An operadic view of Goncharov's bialgebra of iterated ... Galvez... · Seminario de Algebra y Combinatoria ICMAT, 26 November 2013 An operadic view of Goncharov’s bialgebra of iterated

Seminario de Algebra y Combinatoria ICMAT, 26 November 2013

An operadic view ofGoncharov’s bialgebra of iterated integrals

Imma GalvezUniversitat Politecnica de Catalunya

work in progress withRalph Kaufmann (Purdue) and Andy Tonks (London Metropolitan)


Outline

Operadic and simplicial language

Operads and cooperadsSimplicial objectsThe standard simplices as an operadSimplicial sets as cooperads

Cooperads with multiplication give bialgebras

Definition and theoremProof

Examples

Simplicial monoids as bialgebrasBaues’ cobar constructionGoncharov’s bialgebra of iterated integrals


Operads and cooperads in a monoidal category (C,⊗, I )

An non-symmetric operad O is a sequence of objects O(n), n ≥ 0with an identity u : I → O(1) and composition structures

i : O(k)⊗ O(m) −→ O(k + m − 1), 1 ≤ i ≤ k,

that are unital, 1(u ⊗ id) = id = i (id⊗ u), and associative,

(− i −) j − =

− i (− j−i+1 −) if i ≤ j < m + i

((− j −) i+n−1 −)(23) if 1 ≤ j < i ,

where (23) : O(k)⊗ O(m)⊗ O(n) ∼= O(k)⊗ O(n)⊗ O(m).


Example: The endomorphism operad of an object A

If A is an object in a monoidal closed category (C,⊗, I ,HomC),one has the endomorphism operad of all ‘multilinear’ maps on A,

EndA(n) = HomC(A⊗n,A)

with the obvious identity,

u : I → HomC(A,A)

and composition structures

i : End(k)⊗ End(m) −→ End(k + m − 1), 1 ≤ i ≤ k ,

substituting an m-ary map as the ith argument of a k-ary map.


The composition structures i , i = 1, . . . , k , combine to form

γ : O(k)⊗ O(m1)⊗ · · · ⊗ O(mk)

1⊗id−→ O(m1 + k − 1)⊗ O(m2)⊗ · · · ⊗ O(mk)m1+1⊗id−→ · · ·

· · · −→ O(m1 + · · ·+ mk−1 + 1)⊗ O(mk)

−→ O(m1 + · · ·+ mk)

satisfying unital and associative laws.

The i operations can be recovered from γ and the unit u,

i :O(k)⊗ O(m) ∼= O(k)⊗ I⊗i−1O(m)⊗ I⊗k−i

id⊗u⊗i−1⊗id⊗uk−i

−−−−−−−−−−−−→ O(k)⊗ O(1)⊗i−1O(m)⊗ O(1)⊗k−i

γ−−−−−−−−−−−−→ O(k + m − 1)


A cooperad C is a sequence of objects C (n), n ≥ 0,with (counital and coassociative) decomposition structures

i : C (k + m − 1) −→ C (k)⊗ C (m), 1 ≤ i ≤ k,

or, equivalently,

γ : C (m1 + · · ·+ mk) −→ C (k)⊗ C (m1)⊗ · · · ⊗ C (mk).

for all m1, . . . ,mk ≥ 0.


The simplicial category

One denotes by ∆ the category whose objects are thefinite non-empty standard ordinals

[n] := 0, 1, · · · , n, n ≥ 0.

Morphisms are the non-decreasing maps between them.

Among these maps one considers the following generators:for n ≥ 0 and i ∈ [n],

∂ in : [n − 1]→ [n]

is the only increasing injection which skips the value i , and

σin : [n + 1]→ [n]

is the only non-decreasing surjection that repeats i .

One usually writes just ∂ i and σi .


Simplicial objects in a category C

A simplicial object is a (contravariant) functor

X : ∆op → C.

Common notation is

Xn = X ([n]), X• = (Xn)n≥0

The maps ∂ i , σi in ∆ induce the so-called face maps

dni := X (∂in) : Xn → Xn−1, i = 0, . . . , n,

and degeneracy maps

sni := X (σin) : Xn → Xn+1, i = 0, . . . , n.

One writes di and si if there is no danger of confusion.A simplicial map X• → Y• is a natural transformation,that is: a family of morphisms (Xn → Yn)n≥0 in Cthat commute with the face and degeneracy maps.

Simplicial objects in C form a category C∆ := Fun(∆op,C).


The standard simplices

The standard simplices are the simplicial sets

∆[n] = Hom∆(−, [n]) : ∆op → Set

Thus the k-simplices of ∆[n] are non-decreasing functions

[k]→ [n]

By the Yoneda Lemma, for any simplicial set X ,

HomSet∆(∆[k],X ) ∼= Xk

In particular

HomSet∆(∆[k],∆[n]) ∼= ∆[n]k ∼= Hom∆([k], [n])


The standard simplices as an operad

Proposition

The sequence (∆[n])n≥0, forms an operad in (Set∆,∪,∅), with

∅ u−−→ ∆[1]

∆[k] ∪∆[m]i−−→ ∆[k + m − 1]

For i = 1, . . . , k , the partial composition circle-i sends

vertex a of ∆[k] 7−→ vertexa of ∆[k + m − 1] if a ≤ i − 1

a + m − 1 of ∆[k + m − 1] if a ≥ i

vertex a of ∆[m] 7−→ vertex a + i − 1 of ∆[k + m − 1].


Simplicial sets as cooperads

Let X be any simplicial set.Since Xn

∼= HomSet∆(∆[n],X ) and (∆[n]) is an operad,

we have by adjunction:

Proposition

The sequence (Xn)n≥0, forms a cooperad in (Set,×, ∗),with the counit and cocomposition structures given by

X1u−−−−−−−−−−−−−−−→ ∗ (counit)

Xk+m−1i−−−−−−−−−−−−−−−→ Xk × Xm (1 ≤ i ≤ k)

x 7−−−−−→ (x(0,...,i−1, i+m−1,...,k+m−1), x(i−1,...,i+m−1))

If x ∈ Xm then we write x(α(0),...,α(r)) for X(

[r ]α−→ [m]

)(x) ∈ Xr .


As usual the partial cocompositions i determine,and are determined by, the cooperad structure γ,

Xnγ−−−−−−−−→

∏m1+···+mk=n

Xk × Xm1 × Xm2 × . . .Xmk

Rewriting the indices as i0 = 0, i1 = m1, and ir+1 − ir = mr+1

Xnγ−−−−−−−−→

∏0=i0≤i1≤···≤ik=n

Xk × Xm1 × Xm2 × . . .Xmk

x 7−−−−−−−−→ (x(i0,i1,...,ik ),

x(i0,i0+1,...,i1), x(i1,i1+1,...,i2), . . . , x(ik−1,ik−1+1,...,ik ))


Bialgebras from cooperads with multiplication

Definition + Theorem

A (non-symmetric, unital) cooperad (C , γ) with an associativecompatible multiplication µn,n′ : C (n)⊗ C (n′)→ C (n + n′)determines a bialgebra structure (

⊕C (n), µ, δ).

Compatibility means that for all n =k∑

r=1

mr and n′ =k ′∑

r ′=1

m′r ′ ,

C (n)⊗ C (n′)

µn,n′

γ⊗2 // C (k)⊗k⊗

r=1

C (mr )⊗ C (k ′)⊗k ′⊗

r ′=1

C (m′r ′)

µk,k′⊗id

C (n + n′)γ // C (k + k ′)⊗

k⊗r=1

C (mr )⊗k ′⊗

r ′=1

C (m′r ′)


Definition of the comultiplication

The comultiplication δ on⊕

C (n) is defined as follows,using the associative multiplication µ and the cooperad structure γ

C (n)

δ = (id⊗ µ)γ((

γ //⊕k≥1,

n=m1+···+mk

(C (k)⊗

k⊗r=1

C (mr )

)

id⊗µ

⊕k≥1

C (k)⊗ C (n)


Check that⊕

C (n), µ, δ = (id⊗ µ)γ forms a bialgebra

For example, the bialgebra axiom δµ = µ⊗2δ⊗2 may be written:

C (n)⊗ C (n′)

µ

δ⊗2//

compatibility associativity

C (k)⊗ C (n)

⊗ C (k ′)⊗ C (n′)

µ⊗2

C (n + n′)

δ// C (k + k ′)⊗ C (n + n′),

by the compatibility square above and an associativity square for µ:

C (k)⊗k⊗

r=1

C (mr )⊗ C (k ′)⊗k ′⊗

r ′=1

C (m′r ′)

(µ⊗id)(π)

(id⊗µ)⊗2

//C (k)⊗ C (n)

⊗ C (k ′)⊗ C (n′)

µ⊗2

C (k + k ′)⊗

k⊗r=1

C (mr )⊗k ′⊗

r ′=1

C (m′r ′)id⊗µ // C (k + k ′)⊗ C (n + n′).


Simplicial monoids as algebras

Let X be a simplicial set, and let A be the chain complex

A = C∗(X ,Q)

That is, An is the vector space with basis Xn.

If X is a simplicial monoid then A becomes a differentialgraded algebra, using the shuffle product,

µ : A ⊗A = C∗(X ,Q)⊗C∗(X ,Q)shuf−−→ C∗(X×X ,Q)

mult−−→ C∗(X ,Q) = A.


Simplicial monoids as bialgebras

The cooperad structure associated to X that we saw aboveinduces a cooperad structure on the chain complex A,

γ : An −→⊕

m1+···+mk=n

Ak ⊗ Am1 ⊗ · · · ⊗ Amk

We may take the reduced complex A = C ∗(X ) , A0 = 0,so that the sum is finite.

We may take the desuspended complex A = C ∗(X )[1]so that γ is a graded map of degree −1 for all n.

The multiplication µ given by the shuffle map is compatiblewith the cooperad structure γ.


Proposition (From simplicial monoids to bialgebras)

Let A be the chain complex on a simplicial monoid X .Then A• =

⊕An has a bialgebra structure with the multiplication

given by the shuffle product and the comultiplication given by

δ : Anγ−−−→

⊕0=i0<i1<···<ik=n

Ak ⊗ Am1 ⊗ Am2 ⊗ · · · ⊗ Amk

id⊗µ−−−−→⊕k

Ak ⊗ An

x 7−−−−−−−→∑

x(i0,i1,...,ik ) ⊗

x(i0,i0+1,...,i1) · x(i1,i1+1,...,i2) · · · x(ik−1,ik−1+1,...,ik )


Generalisations and observations

Note that this construction does not use the boundary mapsof the chain complex A, only the fact that it is a family ofvector spaces.

The construction does not use all of the monoid structure ofX , in fact the product xy is only needed if the final vertex ofx coincides with the initial vertex of y .

We could therefore state the proposition instead for∆•-categories, where ∆• is the subcategory of ∆ containingonly the end-point-preserving maps.


Adams’ cobar equivalence

Let X be a 1-reduced simplicial set, X0 = X1 = ∗.The chain complex A = (C∗(X ,Q), d) is a differential gradedcoalgebra, with the comultiplication given by

∆(x(0,1,...,n)) =n∑

k=0

x(0,...,k) ⊗ x(k,...,n) (Alexander-Whitney)

The cobar construction on a differential graded coalgebra(A,∆) is the (free) differential graded algebra

ΩA = T (A[1]), dΩ = d + ∆.

Theorem (Adams, 1956)

The homology groups of Ω(C∗X ) are naturally isomorphic to thoseof the loop space on (the geometric realisation of) X


Baues’ cobar comultiplication

The cooperad structure on X induces a one on Ω(C∗X ),and the (free) multiplication µ is compatible with it.

We thus have a comultiplication δ on the cobar construction

δ : T (C ∗X [1]) −−−→ T (C ∗X [1])⊗ T (C ∗X [1])

x 7−−−→∑

x(i0,i1,...,ik ) ⊗

x(i0,i0+1,...,i1)x(i1,i1+1,...,i2) · · · x(ik−1,ik−1+1,...,ik )

This coincides with the d.g. bialgebra structure of Baues.

One can apply the cobar construction to the coalgebra ΩC∗X :

Theorem (Baues, 1981)

If X has trivial 2-skeleton then the homology groups of ΩΩC∗X arenaturally isomorphic to those of the double loop space on X


Goncharov’s bialgebra I•(S)

Let S be any set, and let S be the trivial groupoid with objectset S and exactly one arrow s1 → s2 for all (s1, s2) ∈ S2.

E.g. [n] is the fundamental groupoid of the n-simplex,

S = [n] ⇒ S = π1(∆[n]),

Let X be Ner(S), the simplical nerve of the groupoid S .

Explicitly, the (n + 1)-simplices of X are tuples of elements

Xn+1∼= I(s0; s1, . . . , sn; sn+1) : sr ∈ S ∼= Sn+2.


Let A be the free commutative algebra on X = Ner(S) modulothe 1-skeleton. The (free) multiplication µ is compatible withthe cooperad structure γ on A induced from X .

We therefore have a comultiplication (id⊗ µ)γ : A→ A⊗ A,sending a generator I(s0; s1, . . . ; sn) of A to∑

k≥10=i0≤i1≤···≤ik=n

I(si0 ; si1 , . . . ; sik )

⊗ I(si0 ; si0+1, . . . ; si1)I(si1 ; si1+1, . . . ; si2) · · · I(sik−1; sik−1+1, . . . ; sik )

The bialgebra (A, µ, δ = (id⊗ µ)γ) coincides with thebialgebra (I•(S), ·,∆) of iterated integrals of Goncharov.


Cubical structure

Baues’ and Goncharov’s comultiplications come from path orloop spaces and may be seen having natural cubical structure.

The space of paths P from 0 to n in the n-simplex |∆[n]| is acell complex homeomorphic to the (n − 1)-dimensional cube.

Cubical complexes have a natural diagonal approximation,

δ : P = [0, 1]n−1 ∼=−−→⋃

K∪L=1,...,n−1

∂−K [0, 1]n−1×∂+L [0, 1]n−1 ⊂−−→ P × P

One can identify faces ∂−i of the cube P as the spaces ofpaths through the faces x

(0,...,i ,...,n)of the n-simplex x .

Faces ∂+i are products of a (i − 1)-cube and (n − i − 1)-cube:

the spaces of paths through x(0,...,i) and through x(i ,...,n).

The term for L = i1, . . . , ik−1 under this identification is

x(0,i1,...,ik−1,n) × x(0,1,...,i1) x(i1,i1+1,...,i2) . . . x(ik−1,ik−1+1,...,n).

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An operadic view of Goncharov's bialgebra of iterated ... Galvez... · Seminario de Algebra y Combinatoria ICMAT, 26 November 2013 An operadic view of Goncharov’s bialgebra of iterated