Half-balanced braided monoidal categories and Teichmueller groupoids in genus zero
aa r X i v : . [ m a t h . QA ] S e p HALF-BALANCED BRAIDED MONOIDAL CATEGORIES ANDTEICHM ¨ULLER GROUPOIDS IN GENUS ZERO
BENJAMIN ENRIQUEZ
Abstract.
We introduce the notions of a half-balanced braided monoidal category and ofits contraction. These notions give rise to an explicit description of the action of the Galoisgroup of Q on Teichm¨uller groupoids in genus 0, equivalent to that of L. Schneps. We alsoshow that a prounipotent version of this action is equivalent to a graded action. Introduction and main results
Let M Q g,n be the moduli space of curves of genus g with n marked points. Its fundamentalgroupoid with respect to the set of maximally degenerate curves is called the Teichm¨ullergroupoid T g,n . One of the main features of Grothendieck’s geometric approach to the Galoisgroup G Q of Q is the study of its action on the profinite completions b T g,n ; according to thisphilosophy, G Q could be characterized as the group of automorphisms of the tower of all b T g,n ,compatible with natural operations, such as the Knudsen morphisms. It is therefore importantto describe explicitly the action of G Q on the collection of all the b T ,n . Such a descriptionwas obtained in [Sch]. More precisely, an explicit profinite group d GT was introduced in [Dr],together with a morphism G Q → d GT. The following was then proved in [Sch]:
Theorem 1.
There exists a morphism d GT → Aut( b T ,n ) , such that the morphism G Q → Aut( b T ,n ) factors as G Q → d GT → Aut( b T ,n ) . The first purpose of this paper is to present a variant of the proof of [Sch]. This variantrelies on the notion of a half-balanced braided monoidal category (b.m.c.), which appearedimplicitly recently in [ST] and is here made explicit. We introduce the notion of a (half-)balanced contraction of such a category C : it consists of a functor C → O , satisfying certainproperties. Whereas a balanced b.m.c. gives rise to representations of the framed braid group onthe plane ˜ B n (for n ≥ B n , a (half-)balancedcontraction gives rise to representations of quotients of ˜ B n . This quotient is an abelian extensionof the quotient B n /Z ( B n ) of B n by its center in the case of a balanced contraction, and is anabelian extension of the mapping class group in genus zero Γ ,n (another quotient of B n ) in thecase of a half-balanced contraction.To each set S , we associate an object [ PaB hbalS → \ PaDih S in the category whose objectsare contractions of profinite half-balanced b.m. categories, enjoying universal properties. Thesecontractions may be viewed as the analogues of the universal b.m. categories appearing in [JS].We show that the action of d GT on such categories may be lifted to the half-balanced setup.This defines in particular an action of d GT on \ PaDih S , from which it is is easy to derive anaction of b T ,n .The above profinite theory admits a prounipotent version. The group d GT and the Te-ichm¨uller groupoid b T ,n admit proalgebraic versions k GT( k ) , T ,n ( k ), where k is a Q -ring. We then have morphisms d GT → GT( Q l ), b T ,n → ( T ,n ) l → T ,n ( Q l ), where l is a prime number and ( T ,n ) l is the pro- l completion of T ,n . We construct a group schemeAut T ,n ( − ), together with a morphism Aut T ,n ( k ) → Aut( T ,n ( k )), a group scheme morphismGT( − ) → Aut T ,n ( − ), and a group Aut(( T ,n ) l , T ,n ( Q l )), equipped with morphismsAut(( T ,n ) l ) ← Aut(( T ,n ) l , T ,n ( Q l )) → Aut T ,n ( Q l ) . Theorem 2.
The morphism G Q → Aut(( T ,n ) l ) factors as G Q → Aut(( T ,n ) l , T ,n ( Q l )) → Aut(( T ,n ) l ) , and there exists a morphism GT( − ) → Aut T ,n ( − ) , such that the following dia-gram commutes G Q / / ( ( QQQQQQQQQQQQQQQ (cid:15) (cid:15)
Aut(( T ,n ) l )Aut(( T ,n ) l , T ,n ( Q l )) O O (cid:15) (cid:15) GT( Q l ) / / Aut T ,n ( Q l )We say that an algebraic (resp., prounipotent) group over Q is graded iff its Lie algebra isgraded by Z ≥ (resp., by Z > ). We say that a groupoid G is graded prounipotent if for any s ∈ Ob G , Aut G ( s ) is graded prounipotent. In [Dr], a graded Q -algebraic group GRT( − ) wasconstructed, together with an isomorphism GT( − ) → GRT( − ). Theorem 3.
There exists a graded prounipotent groupoid T gr ,n ( − ) and a graded morphism GRT( − ) → Aut T gr ,n ( − ) , such that the diagram GT( − ) → Aut T ,n ( − ) ↓ ↓ GRT( − ) → Aut T gr ,n commutes. Teichm¨uller groupoids in genus 0
Quotient categories.
Let C be a small category and let G be a group. We define anaction of G on C as the data of: (a) a group morphism G → Perm(Ob C ), (b) for any g ∈ G , anassignment Ob C ∈ X i gX ∈ Iso C ( X, gX ), such that i ghX = i hgX i gX .We then get a group morphism G → Aut C = { autofunctors of C} , where the autofunctorinduced by g ∈ G is the action of g at the level of objects, and gφ := i gY φ ( i gX ) − for φ ∈ Hom C ( X, Y ). Lemma 4.
1) For any α, β ∈ (Ob C ) /G , there is a unique action of G × G on X ( α, β ) := ⊔ X ∈ α,Y ∈ β Hom C ( X, Y ) , such that ( g, h ) Hom C ( X, Y ) = Hom C ( gX, hY ) and ( g, h ) φ = i hY φ ( i gX ) − .2) Set X ( X, β ) := ⊔ Y ∈ β Hom C ( X, Y ) , X ( α, Y ) := ⊔ X ∈ α Hom C ( X, Y ) , then G acts on thesesets (by permutation of β in the first case and of α in the second one) and we have a well-defined map X ( X, β ) G × X ( β, Z ) G → Hom C ( X, Z ) compatible all the maps Hom C ( X, Y ) × Hom C ( Y, Z ) → Hom C ( X, Z ) . Taking the product of these maps over X ∈ α , Z ∈ γ and furtherthe quotient by G × G , we obtain a map X ( α, β ) G × G × X ( β, γ ) G × G → X ( α, γ ) G × G , which isassociative. The proof is straightforward. We then define the quotient category C /G by Ob( C /G ) :=(Ob C ) /G and ( C /G )( α, β ) := X ( α, β ) G × G . Remark 5. If X ∈ α and Y ∈ β , then ( C /G )( α, β ) ≃ C ( X, Y ) G X × G Y , where G X = { g ∈ G | gX = X } . Proposition 6. If D is a small category, then a functor C / Γ → D is the same as a functor F : C → D , such that F ( gX ) = F ( X ) and F ( i gX ) = id F ( X ) for any g ∈ G , X ∈ Ob C . ALF-BALANCED CATEGORIES AND TEICHM¨ULLER GROUPOIDS IN GENUS ZERO 3
The proof is immediate.1.2.
Quotients of the braid group.
Let B n be the braid group of n strands in the plane.It is presented by generators σ , . . . , σ n − subject to the Artin relations σ i σ i +1 σ i = σ i +1 σ i σ i +1 for i = 1 , . . . , n − σ i σ j = σ j σ i for | i − j | >
1. Its center Z n := Z ( B n ) is isomorphic to Z and is generated by ( σ · · · σ n − ) n . There is a morphism B n → S n , uniquely determined by σ i s i := ( i, i + 1); it factors through a morphism B n /Z n → S n . Lemma 7.
Let C n := h g | g n = 1 i be the cyclic group of order n . We have an injection C n ֒ → S n via g (cid:0) ... n ... (cid:1) , which admits a lift C n → B n /Z n , given by g σ · · · σ n − . Let Γ ,n := B n / (( σ · · · σ n − ) n , σ · · · σ n − · · · σ ) be the mapping class group of type (0 , n )(see [Bi]). The relation σ · · · σ n − · · · σ = 1 is called the sphere relation as the quotient B n / ( σ · · · σ n − · · · σ ) is isomorphic to the braid group of n points on the sphere. In thisgroup, the relation ( σ · · · σ n − ) n = 1 holds. The morphism B n → S n factors through amorphism Γ ,n → S n .The dihedral group D n := h r, s | r n = s = ( rs ) = 1 i may be viewed as a subgroup of S n via r (cid:0) ... n ... (cid:1) , s (cid:0) ... nn n − ... (cid:1) . Lemma 8.
There exists a unique morphism D n → Γ ,n , r σ · · · σ n − , s σ ( σ σ ) · · · ( σ n − · · · σ ) ,lifting the injection D n ֒ → S n .Proof. One knows that h n := σ ( σ σ ) · · · ( σ n − · · · σ ) ∈ B n is the half-twist, so that h n = z n = ( σ · · · σ n − ) n = ρ n , where ρ = σ · · · σ n − and z n is the full twist, generating Z ( B n ).Moreover, h n ρ − = im( h n − ∈ B n − → B n ), so ( h n ρ − ) = z n − = z n ( σ n − · · · σ · · · σ n − ) − ,where we identify z n − with its image under B n − → B n . The images of h n , ρ in Γ ,n thereforesatisfy ¯ h n = ¯ ρ n = (¯ h n ¯ ρ − ) = 1, which are equivalent to a presentation of D n . (cid:3) Teichm¨uller groupoids.
Let G be a group and Γ ⊂ S n be a subgroup. Assume that G → S n is a group morphism and let Γ → G be such that G / / S n Γ O O ` ` AAAAAAAA commutes. Let S be aset, with | S | = n .Define a category C G,S by Ob C G,S := Bij([ n ] , S ); for σ, σ ′ ∈ Ob C G,S , Hom( σ, σ ′ ) := G × S n { ( σ ′ ) − σ } ; the composition of morphisms is induced by the product in G .Define an action of Γ on C G,S as follows. For γ ∈ Γ, σ ∈ Bij([ n ] , S ), γ · σ := σγ − , and i γσ ∈ Hom( σ, σγ − ) = G × S n { γ } is im( γ ∈ Γ → G ). We then obtain a quotient category C Γ ,G,S := C G,S / Γ. Example 9.
When G = B n /Z n and Γ = C n , we set Cyc( S ) := C Γ ,G,S ; its set of objects isCyc( S ) := Bij([ n ] , S ) /C n (the set of cyclic orders on S ). Example 10.
When G = Γ ,n and Γ = D n , we set Dih( S ) := C Γ ,G,S ; its set of objects isDih( S ) := Bij([ n ] , S ) /D n = Cyc( S ) / {± } , which we call the set of dihedral orders on S . Definition 11. If C is a small category and T π → Ob C is a map, we define the category π ∗ C by Ob π ∗ C := T and π ∗ C ( t, t ′ ) := C ( π ( t ) , π ( t ′ )) for t, t ′ ∈ T . We have natural maps { planar 3-valent trees with leaves bijectively indexed by S } π cyc → Cyc( S )and { planar 3-valent trees with leaves bijectively indexed by S } / (mirror symmetry) π dih → Dih( S ) . BENJAMIN ENRIQUEZ
We then set T ′ ,S := π ∗ cyc Cyc( S ), T ,S := π ∗ dih Dih( S ).When S = [ n ], T ,S identifies with the fundamental groupoid to the moduli stack M Q ,n withrespect to the set of maximally degenerate real curves (see [Sch]).2. Contractions on (half-)balanced categories (Half-)balanced categories.
Recall that a braided monoidal category (b.m.c.) is a set( C , ⊗ , , β XY , a XY Z ), where C is a category, ⊗ : C × C → C is a bifunctor, β XY : X ⊗ Y → Y ⊗ X and a XY Z : ( X ⊗ Y ) ⊗ Z → X ⊗ ( Y ⊗ Z ) are natural constraints, ∈ Ob C and X ⊗ ∼ → X ∼ ← ⊗ X are natural isomorphisms, satisfying the hexagon and pentagon conditions (see e.g. [Ka]).A balanced structure on the small b.m.c. C is the datum of a natural assignment Ob C ∋ X θ X ∈ Aut C ( X ), such that θ X ⊗ Y = ( θ X ⊗ θ Y ) β Y X β XY for any X, Y ∈ Ob C (see [JS]); the naturality condition is θ X ′ φ = φθ X for any X, X ′ ∈ Ob C and φ ∈ Hom C ( X, X ′ ).Similarly, a half-balanced structure on C is the data of: (a) an involutive autofunctor ∗ : C → C , X X ∗ , such that ( X ⊗ Y ) ∗ = Y ∗ ⊗ X ∗ , ( f ⊗ g ) ∗ = g ∗ ⊗ f ∗ for any X, . . . , Y ′ ∈ Ob C and f ∈ Hom C ( X, X ′ ), g ∈ Hom C ( Y, Y ′ ), a ∗ X = a X ∗ , β ∗ XY = β Y ∗ X ∗ , a ∗ XY Z = a Z ∗ Y ∗ X ∗ ; (b) anatural assignment Ob C ∈ X a X ∈ Iso C ( X, X ∗ ), such that a X ⊗ Y = ( a Y ⊗ a X ) β XY for any X, Y ∈ Ob C ; here naturality means that a Y φ = φ ∗ a X for any φ ∈ Hom C ( X, Y ).Note that a half-balanced structure gives rise to a balanced structure by θ X := a X ∗ a X .2.2. Contractions.Definition 12.
A contraction on the small balanced category C is a functor h−i : C → O , X
7→ h X i , such that:1) for any X, Y ∈ Ob C , h Y ⊗ X i = h X ⊗ Y i (=: h X, Y i ) , and h ( θ Y ⊗ id X ) β XY i = id h X,Y i ;2) for any X, Y, Z ∈ Ob C , h ( X ⊗ Y ) ⊗ Z i = h X ⊗ ( Y ⊗ Z ) i (=: h X, Y, Z i ) and h a XY Z i =id h X,Y,Z i . When needed, we will call such a contraction a “balanced contraction”.
Remark 13.
These axioms imply h θ X ⊗ Y i = id h X,Y i for any X, Y ∈ Ob C , and therefore h θ X i = id h X i by taking Y = . Definition 14.
A contraction on the small half-balanced category C is a functor h−i : C → O ,such that:1) h−i is a balanced contraction on C ;2) for any X ∈ Ob C , h X i = h X ∗ i and h a X i = id h X i . When needed, we will call such a contraction a “half-balanced contraction”.
Lemma 15. If h−i : C → O is a contraction on a half-balanced category, then for any
X, Y ∈ Ob C , h θ X ⊗ θ − Y i = id h X,Y i = h ( θ X ⊗ id Y ) β Y X β XY i .Proof. We have β − XY ( θ − Y ⊗ id X ) a X ∗ ⊗ Y ∗ β − X ∗ Y ∗ ( θ − Y ∗ ⊗ id X ∗ ) a X ⊗ Y = β − XY ( θ − Y ⊗ id X ) a X ∗ ⊗ Y ∗ β − X ∗ Y ∗ a X ⊗ Y (id X ⊗ θ − Y )= β − XY ( θ − Y ⊗ id X )( a Y ∗ ⊗ a X ∗ ) a X ⊗ Y (id X ⊗ θ − Y )= β − XY ( θ − Y ⊗ id X )( a Y ∗ a Y ⊗ a X ∗ a X ) β XY (id X ⊗ θ − Y )= β − XY (id Y ⊗ θ X ) β XY (id X ⊗ θ − Y ) = θ X ⊗ θ − Y . ALF-BALANCED CATEGORIES AND TEICHM¨ULLER GROUPOIDS IN GENUS ZERO 5
Now h a X ⊗ Y i = h a X ∗ ⊗ Y ∗ i = id h X,Y i by the half-balanced contraction axiom, and h β − XY ( θ − Y ⊗ id X ) i = h β − X ∗ Y ∗ ( θ − Y ∗ ⊗ id X ∗ ) i = id h X,Y i by the balanced contraction axiom. It follows that h θ X ⊗ θ − Y i = id h X,Y i . The second statement follows from ( θ X ⊗ id Y ) β Y X β XY = ( θ X ⊗ θ − Y ) θ X ⊗ Y and h θ X ⊗ Y i = id h X,Y i . (cid:3) Relation with braid group representations.
Set ˜ B n := Z n ⋊ B n , where the actionof B n is Z n is via B n → S n → Aut( Z n ); ˜ B n is usually called the framed braid group of theplane. If C is a balanced b.m.c. and X ∈ Ob C , then there is a morphism ˜ B n → Aut C ( X ⊗ n ) (aparenthesization of the n th fold tensor product being chosen), given in the strict case by δ i id X ⊗ i − ⊗ θ X ⊗ id X ⊗ n − i , σ i id X ⊗ i − ⊗ β X,X ⊗ id X ⊗ n − i − . Here δ i is the i th generator of Z n ⊂ ˜ B n .We now define ^ B n /Z n to be the quotient of ˜ B n by its central subgroup (isomorphic to Z )generated by ( Q ni =1 δ i ) z n (recall that z n is a generator of Z n = Z ( B n ); the product in Z n isdenoted multiplicatively). One can prove that there is a (generally non-split) exact sequence1 → Z n → ^ B n /Z n → B n /Z n → Proposition 16.
Let C h−i → O be a balanced contraction of C , then we have a commutativediagram B n ← ˜ B n → Aut C ( X ⊗ n ) ↓ ↓ ↓ h−i B n /Z n ← ^ B n /Z n → Aut O ( h X ⊗ n i ) Proof.
We have im(( Q ni =1 δ i ) z n ∈ ˜ B n → Aut C ( X ⊗ n )) = θ X ⊗ n , so according to Remark 13,the image of this in Aut O ( h X ⊗ n i ) is id h X ⊗ n i . The factorization implied in the right squarefollows. The left square obviously commutes. (cid:3) Set now ˜Γ ,n be the quotient of ˜ B n by the normal subgroup generated by ( Q ni =1 δ i ) z n and δ σ · · · σ n − · · · σ . Then we have an exact sequence 1 → Z n → ˜Γ ,n → Γ ,n → Proposition 17.
Let C be a half-balanced b.m.c., let C h−i → O be a half-balanced contraction andlet X ∈ Ob C . Then we have a comutative diagram B n ← ˜ B n → Aut C ( X ⊗ n ) ↓ ↓ ↓ h−i Γ n ← ˜Γ n → Aut O ( h X ⊗ n i ) Proof.
We have im( δ σ · · · σ n − · · · σ ∈ ˜ B n → Aut C ( X ⊗ n )) = ( θ X ⊗ id Y ) β Y X β XY ( Y = X ⊗ n − ), whose image in Aut O ( h X ⊗ n i ) is id X ⊗ n by Lemma 15. (cid:3) Universal (half-)balanced categories
Universal (strict) braided monoidal categories.
Recall that the small b.m.c. C iscalled strict iff ( X ⊗ Y ) ⊗ Z = X ⊗ ( Y ⊗ Z )(= X ⊗ Y ⊗ Z ) and a X,Y,Z = id X ⊗ Y ⊗ Z for any X, Y, Z ∈ Ob C . Following [JS], we associate a universal strict b.m.c. B S to each set S . Its set of objectsis Ob B S := ⊔ n ≥ S n ; the tensor product is defined by s ⊗ s ′ = ( s , . . . , s n , s ′ , . . . , s ′ n ′ ) ∈ S n + n ′ for s = ( s , . . . , s n ) ∈ S n , s ′ = ( s ′ , . . . , s ′ n ′ ) ∈ S n ′ . If s ∈ S n , s ′ ∈ S n ′ , then Hom B S ( s, s ′ ) = ∅ if n = n ′ , and Hom B s ( s, s ′ ) = B n × S n { f ∈ S n | s ′ f = s } if n = n ′ . The tensor product ofmorphisms is induced by restriction from the group morphism B n × B n ′ → B n + n ′ , ( b, b ′ ) b ∗ b ′ , BENJAMIN ENRIQUEZ uniquely determined by σ i ∗ σ i , 1 ∗ σ i ′ = σ n − i ′ (which corresponds to the juxtapositionof braids). The braiding is β s,s ′ = b nn ′ , where b nn ′ ∈ B n + n ′ is given by b nn ′ = ( σ n ′ · · · σ ) · · · ( σ n + n ′ − · · · σ n ) . The universal property of B S is then expressed as follows: to each strict small b.m.c. C andany map S → Ob C , there corresponds a unique tensor functor B S → C , such that the diagram S / / " " FFFFFFFFF Ob C Ob B S O O commutes.We now describe the universal b.m.c. PaB S associated to S ([JS, Ba]). Define first T n as theset of parenthesizations of a word in n identical letters. Equivalently, this is the set of planar3-valent rooted trees with n leaves, e.g. the tree root ❅❅(cid:0)(cid:0)❆❆✁✁ ❆❆✁✁ corresponds to the word ( •• )( •• ). The concatenation of words is a map T n × T m → T n + m ,( t, t ′ ) t ∗ t ′ (e.g., ( •• , •• ) ( •• )( • ) • ); this is illustrated in terms of trees as follows root ❅❅(cid:0)(cid:0) ... t * root ❅❅(cid:0)(cid:0) ... t ′ = root ❅❅(cid:0)(cid:0)❆❆✁✁ ❆❆✁✁ ... ... t t ′ The set of objects of
PaB S is then defined by Ob PaB S := ⊔ n ≥ T n × S n ; the tensor product isdefined by ( t, s ) ⊗ ( t ′ , s ′ ) := ( t ∗ t ′ , s ⊗ s ′ ). The morphisms are defined by Hom PaB S (( t, s ) , ( t ′ , s ′ )) :=Hom B S ( s, s ′ ). The tensor product of morphisms and the braiding and associativity constraintsare uniquely determined by the condition that the obvious functor PaB S → B S is monoidal.In particular, a XY Z corresponds to 1 ∈ B | X | + | Y | + | Z | , where | ( s, t ) | = n for ( s, t ) ∈ T n × S n .Then PaB S has a universal property with respect to non-necessarily strict braided monoidalcategories, analogous to that of B S .3.2. Universal balanced categories.
For s ∈ Ob B S , set θ s := z | s | ∈ Aut B S ( s ) ⊂ B | s | . Theassignment s θ s equips B S with a balanced structure. We denote by B balS the resultingbalanced strict b.m.c. One checks that it has the following universal property: Lemma 18.
To any balanced strict small b.m.c. C and any map S f → Ob C , such that θ f ( s ) =id f ( s ) for any s ∈ S , there corresponds a unique functor B balS → C compatible with the balancedand monoidal structures, such that the diagram S / / FFFFFFFFF Ob C Ob B balS O O commutes. If now X = ( s, t ) ∈ Ob PaB S , we set θ X := θ s ∈ Aut B S ( s ) = Aut PaB S ( X ). The assignment X θ X equips PaB S with the structure of a balanced b.m.c., denoted PaB balS and enjoyinga universal property with respect to maps S → Ob C , where C is a balanced braided monoidalcategory such that θ f ( s ) = id f ( s ) similar to Lemma 18. ALF-BALANCED CATEGORIES AND TEICHM¨ULLER GROUPOIDS IN GENUS ZERO 7
Universal half-balanced categories.
We define an involution ∗ : B S → B S as follows.It is given at the level of objects by s ∗ := ( s n , . . . , s ) for s = ( s , . . . , s n ) and the level ofmorphisms by restriction of the automorphism σ i σ n − i of B n . For s ∈ Ob B S , we set a s := h | s | ∈ Iso B S ( s, s ∗ ) ⊂ B | s | . This defines the structure of a half-balanced category on B S , denoted B hbalS , whose balanced structure is that described in Subsection 3.2. It has thefollowing universal property: Lemma 19.
For each strict half-balanced small b.m.c. C and each map S f → Ob C such that forany s ∈ S , f ( s ) ∗ = f ( s ) and a f ( s ) = id f ( s ) , there exists a unique functor B hbalS → C , compatiblewith the monoidal and half-balanced structures, and such that the diagram S / / FFFFFFFFF Ob C Ob B balS O O commutes. We now define an involution ∗ of PaB S as follows. At the level of objects, it is given by X ∗ = ( t ∗ , s ∗ ) for X = ( t, s ), where t ∗ is the parenthesized word t , read in the reverse order(in terms of trees, this is the mirror image of t ). At the level of morphisms, it coincideswith the involution ∗ of B S . We define the assignment Ob PaB S ∋ X a X by a X := a s ∈ Iso B S ( s, s ∗ ) = Iso PaB S ( X, X ∗ ) for X = ( t, s ). This equips PaB S with a half-balanced structure;the resulting half-balanced b.m.c. is denoted PaB hbalS . Its underlying balanced b.m.c. is
PaB balS .It has a universal property with respect to half-balanced small braided monoidal categories C and maps S f → Ob C , such that f ( s ) ∗ = f ( s ) and a f ( s ) = id f ( s ) , similar to that of Lemmas 18and 19. 4. Universal contractions for balanced categories
We will construct categories ( Pa ) Cyc S and a diagram PaB balS → PaCyc S ↓ ↓ B balS → Cyc S in which thehorizontal functors are contractions and the left vertical functor is the canonical monoidalfunctor.We construct Cyc S as follows. Define first g Cyc S as the category with the same objects as B balS , and B n replaced by B n /Z n ) in the definition of morphisms. Define an action of Z on g Cyc S by 1 · ( s , · · · , s n ) := ( s n , s , . . . , s n − ) and i s ∈ Iso( s, · s ) ⊂ B n /Z n is the class of σ · · · σ n − .We then set Cyc S := g Cyc S / Z . Note that Ob Cyc S = ⊔ n ≥ Cyc n ( S ), where Cyc n ( S ) = S n /C n .We then define a functor B balS → Cyc S as the composite functor B balS → g Cyc S → Cyc S .Let us show that the functor h−i : B balS → Cyc S satisfies the balanced contraction condition.If s, s ′ ∈ Ob B S , with s = ( s , . . . , s n ) and s ′ = ( s ′ , . . . , s ′ n ′ ), then s ′ ⊗ s = ( s ′ , . . . , s n ) = n ′ · ( s ⊗ s ′ ), which implies that h s ⊗ s ′ i = h s ′ ⊗ s i . Then ( θ s ′ ⊗ id s ) β s,s ′ ∈ Iso B balS ( s ⊗ s ′ , s ′ ⊗ s ) = B n + n ′ corresponds to ( z n ′ ∗ id n ) b nn ′ = ( σ · · · σ n + n ′ − ) n ′ . Its image in g Cyc S is then i n ′ s ⊗ s ′ ∈ g Cyc S ( s ⊗ s ′ , n ′ · ( s ⊗ s ′ )), whose image in Cyc S is id h s,s ′ i .We now prove the universality of this contraction. Proposition 20.
Let C be a strict small balanced b.m.c., equipped with a map S f → Ob C anda balanced contraction C → O . Then there is a functor
Cyc S → O , such that the diagram B balS → Cyc S ↓ ↓C → O commutes. BENJAMIN ENRIQUEZ
Proof.
First note that since h θ X i = id h X i for X = f ( s ) ⊗ · · · ⊗ f ( s n ) and any ( s , . . . , s n ) ∈ Ob B balS , we have a functor F : g Cyc S → O , such that the diagram B balS → g Cyc S ↓ ↓C → O commutes.If ( s , . . . , s n ) ∈ Ob g Cyc S = Ob B balS , then F ( s , . . . , s n ) = h f ( s ) ⊗ · · · ⊗ f ( s n ) i = h f ( s n ) ⊗ f ( s ) ⊗ · · · ⊗ f ( s n − ) i = F ( s n , . . . , s n − ), therefore F ( gX ) = F ( X ) for any X ∈ Ob g Cyc S andany g ∈ Z . Moreover, we have F ( i s ,...,s n ) ) = F ( σ · · · σ n − ) = h ( θ f ( s n ) ⊗ id f ( s ) ⊗···⊗ f ( s n − ) ) β f ( s ) ⊗···⊗ f ( s n − ) ,f ( s n ) i = id F ( s , ··· ,s n ) by the balanced contraction property.According to Proposition 6, this implies that we have a factorization g Cyc S / / $ $ HHHHHHHHH
Cyc S (cid:15) (cid:15) O (cid:3) We now construct the category
PaCyc S as follows. Let P lT n := { planar 3-valent treesequipped with a bijection { leaves } → [ n ], compatible with the cyclic orders } . We first de-fine the category ^ PaCyc S by Ob ^ PaCyc S = ⊔ n ≥ P lT n × S n , Hom ^ PaCyc S (( t, σ ) , ( t ′ , σ ′ )) =Hom g Cyc S ( σ, σ ′ ). We define an action of Z on PaCyc S by 1 · ( t, ( s , . . . , s n )) := ( t ′ , ( s n , s , . . . , s n − )),where if t = ( T, { leaves of T } α → [ n ]), then t ′ := ( T, { leaves of T } α → [ n ] +1 mod n → [ n ]), and i t,σ ) := i σ ; we then set PaCyc S := ^ PaCyc S / Z , so in particular Ob PaCyc S = { (a planar3-valent tree, a map { leaves } → S ) } .We define a map T n → P lT n , t π ( t ) as the operation of (a) assigning labels 1 , . . . , n tothe vertices of the tree t , numbered from left to right; (b) replacing the root and the edgesconnected to it, by a single edge. E.g., we have π (cid:16) root ❅❅(cid:0)(cid:0)❆❆✁✁ ❆❆✁✁ (cid:17) = ❆❆✁✁ ❆❆✁✁ PaB balS → PaCyc S by the condition that (a) at the level of objects,it is given by the map ⊔ n ≥ T n × S n → ⊔ n ≥ ( P lT n × S n ) /C n and by projection, and (b)the diagram PaB balS → PaCyc S ↓ ↓ B balS → Cyc S commutes. Let us check that this defines a contraction. h X ⊗ Y i = h Y ⊗ X i follows from the fact that for t ∈ T n , t ′ ∈ T n ′ , π ( t ⊗ t ′ ) and π ( t ′ ⊗ t ) can beobtained from one another by cyclic permutation of [ n + n ′ ]; here we recall that ( t, t ′ ) t ∗ t ′ is the concatenation map T n × T n ′ → T n + n ′ . The fact that h ( X ⊗ Y ) ⊗ Z i = h X ⊗ ( Y ⊗ Z ) i follows from π (( t ∗ t ′ ) ∗ t ′′ ) = π ( t ∗ ( t ′ ∗ t ′′ )), which is illustrated as follows ❅❅(cid:0)(cid:0) PPPPP❆❆✁✁ ... t ❆❆✁✁ ... t ′ ❆❆✁✁ ... t ′′ = ❅❅(cid:0)(cid:0)✏✏✏✏✏❆❆✁✁ ... t ❆❆✁✁ ... t ′ ❆❆✁✁ ... t ′′ = ✟✟❍❍ ❆❆✁✁ ... t ✁✁❆❆ ··· t ′ ··· t ′′ ALF-BALANCED CATEGORIES AND TEICHM¨ULLER GROUPOIDS IN GENUS ZERO 9
It is then clear that h a XY Z i = id h X,Y,Z i . The proof of h ( θ Y ⊗ id X ) β XY i = id h X,Y i is asabove. We now prove the universality of the contraction h−i : PaB balS → PaCyc S . Proposition 21.
Let C be a balanced small b.m.c., equipped with a contraction C → O and a map S → Ob C . Then there exists a functor PaCyc S → O , such that the diagram PaB balS → PaCyc S ↓ ↓C → O commutes.Proof. We first construct a functor ^ PaCyc S → O , such that PaB balS → ^ PaCyc S ↓ ↓C → O com-mutes. We define a map P lT n × S n → Ob O as follows. Let ( t, ( s , . . . , s n )) ∈ P lT n × S n . Let e be an edge of t . Cutting t at e , we obtain two rooted trees t i ( i = 1 ,
2) equipped with injectivemaps { leaves of t i } → [ n ]. The images of these maps are of the form { a, a + 1 , . . . , a + n } and { a + n + 1 , . . . , a + n + n } (the integers being taken modulo n ). We then define theimage of ( t, ( s , . . . , s n )) to be h ( ⊗ t i ∈ a +[ n ] f ( s i )) ⊗ ( ⊗ t i ∈ a + n +[ n ] f ( s i )) i . The axioms then im-ply that this object do not depend on e . Indeed, if e ′ is another edge, then to the shortestpath e = e → e → · · · → e k = e ′ from e to e ′ there corresponds a sequence of isomor-phisms of the corresponding objects; each isomorphism has the form h A ⊗ ( B ⊗ C ) i h a − ABC i −→h ( A ⊗ B ) ⊗ C i h β − C,A ⊗ B ( θ − C ⊗ id A ⊗ B ) i −−−−−−−→ h C ⊗ ( A ⊗ B ) i , see ♠ C ❍❍✟✟ ♠ A ♠ B e i +1 e i ♠ A ✁✁❆❆ ♠♠ CB → AB ♠♠ ✁✁❆❆ ♠ C → ♠ C ✁✁❆❆ ♠♠ BA or h A ⊗ ( B ⊗ C ) i → h C ⊗ ( A ⊗ B ) i → h B ⊗ ( C ⊗ A ) i , see ♠ C ❍❍✟✟ ♠ A ♠ B e i +1 e i ♠ A ✁✁❆❆ ♠♠ CB → ♠ C ✁✁❆❆ ♠♠ BA → ♠ B ✁✁❆❆ ♠♠ AC One then proves as before that we have a functor ^ PaCyc S → O , which factors through theaction of Z . (cid:3) Universal contractions for half-balanced categories
We now construct categories ( Pa ) Dih S and a commutative diagram PaB hbalS → PaDih S ↓ ↓ B hbalS → Dih S where the horizontal functors are contractions.We first construct Dih S as follows. Define first g Dih S as the category with the same objectsas B hbalS , with B n replaced by its quotient Γ ,n . Let D := Z ⋊ ( Z /
2) be the infinite dihedralgroup presented as D := h r, s | s = ( rs ) = 1 i . We define an action of D on g Dih S as follows. The action on objects is defined by r · ( s , . . . , s n ) := ( s n , s , . . . , s n − ), s · ( s , . . . , s n ) := ( s n , . . . , s ),and i rs = σ · · · σ n − , i ss = h n . We then set Dih S = g Dih S /D .Note that Ob Dih S = ⊔ n ≥ Dih n ( S ), where Dih n ( S ) = S n /D n , and D n is the quotientof D by the relation r n = 1. We define a functor B hbalS h−i → Dih S as the composite functor B hbalS → g Dih S → Dih S . Let us show that it satisfies the half-balanced contraction conditions.We have a commutative diagram B balS h−i → Cyc S ↓ ↓ B hbalS h−i → Dih S Since the left vertical functor is surjec-tive on objects and the bottom functor is a balanced contraction, the upper functor satisfies thebalanced contraction condition. If now s = ( s , . . . , s n ) ∈ Ob B hbalS , then s ∗ = ( s n , . . . , s ) = s · s , so the classes of s and s ∗ are the same in Dih S = g Dih S /D , hence h s i = h s ∗ i . Then h a s i = h i ss i = id h s i . All this shows that B hbalS h−i → Dih S is a half-balanced contraction. We nowprove the universality of this contraction. Proposition 22.
Let C be a strict half-balanced b.m.c., equipped with a map S f → Ob C , suchthat f ( s ) ∗ = f ( s ) for any s ∈ S , and with a half-balanced contraction C → O . Then there existsa functor
Dih S → O , such that the diagram B hbalS → Dih S ↓ ↓C → O commutes.Proof. We define a functor g Dih S → O by the following conditions: it coincides at the level ofobjects with the functor B hbalS → C → O ; since the images by this functor of z n , σ · · · σ n − · · · σ ∈ Aut B hbalS ( s , . . . , s n ) ⊂ B n are respectively h θ f ( s ) ⊗···⊗ f ( s n ) i and h ( θ f ( s ) ⊗ id ⊗ ni =2 f ( s i ) β ⊗ ni =2 f ( s i ) ,f ( s ) β f ( s ) , ⊗ ni =2 f ( s i ) i ∈ Aut O ( h f ( s ) ⊗ · · · ⊗ f ( s n ) i ) , which are the identity by Remark 13 and Lemma 15, the composite functor B hbalS → C → O factorizes as B hbalS → g Dih Sf ↓ ↓C → O We now show as above that F factorizes as g Dih S → Dih SF ց ↓O Indeed, for s = ( s , . . . , s n ) ∈ Ob g Dih S , then F ( s ) = h f ( s ) ⊗ · · · ⊗ f ( s n ) i . Then F ( r · s ) = h f ( s n ) ⊗ · · · ⊗ f ( s n − ) i = F ( s ), using the axiom h X ⊗ Y i = h Y ⊗ X i of balanced contractions,and F ( s · s ) = h f ( s n ) ⊗ · · · ⊗ f ( s ) i = h ( f ( s ) ⊗ · · · ⊗ f ( s n )) ∗ i = h f ( s ) ⊗ · · · ⊗ f ( s n ) i = F ( s )using the axiom h X ∗ i = h X i of half-balanced contraction. If now s = ( s , · · · , s n ) ∈ Ob g Dih S ,then F ( i rs ) = id h X i by the same argument as in Proposition 20, and F ( i ss ) = f ( h n ) = ( a f ( s n ) ⊗ · · · ⊗ a f ( s ) )(id f ( s ) ⊗···⊗ f ( s n − ) ⊗ β f ( s n − ) ,f ( s n ) )(id f ( s ) ⊗···⊗ f ( s n − ) ⊗ β f ( s n − ) ,f ( s n − ) ⊗ f ( s n ) ) · · · β f ( s ) ,f ( s ) ⊗···⊗ f ( s n ) = a f ( s ) ⊗···⊗ f ( s n ) . Hence F ( i ss ) = h a f ( s ) ⊗···⊗ f ( s n ) i = id h s i . So we have the desired factorization of F . (cid:3) We now construct the category
PaDih S as follows. We first define the category ^ PaDih S by Ob ^ PaDih S = ⊔ n ≥ P lT n × S n , ^ PaDih S (( t, s ) , ( t ′ , s ′ )) = g Dih S ( s, s ′ ). The group D acts on ^ PaDih S as follows. The action on objects is g · ( t, s ) = ( g · t, g · s ), where for t = ( T, { leavesof T } α → [ n ]), r · t = ( T, { leaves of T } α → [ n ] +1 mod n → [ n ]), s · t = ( T, { leaves of T } α → [ n ] x n +1 − x → [ n ]), and i g ( t,s ) = i gs ∈ Iso g Dih S ( s, g · s ) for ( t, s ) ∈ Ob ^ PaDih S . We then set PaDih S := ^ PaDih S /D . ALF-BALANCED CATEGORIES AND TEICHM¨ULLER GROUPOIDS IN GENUS ZERO 11
We have Ob
PaDih S = { (a planar 3-valent tree, a map { leaves } → S ) } / (mirror symmetry) = ⊔ n ≥ ( P lT n × S n ) /D n . We define a functor PaB hbalS → PaDih S by the condition that: (a)at the level of objects, it is given by the canonical map T n × S n → ( P lT n × S n ) /D n , (b) thediagram PaB hbalS → PaDih S ↓ ↓ B hbalS → Dih S commutes. One proves as above that this is a half-balancedcontraction. Using the arguments of the proofs of Propositions 20, 21 and 22, one proves: Proposition 23.
Let C be a half-balanced braided monoidal category, equipped with a map S f → Ob C such that f ( s ) ∗ = f ( s ) for any s ∈ S , and a balanced contraction C → O . Thenthere exists a functor
PaDih S → O , such that the diagram PaB S → PaDih S ↓ ↓ B hbalS → Dih S commutes. We then have natural diagrams B S → B balS → B hbalS ↓ ↓ Cyc S → Dih S and PaB S → PaB balS → PaB hbalS ↓ ↓
PaCyc S → PaDih S These diagrams fit in a bigger diagram, with a collection of functors from the left to the right-hand side diagram. 6.
Completions
Let G → S n be a group morphism. One can define the relative pro- l and relative prounipotentcompletions G l and G ( − ) of G → S n . They fit in exact sequences 1 → U l → G l → S n → → U ( − ) → G ( − ) → S n →
1, where U l and U ( − ) are pro- l and Q -prounipotent. We havea morphism G l → G ( Q l ) ([HM], Lemma A.7), fitting in a sequence of morphisms G → b G → G l → G ( Q l ), where b G is the profinite completion of G . Applying this to B n are any of thisquotients B n /Z n , Γ ,n considered above, we obtain for each of the categories C = ( Pa ) B ( h )( bal ) S ,( Pa ) Cyc S , ( Pa ) Dih S , completed categories b C , C l , C ( − ), and functors C → b C → C l → C ( Q l ).Let us say that a pro- l (resp., prounipotent) b.m.c. is a b.m.c. C , equipped with an assignmentOb C ∋ X
7→ U X ⊳ Aut C ( X ), such that U X is pro- l (resp., prounipotent) for any X , and forany X, Y ∈ Ob C and f ∈ Iso C ( X, Y ), f U X f − = U Y and im( P n → Aut C ( X ⊗ · · · ⊗ X n )) ⊂U X ⊗···⊗ X n (here P n = Ker( B n → S n ) is the pure braid group with n strands). Similarly, C iscalled profinite if Aut C ( X ) is profinite for any X ∈ Ob C .Then the completions \ ( Pa ) B S , ( Pa ) B S,l and ( Pa ) B S ( − ) are profinite, pro- l and prounipo-tent (strict) braided monoidal categories and are universal for such braided monoidal categories C , equipped with a map S → Ob C .7. Actions of the Grothendieck–Teichm¨uller group
Grothendieck-Teichm¨uller semigroups.
Recall that the Grothendieck–Teichm¨uller semi-group is defined ([Dr]) asGT = { ( λ, f ) ∈ (1 + 2 Z ) × F | f ( Y, X ) = f ( X, Y ) − ,f ( X , X ) X m f ( X , X ) X m f ( X , X ) X m = 1 , ∂ ( f ) ∂ ( f ) = ∂ ( f ) ∂ ( f ) ∂ ( f ) } , where F is the free group with two generators X, Y , ∂ , . . . , ∂ : F → P are simplicialmorphisms, X X X = 1, m = ( λ − /
2. It is a semigroup with ( λ, f )( λ ′ , f ′ ) = ( λ ′′ , f ′′ ),where λ ′′ = λλ ′ and f ′′ = θ ( λ ′ ,f ′ ) ( f ) f ′ , where θ ( λ ′ ,f ′ ) ∈ End( F ) is given by ( X, Y ) ( f ′ X λ ′ f ′− , Y λ ′ ). Then GT → End( F ) op , ( λ, f ) θ ( λ,f ) is a semigroup morphism. The profinite, pro- l and prounipotent analogues d GT, GT l and GT( − ) of GT are defined by replac-ing ( Z , F ) by ( b Z , b F ), ( Z l , ( F ) l ), and k ( k , F ( k )) where k is a Q -ring. We then havemorphisms of semigroups GT → d GT → GT l → GT( Q l ); the associated groups are denotedGT , d GT , GT l , GT( − ).7.2. Action on (half-)braided monoidal categories.
The semigroup GT acts on { braidedmonoidal categories } as follows: ( λ, f ) ∗ ( C , ⊗ , β XY , a XY Z ) = ( C , ⊗ , β ′ XY , a ′ XY Z ), where β ′ XY = β XY ( β Y X β XY ) m and a ′ XY Z = a XY Z f ( β Y X β XY ⊗ id Z , a − XY Z (id X ⊗ β ZY β Y Z ) a XY Z ) . In the same way, d GT acts on { braided monoidal categories C , such that Aut C ( X ) is finitefor any X ∈ Ob C} , GT l acts on { pro- l braided monoidal categories } and GT( k ) acts on { k -prounipotent braided monoidal categories } .We have natural functors { half-balanced braided monoidal categories } → { balanced braidedmonoidal categories } → { braided monoidal categories } . Proposition 24.
The action of GT on { braided monoidal categories } lifts to compatible actionson { (half-)balanced braided monoidal categories } . Similarly, the actions of d GT , . . . , GT( k ) lift tocompatible actions on { (half-)balanced finite braided monoidal categories } , ..., { (half-)balanced k -prounipotent braided monoidal categories } .Proof. This lift is given by ( λ, f ) ∗ ( C , ⊗ , β XY , a XY Z , θ X ) := ( C , ⊗ , β ′ XY , a ′ XY Z , θ ′ X ), where θ ′ X := θ λX and ( λ, f ) ∗ ( C , ⊗ , β XY , a XY Z , a X ) := ( C , ⊗ , β ′ XY , a ′ XY Z , a ′ X ), where a ′ X := ( a X ∗ a X ) m a X ,where m = ( λ − / (cid:3) Proposition 25.
Let C be a half-balanced category and let C h−i → O be a half-balanced contrac-tion. Then for any ( λ, f ) ∈ GT , the composite functor ( λ, f ) ∗ C ∼ → C h−i → O is a half-balancedcontraction on ( λ, f ) ∗ C . Here ( λ, f ) ∗ C ∼ → C is the identity functor (which is not tensor). Samestatements with C finite, ..., k -unipotent and GT replaced by d GT , ..., GT( k ) .Proof. Assume that ( C , β XY , a X ) is half-balanced; we set θ X := a X ∗ a X . Then ( C , β XY , θ X )is balanced and θ ′ X = θ λX . Then ( θ ′ Y ⊗ id X ) β ′ XY = ( θ Y ⊗ id X ) β XY ( θ − X ⊗ θ Y ) m θ mX ⊗ Y . Theidentities h θ X i = id h X i , h θ − X ⊗ θ Y i = id h X,Y i (see Lemma 15) and h ( θ Y ⊗ id X ) β XY i = id h X,Y i (as h−i is a half-balanced contraction) imply that h ( θ ′ Y ⊗ id X ) β ′ XY i = id h X,Y i , so h−i is abalanced contraction for ( λ, f ) ∗ C . Moreover, a ′ X = a X ( a X ∗ a X ) m = a X θ mX , so h θ X i = id h X i implies h a ′ X i = h a X i = id h X i . (cid:3) Action on PaDih S . For ( λ, f ) ∈ GT, let i ( λ,f ) be the endomorphism of PaB ( h ) balS definedas the composite functor PaB ( h ) balS α ( λ,f ) → ( λ, f ) ∗ PaB ( h ) balS ∼ → PaB ( h ) balS , where the first functoris the unique (half-)balanced monoidal functor which is the identity on objects, and the secondfunctor is the identity functor (which is not monoidal). As in [E], Proposition 80, one showsthat ( λ, f ) i ( λ,f ) is a morphism GT → End(
PaB ( h ) balS ) op . One similarly defines morphisms d GT → End( [ PaB ( h ) balS ) op , ..., GT( k ) → End(
PaB ( h ) balS, k ) op .For ( λ, f ) ∈ GT, we define an endofunctor j ( λ,f ) of PaDih S as follows: according to Propo-sition 25, the composite functor ( λ, f ) ∗ PaB hbalS ∼ → PaB hbalS h−i → PaDih S is a half-balancedcontraction. By universality of the contraction PaB hbalS h−i → PaDih S , there exists a unique ALF-BALANCED CATEGORIES AND TEICHM¨ULLER GROUPOIDS IN GENUS ZERO 13 endofunctor j ( λ,f ) of PaDih S , such that the following diagram commutes PaB hbalS α ( λ,f ) / / h−i (cid:15) (cid:15) i ( λ,f ) , , ( λ, f ) ∗ PaB hbalS ∼ / / (cid:15) (cid:15) PaB hbalS h−i w w nnnnnnnnnnnn PaDih
S j ( λ,f ) / / PaDih S Proposition 26.
The map ( λ, f ) j ( λ,f ) defines a morphism GT → End(
PaDih S ) op ; onesimilarly defines morphisms d GT → End( \ PaDih S ) op , etc.Proof. We have a commutative diagram(1)
PaB hbalS α ( λ ′ ,f ′ ) / / h−i (cid:15) (cid:15) ( λ ′ , f ′ ) ∗ PaB hbalS ∼ (cid:15) (cid:15) PaB hbalS h−i (cid:15) (cid:15)
PaDih
S j ( λ ′ ,f ′ ) / / PaDih S which gives rise to ( λ, f ) ∗ PaB hbalS ( λ,f ) ∗ α ( λ ′ ,f ′ ) / / h−i (cid:15) (cid:15) ( λ, f )( λ ′ , f ′ ) ∗ PaB hbalS ∼ (cid:15) (cid:15) PaB hbalS h−i (cid:15) (cid:15)
PaDih
S j ( λ ′ ,f ′ ) / / PaDih S Composing it with the analogue of (1) with ( λ ′ , f ′ ) replaced by ( λ, f ), we get a commutativediagram PaB hbalS (( λ,f ) ∗ α ( λ ′ ,f ′ ) ) ◦ α ( λ,f ) / / h−i (cid:15) (cid:15) ( λ, f )( λ ′ , f ′ ) ∗ PaB hbalS ∼ (cid:15) (cid:15) PaB hbalS h−i (cid:15) (cid:15)
PaDih
S j ( λ ′ ,f ′ ) ◦ j ( λ,f ) / / PaDih S On the other hand, both (( λ, f ) ∗ α ( λ ′ ,f ′ ) ) α ( λ,f ) and α ( λ,f )( λ ′ ,f ′ ) are tensor functors PaB hbalS → ( λ, f )( λ ′ , f ′ ) ∗ PaB hbalS of half-balanced braided monoidal categories, inducing the identity atthe level of objects, and by the uniqueness of such functors, they coincide. The above diagram may therefore be rewritten as
PaB hbalS α ( λ,f )( λ ′ ,f ′ ) / / h−i (cid:15) (cid:15) ( λ, f )( λ ′ , f ′ ) ∗ PaB hbalS ∼ (cid:15) (cid:15) PaB hbalS h−i (cid:15) (cid:15)
PaDih
S j ( λ ′ ,f ′ ) ◦ j ( λ,f ) / / PaDih S which may be viewed as a functor between half-balanced categories with a contraction.On the other hand, another such a functor is PaB hbalS α ( λ,f )( λ ′ ,f ′ ) / / h−i (cid:15) (cid:15) ( λ, f )( λ ′ , f ′ ) ∗ PaB hbalS ∼ (cid:15) (cid:15) PaB hbalS h−i (cid:15) (cid:15)
PaDih
S j ( λ,f )( λ ′ ,f ′ ) / / PaDih S By the universality of the contraction
PaB hbalS h−i → PaDih S , we then have j ( λ,f )( λ ′ ,f ′ ) = j ( λ ′ ,f ′ ) j ( λ,f ) . (cid:3) Action on Teichm¨uller groupoids and proof of Theorem 1. T ,S may be viewed asthe full subcategory of PaDih S whose objects are the classes modulo D of P lT | S | × Bij( | S | , S ).The action of GT then restricts to T ,S , and similarly in the completed cases. In the profinitecase, one checks that that resulting action coincides with that defined in in [Sch]. This provesTheorem 17.5. Proof of Theorem 2.
We define T ,n ( k ) by Ob T ,n ( k ) = Ob T ,n and for b, c ∈ Ob T ,n ,Hom T ,n ( k ) ( b, c ) = Aut T ,n ( b )( k ) × Aut T ,n ( b ) Hom T ,n ( b, c ), where for G a finitely generatedgroup, G ( k ) is its prounipotent completion.If π is a finitely generated group, we define the group scheme Aut π ( − ) by Aut π ( k ) :=Aut((Lie π ) k ), where for Lie π is the Lie algebra of the prounipotent completion of π , g k =lim ← ( g / g n ) ⊗ k , and g = g , g n +1 = [ g , g n ]. We then have a morphism Aut π ( k ) → Aut( π ( k )), θ θ ∗ . Aut( π l , π ( Q l )) is then defined as { ( θ, θ l ) ∈ Aut π ( Q l ) × Aut( π l ) | θ ∗ i = iθ l } , where i isthe morphism π l → π ( Q l ).If G is a groupoid such that Iso G ( b, c ) = ∅ for any b, c ∈ Ob G , then the choice of b ∈ Ob G gives rise to an isomorphism Aut G ≃ π Ob G −{ b } ⋊ Aut π , where π = Aut G ( b ); we then definethe group scheme Aut G ( − ) by Aut G ( k ) := π ( k ) Ob G −{ b } ⋊ Aut π ( k ). We define as aboveAut( G l , G ( Q l )) and the morphisms Aut( G l ) ← Aut( G l , G ( Q l )) → Aut G ( Q l ).We have morphisms G Q → GT l → GT( Q l ) and a functor PaB
S,l → PaB S ( Q l ). Theorem 2follows from the fact that this functor is compatible with the actions of GT l , GT( Q l ) on PaB
S,l , PaB S ( Q l ). ALF-BALANCED CATEGORIES AND TEICHM¨ULLER GROUPOIDS IN GENUS ZERO 15 Graded aspects
Let t n be the graded Lie algebra with generators t ij , i = j ∈ [ n ] and relations t ji = t ij ,[ t ij , t ik + t jk ] = 0, [ t ij , t kl ] = 0 for i, j, k, l distinct. Let p n be the quotient of t n by the relations P j | j = i t ij = 0, for any i ∈ [ n ]. Equivalently, p n is presented by generators t ij are relations t ji = t ij , P j | j = i t ij = 0 for any i , and [ t ij , t kl ] = 0 for i, j, k, l distinct.Let k be a Q -ring, then the set M ( k ) of Drinfeld associators defined over k is the set of pairs( µ, Φ) ∈ k × exp(ˆ f k ), satisfying the duality, hexagon and pentagon conditions (see [Dr]). Thedata of t ∈ T n and ( µ, Φ) ∈ M ( k ) gives rise to a morphism B n i t, Φ → exp(ˆ t k n ) ⋊ S n , which extendsto an isomorphism B n ( k ) ∼ → exp(ˆ t k n ) ⋊ S n (see e.g. [AET]) if µ ∈ k × . Proposition 27.
There exists a unique morphism Γ ,n → exp(ˆ p k n ) ⋊ S n , such that the diagram B n i t, Φ → exp(ˆ t k n ) ⋊ S n ↓ ↓ Γ ,n → exp(ˆ p k n ) ⋊ S n commutes. It gives rise to an isomorphism Γ ,n ( k ) ∼ → exp(ˆ p k n ) ⋊ S n .Proof. One checks that i t, Φ takes z n to exp( µ P ≤ i PaDih grS similarly to PaDih S , i.e., as the quotient by D of an inter-mediate category ^ PaDih grS obtained from ^ PaDih S by replacing Γ ,n by exp(ˆ p k n ) ⋊ S n , and themorphism D → D n → Γ ,n by D → D n → S n → exp(ˆ p k n ) ⋊ S n .If ( µ, Φ) ∈ M ( k ), recall that a braided monoidal category PaCD Φ S may be defined as fol-lows: Ob PaCD Φ S = Ob PaB S ; Hom PaCD Φ S (( s, t ) , ( s ′ , t ′ )) is empty if | s | 6 = | s ′ | , and is equalto exp(ˆ t k n ) ⋊ { f ∈ S n | s ′ f = s } ; the composition is induced by the product in exp(ˆ t k n ) ⋊ S n ;and the tensor product is obtained by restriction from the group morphism (exp(ˆ t k n ) ⋊ S n ) × (exp(ˆ t k n ′ ) ⋊ S n ′ ) → exp(ˆ t k n + n ′ ) ⋊ S n + n ′ , induced by the Lie algebra morphism ˆ t k n × ˆ t k n ′ → ˆ t k n + n ′ ,( t ij , t ij , (0 , t ij ) t n + i,n + j , and the group morphism S n × S n ′ → S n + n ′ , ( σ, σ ′ ) σ ∗ σ ′ ,such that ( σ ∗ σ ′ )( i ) = σ ( i ) for i ∈ [ n ], and ( σ ∗ σ ′ )( n + i ) = n + σ ′ ( i ) for i ∈ [ n ′ ]. The braidingconstraint is defined by β XY = ( e µt / ) [ n ] ,n +[ n ′ ] s n,n ′ and the associativity constraint is definedby a XY Z = (Φ( t , t )) [ n ] ,n +[ n ′ ] ,n + n ′ +[ n ′′ ] for | X | = n , | Y | = n ′ , | Z | = n ′′ , s n,n ′ ∈ S n + n ′ isdefined by s n,n ′ ( i ) = n ′ + i for i ∈ [ n ] and s n,n ′ ( n + i ) = i for i ∈ [ n ′ ], and for I , . . . , I n ⊂ [ m ]disjoint subsets, the morphism t n → t m , x x I ,...,I n is defined by t ij P α ∈ I i ,β ∈ I j t αβ . Then If g is a graded Lie algebra, then ˆ g k is the degree completion of g ⊗ k . PaCD Φ S is a braided monoidal category; it follows that there is a unique monoidal functor j Φ : PaB S → PaCD Φ S , which induces the identity on objects.We then define a functor PaCD Φ S → PaDih grS as the composite functor PaCD Φ S → ^ PaDih grS → PaDih grS , where the first functor is induced by the projection morphisms t n → p n and the sec-ond functor is the quotient functor ^ PaDih grS → ^ PaDih grS /D ≃ PaDih grS . Proposition 28. The functor PaCD Φ S → PaDih grS is a half-balanced contraction.Proof. We first show: Lemma 29. Let X ∈ Ob PaB {•} be of degree n , then im( h n ∈ PaB {•} ( X, X ∗ ) → PaCD Φ {•} ( X, X ∗ )) = exp( µ X ≤ i It suffices to prove this for a particular X ∈ Ob PaB {•} of degree n , say X = • ( • ( · · · ( •• ))). Indeed, if we denote by h Xn ∈ PaB {•} ( X, X ∗ ) the element corresponding to h n and if we have im( h X n ) = exp( µ P ≤ i 2. Assume itat order n − 1. Then h X n n ∈ PaB {•} ( X n , X ∗ n ) may be viewed as the composite morphism X n = • ⊗ X n − • ⊗ h n − → • ⊗ X ∗ n − β • ,X ∗ n − → X ∗ n − ⊗ • = X ∗ n , whose image in PaCD Φ {•} is s exp( µ P ni =2 t i ) exp( µ P ≤ i Lemma 30. Let X, Y ∈ Ob PaB {•} be of degrees n, m , then im(( θ Y ⊗ id X ) β XY ∈ PaB {•} ( X ⊗ Y, Y ⊗ X ) → PaCD Φ {•} ( X ⊗ Y, Y ⊗ X ))= (cid:0) ··· n n +1 ··· n + mm +1 ··· m + n ··· m (cid:1) exp( µ X j ∈ n +[ m ] X α ∈ [ n + m ] −{ j } t αj ) . Proof. The image of β XY is (cid:0) ··· n n +1 ··· n + mm +1 ··· m + n ··· m (cid:1) exp( µ P i ∈ [ n ] ,j ∈ n +[ m ] t ij ), while theimage of θ Y ⊗ id X is exp( µ P j PaCD Φ S ) have degrees n, m , then the image of ( θ Y ⊗ id X ) β XY ∈ PaCD Φ S ( X ⊗ Y, Y ⊗ X ) in ^ PaDih grS ( X ⊗ Y, Y ⊗ X ) = exp(ˆ p k n + m ) ⋊ S n + m is c m , where c = (cid:0) ··· n + m ··· (cid:1) as im( P α ∈ [ n + m ] −{ j } t αj ∈ t n + m → p n + m ) = 0 for any j . It follows that thisimage coincides with i r m X ⊗ Y , whose image in Aut PaDih grS ( h X ⊗ Y i ) is id h X ⊗ Y i . It follows that h ( θ Y ⊗ id X ) β XY i = id h X ⊗ Y i ∈ Aut PaDih grS ( h X ⊗ Y i ). All this implies that PaCD Φ S → PaDih grS satisfies the half-balanced contraction conditions. (cid:3) Proposition 28 immediately implies: ALF-BALANCED CATEGORIES AND TEICHM¨ULLER GROUPOIDS IN GENUS ZERO 17 Corollary 31. There exists a unique functor PaDih S k Φ → PaDih grS , with is the identity onobjects and such that the diagram PaB S j Φ / / h−i (cid:15) (cid:15) PaCD Φ S h−i (cid:15) (cid:15) PaDih S k Φ / / PaDih grS commutes. Recall that the graded Grothendieck-Teichm¨uller group GRT( k ) is defined as GRT( k ) =GRT ( k ) ⋊ k × , where GRT ( k ) is the set of all g ∈ exp(ˆ f k ) ⊂ exp(ˆ t k ) ( f ⊂ t being the Liesubalgebra generated by t , t ), such that g , , = g − , t + Ad( g , , ) − ( t ) + Ad( g , , ) − ( t ) = t + t + t ,g , , g , , g , , = g , , g , , , equipped with the group law ( g ∗ g )( A, B ) := g (Ad( g ( A, B ))( A ) , B ) g ( A, B ), on which k × acts by ( c · g )( A, B ) := g ( c − A, c − B ).We now construct an action of this group on PaDih grS . For this, we recall from [E] thenotion of infinitesimally braided monoidal category (i.b.m.c.). Definition 32. An i.b.m.c. is a braided monoidal category ( C , ⊗ , c XY , a XY Z ) , which is(1) symmetric, i.e., such that c Y X c XY = id X ⊗ Y for any X, Y ∈ Ob C ,(2) prounipotent (see Section 6), i.e., equipped with an assignment Ob C ∋ X 7→ U X ⊳ Aut C ( X ) , such that f U X f − = U Y for f ∈ Iso C ( X, Y ) ,(3) equipped with a functorial assignment (Ob C ) ∋ ( X, Y ) t XY ∈ Lie U X ⊗ Y , such that t Y X = c Y X t Y X c XY and t X ⊗ Y,Z = a XY Z (id X ⊗ t Y Z ) a − XY Z + ( c Y X ⊗ id Z ) a Y XZ (id Y ⊗ t XZ ) a − Y XZ ( c Y X ⊗ id Z ) − . According to [Dr], GRT( k ) acts on { i.m.b. categories } from the right as follows: g ∈ GRT ( k ) ⊂ exp(ˆ f k ) acts by ( C , ⊗ , c XY , a XY Z , t XY ) · g := ( C , ⊗ , c XY , a ′ XY Z , t XY ), where a ′ XY Z := g ( t XY ⊗ id Z , a XY Z (id X ⊗ t Y Z ) a − XY Z ) a XY Z and c ∈ k × acts by ( C , . . . ) · g := ( C , ⊗ , c XY , a XY Z , ct XY ).Moreover, PaCD S , equipped with c XY := s | X | , | Y | , a XY Z := id | X | + | Y | + | Z | and t XY := t [ | X | ] , | X | +[ | Y | ]12 is universal among i.b.m.cs C , equipped with a map S → Ob C . We derive from this, as in [E],Proposition 80, a morphism GRT( k ) → Aut( PaCD S ).We now introduce the notion of a balanced i.b.m.c. Definition 33. A balanced structure on the i.b.m.c. C is a functorial assignment Ob C ∋ X t X ∈ Lie U X , such that for any X, Y ∈ Ob C , t X ⊗ Y − t X ⊗ id Y − id X ⊗ t Y = t XY . Definition 34. A contraction on the small balanced i.b.m.c. C is a functor C h−i → O , such that forany X, Y, Z ∈ Ob C , h X ⊗ Y i = h Y ⊗ X i (=: h X, Y i ) , h ( X ⊗ Y ) ⊗ Z i = h X ⊗ ( Y ⊗ Z ) i (=: h X, Y, Z i ) , h c XY i = id h X,Y i , h a XY Z i = id h X,Y,Z i , and h t XY + 2 id X ⊗ t Y i = 0 . Remark 35. We derive from the latter condition that h t X i = 0 for any X ∈ Ob C . Indeed, itgives by symmetrization h t X ⊗ Y i = 0, and therefore h t X i = 0 by taking Y = . By antisym-metrization, this condition also implies h t X ⊗ id Y − id X ⊗ t Y i = 0.We now construct a universal contraction on balanced i.b.m. categories. Proposition 36. The i.b.m.c. PaCD S is equipped with a balanced structure given by t X = P ≤ i For | X | = n , | Y | = m , t XY + 2 id X ⊗ t Y = P j ∈ n +[ m ] P α ∈ [ n + m ] −{ j } t jα , so h t XY +2 id X ⊗ t Y i = 0 as P α ∈ [ n + m ] −{ j } t jα = 0 in p n + m for any j ∈ n + [ m ]. (cid:3) Proposition 37. Let C be a balanced i.b.m.c. and let C h−i → O be a contraction. Let S f → Ob C be a map such that for any s ∈ S , t f ( s ) = 0 . Then we have a commutative diagram PaCD S → C↓ ↓ PaDih grS → O Proof. As C is an i.b.m.c., there exists a unique functor PaCD S → C of i.b.m. categories,extending f . As t f ( s ) = 0 for s ∈ S , it is compatible with the balanced structures. Theconstruction of the commutative diagram is similar to the proof of Propositions 21, 22. (cid:3) Proposition 38. 1) The action of GRT( k ) on { i.b.m. categories } lifts to { balanced i.b.m. categories } as follows: for ( C , ⊗ , c XY , a XY Z , t XY , t X ) a balanced i.b.m.c., and g ∈ GRT( k ) , C · g =( C , . . . , t ′ X ) , where t ′ X = ct X and c = im( g ∈ GRT( k ) → k × ) .2) If C F → O is a contraction of the balanced i.b.m.c. C , then C · g ∼ → C F → O is a contractionof the balanced i.b.m.c. C (where C · g ∼ → C is the identity of the underlying categories). The proof is immediate.We now construct an action of GRT( k ) on PaCD S → PaDih grS . A morphism GRT( k ) → Aut( PaCD S ), g a g is defined by a g : PaCD S → PaCD S ∗ g ∼ → PaCD S , where the firstmorphism is the unique functor of i.m.b. categories, inducing the identity on objects, and thesecond morphism is the identification of the underlying categories.We define a morphism GRT( k ) → Aut( PaDih grS ), g j g by the condition that the diagram PaCD S → PaCD S ∗ g h−i↓ ↓ h−i PaDih grS j g → PaDih grS is a functor of balanced i.b.m. categories with contractions. Wethen have a commutative diagram PaCD S a g → PaCD S h−i↓ ↓ h−i PaDih grS j g → PaDih grS Let now ( µ, Φ) ∈ M ( k ), where µ ∈ k × , be an associator. It gives rise to an isomorphism i Φ : GT( k ) → GRT( k ), defined by the condition that g ∗ Φ = Φ ∗ i Φ ( g ) for any g ∈ GT( k ). Inthe diagram PaB S j Φ / / h−i + + g (cid:15) (cid:15) PaCD Si Φ ( g ) (cid:15) (cid:15) h−i + + PaDih Sg (cid:15) (cid:15) k Φ / / PaDih grSi Φ ( g ) (cid:15) (cid:15) PaB S j Φ / / h−i PaCD S h−i PaDih S k Φ / / PaDih grS all the squares except perhaps the rightmost one commute. But this last square has to commuteby the uniqueness of the morphism PaDih S → O in Proposition 23 (the existence in thisproposition implies uniqueness by abstract nonsense).All this implies that the isomorphism PaDih S k Φ → PaDih grS gives rise to a commutativediagram GT( k ) → Aut PaDih Si Φ ↓ ↓ GRT( k ) → Aut PaDih grS The isomorphism PaDih S k Φ → PaDih grS and the actions of G(R)T( k ) on these categoriesinduce the identity at the level of objects. We then define T gr ,n to be the full subcategory of ALF-BALANCED CATEGORIES AND TEICHM¨ULLER GROUPOIDS IN GENUS ZERO 19 PaDih gr [ n ] , whose set of objects is ( P lT n × Bij([ n ] , [ n ])) /D n , and obtain this way an isomorphism T ,n ( k ) → T gr ,n inducing a commutative diagram GT( k ) → Aut T ,n ( k ) ↓ ↓ GRT( k ) → Aut T gr ,n ( k )This proves Theorem 3. Remark 39. T gr ,n could alternatively be defined as π ∗ dih C Γ ,G,S , where Γ = D n , G = exp(ˆ p k n ) ⋊ S n , and S = [ n ] (see Section 1). References [AET] A. Alekseev, B. Enriquez, C. Torossian, Drinfeld associators, braid groups and explicit solutions of theKashiwara–Vergne equations, arXiv:0903.4067[Ba] D. Bar-Natan, On associators and the Grothendieck-Teichm¨uller group. I. Selecta Math. (N.S.) 4 (1998),no. 2, 183–212.[Bi] J. Birman, Braids, links, and mapping class groups, Ann. of Math. Studies, vol. 82, Princeton Univ. Press,1975.[Dr] V. 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