Little disks and braids The Swiss-Cheese operad Chord diagrams Swiss-Cheese operad and Drinfeld center Najib Idrissi June 3rd, 2016 @ ETH Zürich
Little disks and braids The Swiss-Cheese operad Chord diagrams Outline 1 Background: Little disks and braids 2 The Swiss-Cheese operad 3 Rational model: Chords diagrams and Drinfeld associators
Little disks and braids The Swiss-Cheese operad Chord diagrams Outline 1 Background: Little disks and braids 2 The Swiss-Cheese operad 3 Rational model: Chords diagrams and Drinfeld associators
Little disks and braids The Swiss-Cheese operad Chord diagrams Little disks operad The topological operad D n [Boardman–Vogt, May] of little n -disks governs homotopy associative and commutative algebras: 1 1 1 2 2 3 2 ◦ 2 = 2 3 1 2 1 2 D 2 (3) 1 D 2 (2) D 2 (2)
Little disks and braids The Swiss-Cheese operad Chord diagrams Braid groups Recall: pure braid group P r Proposition D 2 ( r ) ≃ Conf r ( R 2 ) ≃ K ( P r , 1)
Little disks and braids The Swiss-Cheese operad Chord diagrams Braid groups Recall: pure braid group P r Proposition D 2 ( r ) ≃ Conf r ( R 2 ) ≃ K ( P r , 1) = ⇒ D 2 ≃ B( π D 2 )
Little disks and braids The Swiss-Cheese operad Chord diagrams Braid group oid s 1 1 3 3 2 2 “Extension” of P r : colored braid groupoid CoB ( r ) End CoB ( r ) ( σ ) ∼ ob CoB ( r ) = Σ r , = P r
Little disks and braids The Swiss-Cheese operad Chord diagrams Cabling “Cabling”: insertion of a braid inside a strand 3 1 2 1 2 4 1 2 3 ◦ 2 = =
Little disks and braids The Swiss-Cheese operad Chord diagrams Cabling “Cabling”: insertion of a braid inside a strand 3 1 2 1 2 4 1 2 3 ◦ 2 = = = ⇒ { CoB ( r ) } r ≥ 1 is a symmetric operad in groupoids: ◦ i : CoB ( k ) × CoB ( l ) → CoB ( k + l − 1) , 1 ≤ i ≤ k
Little disks and braids The Swiss-Cheese operad Chord diagrams Little disks and braids � CoB ( r ) ∼ = subgroupoid of π D 2 ( r )
Little disks and braids The Swiss-Cheese operad Chord diagrams Little disks and braids � CoB ( r ) ∼ = subgroupoid of π D 2 ( r ) Problem : inclusion not compatible with operad structure
Little disks and braids The Swiss-Cheese operad Chord diagrams Little disks and braids (2) Solution: parenthesized braids PaB 1 2 3 �
Little disks and braids The Swiss-Cheese operad Chord diagrams Little disks and braids (2) Solution: parenthesized braids PaB 1 2 3 � Theorem (Fresse; see also results of Fiedorowicz, Tamarkin...) Operads π D 2 and CoB are weakly equivalent. ∼ − PaB ∼ π D 2 ← − → CoB is a zigzag of weak equivalences of operads .
Little disks and braids The Swiss-Cheese operad Chord diagrams Algebras over categorical operads P ∈ CatOp = ⇒ a P -algebra is given by: • A category C; x : C × r → C; • For every object x ∈ ob P ( r ), a functor ¯ • For every morphism f ∈ Hom P ( r ) ( x , y ), a natural transformation ¯ x C × r ¯ C f ¯ y • + compatibility with the action of symmetric groups and operadic composition.
Little disks and braids The Swiss-Cheese operad Chord diagrams Algebras over CoB For P = CoB , algebras are given by: • A category C; • σ ∈ ob CoB ( r ) = Σ r � ⊗ σ : C × r → C s.t. ⊗ id 1 = id C ;
Little disks and braids The Swiss-Cheese operad Chord diagrams Algebras over CoB For P = CoB , algebras are given by: • A category C; • σ ∈ ob CoB ( r ) = Σ r � ⊗ σ : C × r → C s.t. ⊗ id 1 = id C ; • ⊗ σ ( X 1 , . . . , X n ) = ⊗ id r ( X σ (1) , . . . , X σ ( n ) );
Little disks and braids The Swiss-Cheese operad Chord diagrams Algebras over CoB For P = CoB , algebras are given by: • A category C; • σ ∈ ob CoB ( r ) = Σ r � ⊗ σ : C × r → C s.t. ⊗ id 1 = id C ; • ⊗ σ ( X 1 , . . . , X n ) = ⊗ id r ( X σ (1) , . . . , X σ ( n ) ); • ⊗ id 2 ( ⊗ id 2 ( X , Y ) , Z ) = ⊗ id 3 ( X , Y , Z ) = ⊗ id 2 ( X , ⊗ id 2 ( Y , Z ))...
Little disks and braids The Swiss-Cheese operad Chord diagrams Algebras over CoB For P = CoB , algebras are given by: • A category C; • σ ∈ ob CoB ( r ) = Σ r � ⊗ σ : C × r → C s.t. ⊗ id 1 = id C ; • ⊗ σ ( X 1 , . . . , X n ) = ⊗ id r ( X σ (1) , . . . , X σ ( n ) ); • ⊗ id 2 ( ⊗ id 2 ( X , Y ) , Z ) = ⊗ id 3 ( X , Y , Z ) = ⊗ id 2 ( X , ⊗ id 2 ( Y , Z ))... • β ∈ Hom CoB ( r ) ( σ, σ ′ ) colored braid � natural transformation β ∗ : ⊗ σ → ⊗ σ ′ . For example: 1 2 � τ X , Y : X ⊗ Y → Y ⊗ X
Little disks and braids The Swiss-Cheese operad Chord diagrams Algebras over CoB For P = CoB , algebras are given by: • A category C; • σ ∈ ob CoB ( r ) = Σ r � ⊗ σ : C × r → C s.t. ⊗ id 1 = id C ; • ⊗ σ ( X 1 , . . . , X n ) = ⊗ id r ( X σ (1) , . . . , X σ ( n ) ); • ⊗ id 2 ( ⊗ id 2 ( X , Y ) , Z ) = ⊗ id 3 ( X , Y , Z ) = ⊗ id 2 ( X , ⊗ id 2 ( Y , Z ))... • β ∈ Hom CoB ( r ) ( σ, σ ′ ) colored braid � natural transformation β ∗ : ⊗ σ → ⊗ σ ′ . For example: 1 2 � τ X , Y : X ⊗ Y → Y ⊗ X Theorem (MacLane, Joyal–Street) An algebra over CoB is a braided monoidal category (strict, no unit).
Little disks and braids The Swiss-Cheese operad Chord diagrams Remarks Extension of the theorem for parenthesized braids: Theorem An algebra over PaB is a braided monoidal category (no unit). Unital versions CoB + and PaB + : Theorem An algebra over CoB + (resp. PaB + ) is a strict (resp. non-strict) braided monoidal category with a strict (in both cases) unit.
Little disks and braids The Swiss-Cheese operad Chord diagrams Outline 1 Background: Little disks and braids 2 The Swiss-Cheese operad 3 Rational model: Chords diagrams and Drinfeld associators
Little disks and braids The Swiss-Cheese operad Chord diagrams Definition of the Swiss-Cheese operad The Swiss-Cheese operad SC [Voronov, 1999] governs a D 2 -algebra acting on a D 1 -algebra. It’s a colored operad, with two colors c (“closed” ↔ D 2 ) and o (“open” ↔ D 1 ).
Little disks and braids The Swiss-Cheese operad Chord diagrams Definition of the Swiss-Cheese operad The Swiss-Cheese operad SC [Voronov, 1999] governs a D 2 -algebra acting on a D 1 -algebra. It’s a colored operad, with two colors c (“closed” ↔ D 2 ) and o (“open” ↔ D 1 ). 1 1 1 2 2 ◦ c = 1 1 2 1 2 SC c (0 , 2) = D 2 (2) SC o (2 , 1) SC o (2 , 2)
Little disks and braids The Swiss-Cheese operad Chord diagrams Definition of the Swiss-Cheese operad The Swiss-Cheese operad SC [Voronov, 1999] governs a D 2 -algebra acting on a D 1 -algebra. It’s a colored operad, with two colors c (“closed” ↔ D 2 ) and o (“open” ↔ D 1 ). 1 1 1 2 2 ◦ c = 1 1 2 1 2 SC c (0 , 2) = D 2 (2) SC o (2 , 1) SC o (2 , 2) 1 1 1 ◦ o = 1 1 1 2 2 SC o (2 , 1) SC o (1 , 1) SC o (1 , 2)
Little disks and braids The Swiss-Cheese operad Chord diagrams The operad CoPB Idea Extend CoB to build a colored operad weakly equivalent to π SC . 1 3 1 2 2 � CoPB (2 , 3)
Little disks and braids The Swiss-Cheese operad Chord diagrams The operad CoPB Idea Extend CoB to build a colored operad weakly equivalent to π SC . 1 3 1 2 2 � CoPB (2 , 3) Theorem (I.) ∼ − PaPB ∼ π SC ← − → CoPB .
Little disks and braids The Swiss-Cheese operad Chord diagrams Braidings and semi-braidings In D 2 / CoB : braiding = homotopy commutativity 1 2
Little disks and braids The Swiss-Cheese operad Chord diagrams Braidings and semi-braidings In D 2 / CoB : braiding = homotopy commutativity 1 2 In SC / CoPB : half-braiding = “central” morphism 1 1
Little disks and braids The Swiss-Cheese operad Chord diagrams Drinfeld center C: monoidal category � ΣC bicategory with one object � Drinfeld center Z (C) := End(id ΣC ) ∼ = • objects: ( X , Φ) with X ∈ C and Φ : ( X ⊗ − ) − → ( − ⊗ X ) (“half-braiding”) ; • {morphisms ( X , Φ) → ( Y , Ψ)} = {morphisms X → Y compatible with Φ and Ψ}. Theorem (Drinfeld, Joyal–Street 1991, Majid 1991) Z (C) is a braided monoidal category with: � X ⊗ Y , (Ψ ⊗ 1) ◦ (1 ⊗ Φ) � , ( X , Φ) ⊗ ( Y , Ψ) = τ ( X , Φ) , ( Y , Ψ) = Φ Y .
Little disks and braids The Swiss-Cheese operad Chord diagrams Voronov’s theorem Recall: H ∗ ( D 1 ) = Ass , H ∗ ( D 2 ) = Ger Theorem (Voronov, Hoefel) An algebra over H ∗ ( SC ) is given by: • An associative algebra A ; • A Gerstenhaber algebra B ; • A central morphism of commutative algebras B → Z ( A ). (Voronov’s original version: B ⊗ A → A instead B → A )
Little disks and braids The Swiss-Cheese operad Chord diagrams Algebras over CoPB Theorem (I.) An algebra over CoPB is given by: • A (strict non-unital) monoidal category N ; • A (strict non-unital) braided monoidal category M ; • A (strict) braided monoidal functor F : M → Z (N). → categorical version of Voronov’s theorem
Little disks and braids The Swiss-Cheese operad Chord diagrams Algebras over CoPB Theorem (I.) An algebra over CoPB is given by: • A (strict non-unital) monoidal category N ; • A (strict non-unital) braided monoidal category M ; • A (strict) braided monoidal functor F : M → Z (N). → categorical version of Voronov’s theorem Like CoB : non-strict and/or unitary versions of the theorem.
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