Homotopy Automorphisms of Operads & Grothendieck-Teichm¨ullerGroups

Benoit Fresse

Universit´eLille 1

GDO, Isaac Newton Institute March 5, 2013 Reminder:

I The notion of an En-operad refers to a class of operads defined by a reference model, the operad of little n-discs Dn. Goal:

I The Grothendieck-Teichm¨ullergroup is the of homotopy automorphisms of E2-operads over the rationals. I The torsor of Drinfeld’s associators is the set of homotopy classes of rational formality quasi-isomorphisms of E2-operads. Plan

1. The fundamental operad of little 2-discs and braided monoidal structures 2. Homotopy automorphisms 3. The topological interpretation of the Grothendieck-Teichm¨ullergroup 1. The fundamental groupoid operad of little 2-discs and braided § monoidal structures

˚2 I The homotopy equivalence D2(r) F(D , r) implies that the ∼ space D2(r) is an Eilenberg-MacLane space K(Pr , 1), where ˚2 Pr = π1 F(D , r) is the pure on r strands. I The idea is to use the fundamental π D2(r) in order to get a combinatorial model of the operad D2. Construction:

I The fundamental groupoid π D2(r) has Ob π D2(r) = D2(r) as object set.

I The morphisms of Morπ D2(r)(a, b) are homotopy classes of paths γ : [0, 1] D (r) going from γ(0) = a to γ(1) = b. → 2 I The map

disc centers Morπ r (a, b) Mor ˚2 (a, b) D2( ) −−−−−−−→' π F(D ,r) identifies the morphism sets of this groupoid with cosets of the pure braid group Pr inside the braid group Br . Thus, a morphism in this groupoid can be represented by a picture of the form: 2 1 = a γ = Mor π D (2) ∈ 2 2 1 = b The structure of the fundamental groupoid operad:

I The groupoids π D2(r) inherit a symmetric structure, as well as operadic composition products

i : π D (m) π D (n) π D (m + n 1), ◦ 2 × 2 → 2 − and hence form an operad in the of groupoids.

I In the braid picture, the operadic composition products can be depicted as cabling operations:

1 2 1 2 1 2 3

= ◦1 2 1 1 2 3 1 2

Mor π D (2) Mor π D2(2) Mor π D2(3) ∈ 2 ∈ ∈ Ideas:

I There is no need to consider the whole D2(r) as object set. I The groupoids of parenthesized braids PaB(r) are full subgroupoids of the fundamental groupoid π D2(r) defined by appropriate subsets of little 2-discs configurations Ω(r) D (r) as object sets. ⊂ 2 I 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 D2. ∈ I 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,... ). I 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 D2 generated by this element. I Proposition: This operad Ω is identified with the Magma operad (the free operad on one ). Fundamental morphisms of PaB include I the associator

1 2 3 = µ µ ◦1 α = Mor PaB(3) ∈ 1 2 3 = µ µ ◦2

I and the braiding

1 2 = µ

τ = Mor PaB(2) ∈ 2 1 = tµ

where t = (12) Σ . ∈ 2 In the morphism set of the operad PaB:

I the associator satisfy the pentagon equation

1 2 3 4

1 2 3 4

≡

I and we have two hexagon equations combining associators and braidings. I The first one reads:

1 2 3 1 2 3

≡ I and the second one reads:

1 2 3 1 2 3

≡ 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:

I an object m Ob P(2), which represents the image of the ∈ little 2-disc configuration µ D (2) under the map ∈ 2 φ : Ob PaB(2) Ob P(2), → I an isomorphism a Mor (m m, m m), which represents ∈ P(3) ◦1 ◦2 the image of the associator α Mor (µ µ, µ µ), ∈ PaB(3) ◦1 ◦2 I and an isomorphism c Mor (m, tm), which represents the ∈ P(2) image of the braiding τ Mor (µ, tµ), ∈ PaB(2) I 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 (without unit). I 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 D2. This operads govern braided monoidal category structures with strict units.

I Theorem (variant of a result of Fiedorowicz): The operad B(PaB)+ formed by the classifying spaces of the groupoids PaB+(r) is an E2-operad. I Corollary: The classifying space B(M) of a braided monoidal category M is weakly-equivalent to a double Ω2 X. 2. Homotopy automorphisms of operads in topological spaces §

I 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).

I 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.

I 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). I Theorem: The category of operads in topological spaces inherits a cofibrantly generated model category structure such that:

I 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, I the cofibrations are characterized by the left-lifting-property with respect to the class of acyclic fibrations. ∼ I 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 (P, Q) = MorTopOp(R, Q)/ op, Ho(TopOp) '

where op refers to an operadic homotopy relation. ' Definitions:

I 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 φt :P Q, t [0, 1], → ∈ such that φ0 = φ and φ1 = ψ. I 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.

I In the case of cofibrant operads, we have an identity between ' the homotopy-equivalences φ :P Q and the ∼ −→ weak-equivalences φ :P Q. −→ I For an operad P, we accordingly have an identity:

' Aut (P) = φ :R R / op, Ho(TopOp) { −→ } ' where we consider the group of operadic homotopy classes of the homotopy self-equivalences of a cofibrant resolution R of the operad P.

I In fact, we have a whole space hAut(R) such that

' π hAutTopOp(R) = φ :R R / op, 0 { −→ } ' where we consider:

I a group formed by the path-connected components of this space on one hand, I and this group of homotopy classes of homotopy ' self-equivalences φ :R R on the other hand. −→ 3. The Grothendieck-Teichm¨ullergroup § Definition: I Form the Malcev completion PaB(d r) of the parenthesized braid groupoids PaB(r). The collection of these completed groupoids forms an operad PaB,d which also admits a unitary extension PaBd +. 1 I The Grothendieck-Teichm¨ullergroup GT (Q) is the group formed by the automorphisms of operads in groupoids

=∼ γ : PaBd + PaBd + −→ which are the identity on object sets, and fix the braiding τ Mor PaB(2). ∈ I Reminder (Fiedorowicz’s theorem): The operad of little 2-discs D2 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).

I Construction: The topological operad

B(PaB)d + = B(PaB(d 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

∼ D2 B(PaB)d + ←−· → inducing the rationalization on homotopy groups. I Construction: Let Dc2 denote a cofibrant replacement of the operad B(PaB)d +. By functoriality of the classifying space =∼ construction, any automorphism γ : PaBd + PaBd + induces −→ an automorphism

=∼ B(γ): B(PaB)d + B(PaB)d +, −→ and yields an automorphism of the rationalization of the little 2-disc operad D2 in the homotopy category of operads Ho(TopOp). This automorphism lifts to a self-homotopy equivalence on the cofibrant replacement Dc2. I Theorem (BF): The mapping γ B(γ) induces a group 7→ isomorphism

∼ 1 =  GT (Q) ker AutHo(TopOp)(Dc2) Autgr Op(H∗(D2)) −→ →

and we have π∗ hAutTopOp(Dc2) = 0 for > 0. ∗ See: http://math.univ-lille1.fr/~fresse/operads2012.html

Thank you for your attention!