograph y . 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