Homotopy Automorphisms of Operads & Grothendieck-Teichmüller Groups

Transcription

1 Homotopy Automorphisms of Operads & Grothendieck-Teichmüller Groups Benoit Fresse Université Lille 1 GDO, Isaac Newton Institute March 5, 2013

2 Reminder: The notion of an E n -operad refers to a class of operads defined by a reference model, the operad of little n-discs D n. Goal: The Grothendieck-Teichmüller group is the group of homotopy automorphisms of E 2 -operads over the rationals. The torsor of Drinfeld s associators is the set of homotopy classes of rational formality quasi-isomorphisms of E 2 -operads.

3 Plan 1. The fundamental groupoid operad of little 2-discs and braided monoidal structures 2. Homotopy automorphisms 3. The topological interpretation of the Grothendieck-Teichmüller group

4 1. The fundamental groupoid operad of little 2-discs and braided monoidal structures The homotopy equivalence D 2 (r) F( D 2, r) implies that the space D 2 (r) is an Eilenberg-MacLane space K(P r, 1), where P r = π 1 F( D 2, r) is the pure braid group on r strands. The idea is to use the fundamental groupoids π D 2 (r) in order to get a combinatorial model of the operad D 2.

5 Construction: The fundamental groupoid π D 2 (r) has Ob π D 2 (r) = D 2 (r) as object set. The morphisms of Mor π D2 (r)(a, b) are homotopy classes of paths γ : [0, 1] D 2 (r) going from γ(0) = a to γ(1) = b. The map Mor π D2 (r)(a, b) disc centers Mor π F( D 2,r)(a, b) identifies the morphism sets of this groupoid with cosets of the pure braid group P r inside the braid group B r. Thus, a morphism in this groupoid can be represented by a picture of the form: γ = 1 2 = a Mor π D 2 (2) 2 1 = b

6 The structure of the fundamental groupoid operad: The groupoids π D 2 (r) inherit a symmetric structure, as well as operadic composition products i : π D 2 (m) π D 2 (n) π D 2 (m + n 1), and hence form an operad in the category of groupoids. In the braid picture, the operadic composition products can be depicted as cabling operations: = Mor π D 2 (2) Mor π D 2 (2) Mor π D 2 (3)

7 Ideas: There is no need to consider the whole D 2 (r) as object set. The groupoids of parenthesized braids PaB(r) are full subgroupoids of the fundamental groupoid π D 2 (r) defined by appropriate subsets of little 2-discs configurations Ω(r) D 2 (r) as object sets. These object sets Ω(r) are preserved by the operadic composition structure of little 2-discs so that the collection of groupoids PaB(r), r N, forms a suboperad of π D 2.

8 The sets Ω(r), r = 2, 3, 4,..., prescribing the origin and end-points of paths in PaB(r), consist of little 2-disc configurations of the following form: i j i j k i j k i j k l (the indices i, j,... run over all permutations of 1, 2,... ). These configurations represents the iterated operadic composites of the following element µ = 1 2 D 2 (2) and Ω is formally defined as the suboperad of D 2 generated by this element. Proposition: This operad Ω is identified with the Magma operad (the free operad on one generator).

9 Fundamental morphisms of PaB include the associator = µ 1 µ α = Mor PaB(3) = µ 2 µ and the braiding 1 2 = µ τ = Mor PaB(2) 2 1 = tµ where t = (12) Σ 2.

10 In the morphism set of the operad PaB: the associator satisfy the pentagon equation and we have two hexagon equations combining associators and braidings.

11 The first one reads:

12 and the second one reads:

13 Theorem (Mac Lane + Joyal-Street): An operad morphism φ : PaB P, where P is any operad in the category of categories, is uniquely determined by: an object m Ob P(2), which represents the image of the little 2-disc configuration µ D 2 (2) under the map φ : Ob PaB(2) Ob P(2), an isomorphism a Mor P(3) (m 1 m, m 2 m), which represents the image of the associator α Mor PaB(3) (µ 1 µ, µ 2 µ), and an isomorphism c Mor P(2) (m, tm), which represents the image of the braiding τ Mor PaB(2) (µ, tµ), so that a and c satisfy the analogue of the usual pentagon and hexagon relations of braided monoidal categories in Mor P. This result implies that PaB is generated by operations defining the structure of a braided monoidal category (without unit).

14 Observation: The operad PaB has a unitary extension PaB + with PaB + (0) =, and where the operadic composition with the additional arity zero element PaB + (0) is given by a rectification process in the little 2-discs space D 2. This operads govern braided monoidal category structures with strict units. Theorem (variant of a result of Fiedorowicz): The operad B(PaB) + formed by the classifying spaces of the groupoids PaB + (r) is an E 2 -operad. Corollary: The classifying space B(M) of a braided monoidal category M is weakly-equivalent to a double loop space Ω 2 X.

15 2. Homotopy automorphisms of operads in topological spaces Reminder: A weak-equivalence of operads (in topological spaces) is an operad morphism φ : P Q of which components φ(r) : P(r) Q(r) are weak-equivalences (of topological spaces). We use the notation to mark any distinguished class of weak-equivalences in a category (e.g. in the category of topological spaces, in the category of operads). Definition (homotopy category): The homotopy category of operads Ho(TopOp) is the category defined by formally inverting the class of weak-equivalences in the category of operads. Definition (homotopy automorphism groups): The group of homotopy automorphisms of an operad in topological spaces P is the group of automorphisms of this object in the homotopy category Ho(TopOp).

16 Theorem: The category of operads in topological spaces inherits a cofibrantly generated model category structure such that: the weak-equivalences (fibrations) are the operad morphism φ : P Q of which components φ(r) : P(r) Q(r) are weak-equivalences (respectively, Serre fibrations) in the category of topological spaces, the cofibrations are characterized by the left-lifting-property with respect to the class of acyclic fibrations. Corollary: Any operad P inherits a cofibrant resolution R P (an operad so that the unit morphism η : I R defines a cofibration in the category of operads), and we have Mor Ho(TopOp) (P, Q) = Mor TopOp (R, Q)/ op, where op refers to an operadic homotopy relation.

17 Definitions: Morphisms of topological operads φ, ψ : P Q are homotopic as operad morphisms φ op ψ if we have a continuous family of operad morphisms φ t : P Q, t [0, 1], such that φ 0 = φ and φ 1 = ψ. A morphism of operads φ : P Q is a homotopy equivalence (in the category of operads) if we have an operad morphism in the converse direction ψ : Q P such that φψ op id and ψφ op id. I will use the notation to mark homotopy equivalences of operads. In the case of cofibrant operads, we have an identity between the homotopy-equivalences φ : P Q and the weak-equivalences φ : P Q.

18 For an operad P, we accordingly have an identity: Aut Ho(TopOp) (P) = {φ : R R}/ op, where we consider the group of operadic homotopy classes of the homotopy self-equivalences of a cofibrant resolution R of the operad P. In fact, we have a whole space haut(r) such that π 0 haut TopOp (R) = {φ : R R}/ op, where we consider: a group formed by the path-connected components of this space on one hand, and this group of homotopy classes of homotopy self-equivalences φ : R R on the other hand.

19 Definition: 3. The Grothendieck-Teichmüller group Form the Malcev completion PaB(r) of the parenthesized braid groupoids PaB(r). The collection of these completed groupoids forms an operad PaB, which also admits a unitary extension PaB +. The Grothendieck-Teichmüller group GT 1 (Q) is the group formed by the automorphisms of operads in groupoids γ : PaB + = PaB+ which are the identity on object sets, and fix the braiding τ Mor PaB(2).

20 Reminder (Fiedorowicz s theorem): The operad of little 2-discs D 2 is weakly-equivalent, as an operad in spaces, to an operad B(PaB) + formed by the classifying spaces of the parenthesized braid groupoids PaB(r). Construction: The topological operad B( PaB) + = {B( PaB(r)), r N} formed from the Malcev completion of the parenthesized braid groupoids PaB(r) defines a rationalization of the topological operad of little 2-discs in the sense that we have a chain of operad morphisms D 2 B( PaB)+ inducing the rationalization on homotopy groups.

21 Construction: Let D 2 denote a cofibrant replacement of the operad B( PaB) +. By functoriality of the classifying space construction, any automorphism γ : PaB = + PaB+ induces an automorphism B(γ) : B( PaB) + = B( PaB)+, and yields an automorphism of the rationalization of the little 2-disc operad D 2 in the homotopy category of operads Ho(TopOp). This automorphism lifts to a self-homotopy equivalence on the cofibrant replacement D 2. Theorem (BF): The mapping γ B(γ) induces a group isomorphism GT 1 (Q) = ker { AutHo(TopOp) ( D 2 ) Aut gr Op (H (D 2 )) } and we have π haut TopOp ( D 2 ) = 0 for > 0.

22 See: Thank you for your attention!