Deformation theory of morphisms Bruno Vallette

Deformation theory of morphisms

Bruno Vallette

Trends in Noncommutative Geometry

0-0

1 Paradigm : Associative algebras

•

Let

V

be a

K

-module, consider

EndV := {Hom(V

^⊗n

, V )}

_n≥0. For

f ∈ Hom(V

^⊗n

, V )

and

g ∈ Hom(V

^⊗m

, V )

, binary product

f ? g :=

X

n

i=1

±

g

??? ÄÄ Ä

i

f

::: : ¥ ¥ ¥ ¥ ⁼

X

n

i=1

±f ◦

_i

g.

Degree convention :

|f | = n − 1

,

|g| = m − 1

, so

|f ? g| = |f | + |g|

, that is

| ? | = 0

. Theorem (Gerstenhaber).

(f ? g) ? h − f ? (g ? h) = (f ? h) ? g − f ? (h ? g)

Assoc

(f, g, h) =

Assoc

(f, h, g) (EndV, ?)

is a preLie algebra.

= ⇒

with

[f, g ] := f ? g − (−1)

^|f|.|g|

g ? f

,

(EndV, [ ])

is a Lie algebra.

•

Associative algebra structure on

V

:

µ : V

^⊗2

→ V , ? ? ÄÄ

? ?

? ?

ÄÄ ÄÄ ÄÄ ₋ ? ? ? ?

? ÄÄ

? ÄÄ ^{= 0}

ⁱⁿ

^Hom(V

^⊗3

^{, V )}

⇐⇒ µ ? µ = 0 ⇐⇒ [µ, µ] = 0

In this case,

d

_µ

(f ) := [µ, f ]

verifies

d

_µ

(f )

²

= 0.

Explicitly, for

f ∈ Hom(V

^⊗n

, V )

d

_µ

(f ) = X

n

i=1

±

µ

??? ÄÄ Ä

i

f

::: : ¥ ¥ ¥ ¥ ^±

f

== = } } } }

µ

>>>> ¥¥ ¥ ¥ ^±

f

¢ ¢ AAA A ¢

µ

::: : ¡ ¡ ¡ ¡

∈ Hom(V

^⊗n+1

, V )

Hom(K, V ) −−→

^dµ

Hom(V, V ) −−→

^dµ

Hom(V

^⊗2

, V ) −−→ · · ·

^dµ

Hochschild cohomology of the associative algebra

V

“with coefficients into itself”

( C

^•

(Ass, V ), d

_µ

, [ , ])

dg Lie algebra (twisted by

µ

).

Deformation complex of the associative structure

µ

(Interpretation of H⁰

,

H¹

,

H² in terms of formal deformations : see Konstevich) Operations on

C

^•

(Ass, V )

:

.

Cup product

∪

: associative operation

.

Deligne Conjecture

• (V, d)

dg module,

EndV

is a dg module

D(f ) :=

X

n

i=1

d

i

f

¦ ¦

¦ ¦ 999 9

− (−1)

^|f|

f

¢ ¢

== = ¢

d

(EndV, D, ?)

is a dg preLie algebra and

(EndV, D, [ ])

is a dg Lie algebra.

(V, d, µ)

is a dg associative algebra

⇐⇒

Dµ + µ ? µ = 0 ⇐⇒ Dµ + 1

2 [µ, µ] = 0 :

Maurer-Cartan equation General solutions :

µ ∈ {Hom(V

^⊗n

, V )}

_n≥1,

µ

_n

: V

^⊗n

→ V

with

µ

₁

= d

.

Dµ + µ ? µ = 0 ⇐⇒

n = 2 :

d > > >

¦¦ ¦¦ ₊ 9 9 9 9 ^d

¡¡ ¡

= D D D zz z

d n = 3 : ? ? ? ?

ÄÄ ÄÄ ÄÄ ₋ ? ? ? ?

? ÄÄ

? ÄÄ ^{= D(µ}

³

⁾

µ

₂ is associative up to the homotopy

µ

₃

n : X

i+j=n+1 i,j≥2

± ? ? ? ?

_i

? ?

_j

ÄÄ Ä

ÄÄ ÄÄ ÄÄ

= D(µ

_n

)

Definition (Stasheff).

A Maurer-Cartan element

µ

is an associative algebra up to homotopy or

A

∞-algebra structure on

(V, d, µ = {µ

_n

}

_n

)

.

Viewpoint : An associative algebra

=

very particular

A

∞-algebra.

Once again,

d

_µ

(f ) := D(f ) + [µ, f ]

verifies

d

²_µ

= 0

.

( C

^•

(Ass, V ), d

_µ

, [ , ])

dg Lie algebra (twisted by

µ

) defines the cohomology of an

A

∞-algebra.

Same interpretation of all the

H

^• in terms of deformations of

µ

.

•

Homological perturbation lemma

(V, d

_V

)

i

//

(W, d

_W

)

oo

p

ss

^h

p ◦ i = Id

_V and

i ◦ p − Id

_W

= d

_W

◦ h + h ◦ d

_W

V

is a deformation retract of

W

.

Theorem (Kadeishvili, Merkulov, Kontsevich-Soibelman, Markl).

If

ν = {ν

_n

}

_n is an

A

∞-algebra structure on

W

, than

µ

_n

= X

planar trees with

nleaves

i i i i i

ν

₂

y y y y

@@@@

ν

₃

~ ~

~ ~ EEEE

ν

₂

FFFF

h ^h

y y y y

p

defines an

A

∞-algebra on

V

such that

i

,

p

and

h

extends to morphisms and homotopy in the category of

A

∞-algebras.

•

Other kind of algebraic structures :

.

Lie, commutative, Poisson, Gerstenhaber, PreLie, BV algebras.

.

Lie bialgebras, associative bialgebras.

For any type of (bi)algebras

dg module

V

²²

dg Lie algebra

²²

Maurer-Cartan elements

=

(bi)algebra∞structure on

V

²² //

Homological perturbation lemma

twisted dg Lie : deformation complex (cohomology)

2 ..., ..., ...

Operations

•

@ @

• ~~

no symmetry

@ @

• ~~

Composition

•

•

@ @

~~ @ @

• L L L • • ~~

• rrr

planar

@ @

~~ @ @

• L L L • • ~~

• rrr

non-planar

Monoidal category

(

Vect

, ⊗) (

gVect

, ◦) (S

-Mod

, ◦)

Monoid

A ⊗ A → A P ◦ P → P P ◦ P → P

Modules Modules Non-symmetric

algebras Algebras

Examples associative algebras Lie, commutative, Gerstenhaber algebras

Free monoid ^Ladders

(Tensor module)

Planar

trees Trees

Operations

@ @

• ~~

~~ @ @ @ @

• ~~

~~ @ @

Composition

@ @

• ~~

@ @

@ @ ^•

~~ ~~

• •

~~ @ @

@ @

~~ @ @

• ~~

@ @

@ @ ^•

~~ ~~ ^•

• •

~~ @ @ •

Monoidal category

(S

-biMod

, £

_c

) (S

-biMod

, £)

Monoid

P £

_c

P → P P £ P → P

Modules (Bial)gebras (Bial)gebras Examples Lie, associative

bialgebras

Free monoid ^Connected

graphs Graphs

2 Operads, properads, props

Operations

•

@ @

• ~~

no symmetry

@ @

• ~~

Composition

•

•

@ @

~~ @ @

• L L L • • ~~

• rrr

planar

@ @

~~ @ @

• L L L • • ~~

• rrr

non-planar

Monoidal category

(

Vect

, ⊗) (

gVect

, ◦) (S

-Mod

, ◦)

Monoid Associative

algebras

Non-symmetric

operads Operads

Modules Modules Non-symmetric

algebras Algebras

Examples associative algebras Lie, commutative, Gerstenhaber algebras

Free monoid ^Ladders

(Tensor module)

Planar

trees Trees

Operations

@ @

• ~~

~~ @ @ @ @

• ~~

~~ @ @

Composition

@ @

• ~~

@ @

@ @ ^•

~~ ~~

• •

~~ @ @

@ @

~~ @ @

• ~~

@ @

@ @ ^•

~~ ~~ ^•

• •

~~ @ @ ^•

Monoidal category

(S

-biMod

, £

_c

) (S

-biMod

, £)

Monoid Properads Props

Modules (Bial)gebras (Bial)gebras Examples Lie, associative

bialgebras

Free monoid ^Connected

graphs Graphs

3 Homological algebra for prop(erad)s

•

Recall for associative (co)algebras [Cartan, Eilenberg, MacLane, Moore, ...].

Pair of adjoint functors :

bar construction

B : {

dg algebras

} ­ {

dg coalgebras

} : Ω

cobar construction

B(A) := (T

^c

(s A), δ) ¯

, where

• T

^c : cofree connected coalgebra (tensor module)

• s

homological suspension

• A ¯

augmentation ideal

• δ

unique coderivation which extends the product of

A

T

^c

(s A) ¯ ³ (s A) ¯

^⊗2

−−−→

^s−1

s( ¯ A ⊗ A) ¯ −−→

^sµ

s A ¯

Explicitly,

δ(a

₁

⊗ · · · a

_n

) = X

i

±a

₁

⊗ · · · ⊗ µ(a

_i

, a

_i+1

) ⊗ · · · ⊗ a

_n

.

δ



 

 

 

 

 

 

 

 

 

 

 

 

 

 

 a

_n

.. .

a

_i

a

_i+1

.. .

a

₁



 

 

 

 

 

 

 

 

 

 

 

 

 

 



= X

i

±

a

_n

.. .

µ(a

_i

, a

_i+1

)

.. .

a

₁

Contracting internal edges : Graph homology `a la Kontsevich

•

For operads, pair of adjoint functors [Ginzburg-Kapranov, Getzler-Jones]

bar construction

B : {

dg operads

} ­ {

dg cooperads

} : Ω

cobar construction

B(P ) := (F

^c

(s P ¯ ), δ )

, where

• F

^c : cofree connected cooperad (trees)

• s

homological suspension

• P ¯

augmentation ideal

• δ

unique coderivation which extends the partial product of

P

(composition of two operations)

F

^c

(s P ¯ ) ³ (s P ¯ )

^⊗2

−−−→

^s−1

s( ¯ P ⊗ P ¯ ) −−→

^sγ

s P ¯

Explicitely,

δ



 

 

 

 

p

₂

w w w w BBBB

p

₃

| |

| | GGGG

p

₁

DDDD { { { {



 

 

 

 

= X

±

p

₃

| |

| | PPPPP PP

γ(p

₁

⊗ p

₂

)

KKKK KK q q q q q

Contracting internal edges : Graph homology `a la Kontsevich

•

Where do these constructions come from conceptually ?

(C, ∆)

coalgebra,

(A, µ)

algebra;

f, g : C → A

f ? g :=

C

^g

// _A

C

^∆

// _{⊗ ⊗}

^µ

// _A

C

^f

// _A

(Hom(C, A), ?)

associative convolution algebra.

Theorem (Merkulov-V.).

For

C

a dg coprop(erad) and

P

a dg prop(erad),

Hom(C , P )

is a dg prop(erad) called the convolution prop(erad).

Corollary (Merkulov-V.).

(Hom(C , P ), [ , ])

is a dg Lie algebra.

Tw

(C, P ) :=

set of Maurer-Cartan elements in

(Hom(C, P ), [ , ])

: set of Twisting morphisms (cochains).

Tw

(−, −)

is a bifunctor, try to represent it.

Definition. Bar construction of a prop(erad) :

B(P ) := (F

^c

(s P ¯ ), δ)

, where

• F

^c : cofree connected coprop(erad) (graphs)

• s

homological suspension

• P ¯

augmentation ideal

• δ

unique coderivation which extends the partial product of

P

, composition of two operations

p

₂

F F xx xx x

xx xx x

x x

p

₁

FF

F F xx

− →

γ

γ(p

₁

, p

₂

)

NNNN p p p p N N N N pppp

Remark : The number of internal edges is not relevant.

Explicitly,

d



 

 

 

 

 



p

₃

x x FF

F F F F F

p

₂

F F xx xx x

xx xx x

x x

p

₁

F F xx



 

 

 

 

 



= X

±

p

₃

w w w w w w GGGG GG

γ(p

₁

⊗ p

₂

)

E E

E E

E E

yyy yyy

Recover particular cases : Associative algebras, operads.

Cobar construction

Ω(C)

is dual.

Theorem (Merkulov, V.).

Hom

dg prop(erad)s

(Ω(C), P ) ∼ =

^Tw

(C, P ) ∼ = Hom

dg coprop(erad)s

(C, B(P ))

Representation of Tw

(−, −)

and adjunction.

PROOF.

Hom

prop(erad)s

(Ω(C), P ) = Hom

prop(erad)s

(F (s

⁻¹

C ¯ ), P )

∼ = Hom

^S₋₁

( ¯ C , P ) ⊂ Hom

^S₋₁

(C, P )

Hom

dg prop(erad)s

(Ω(C), P ) =

MC

(Hom

^S

(C , P )) =

Tw

(C , P )

¤

Theorem (V.). Canonical bar-cobar resolution

Ω(B(P )) −→ P

^∼

•

Minimal models

Definition. Minimal model for

P

(F (X), ∂) −→ P

^∼

where

. F (X )

is a (quasi)-free properad

. ∂

is a derivation

←→ ∂

_|X

: X → F (X)

∂

_|X

: @ @

• ~~

~~ @ @ 7→

@ @

• ~~

@ @

@ @ ^•

~~ ~~

• •

~~ @ @ ^:

Vertex expansion

Definition. A quadratic model is a minimal model

(F (X), ∂) −→ P

^∼ such that

∂

_|X

: X → F (X)

⁽²⁾

:

graphs with 2 vertices

@ @

• ~~

~~ @ @ 7→

@ @

• ~~

@ @

@ @

@ @

@ @

~~ @ • @

The number of vertices is relevant, not the number of internal edges.

If

P

has a quadratic model, it is called a Koszul properad.

In this case,

X = C

is a coproperad and

(F (X), ∂) = Ω(C )

•

Koszul duality theory

(associative algebras [Priddy], operads [Ginzburg-Kapranov], properads [V.]) provides a method to

.

compute

X = P

^¡ : Koszul dual

.

make

∂

explicit

.

criterion to prove the quasi-isomorphism

F (X ) = Ω(P

^¡

) −→ P

^∼ . (acyclicity of a small chain complex : the Koszul complex)

= ⇒

Graph homology [Kontsevich, Markl-Voronov]

4 Deformation complex of a morphism

P − → Q

^f morphism of prop(erad)s (

Q

is a representation of

P

).

Example.

V

dg module,

EndV := {Hom(V

^⊗n

, V

^⊗m

)}

_n,m is a dg prop(erad) (composition of multilinear functions)

Definition.

A structure of

P

-gebra on

V

is a morphism of prop(erad)s

P → EndV

. Recall Quillen : “(commutative) algebraic geometry”

deformation complex of morphisms of commutative algebras (cotangent complex, model category structure).

comm. algebras

→

ass. algebras

→

operads

→

prop(erad)s

7→

Noncommutative geometry [Nonlinear]

Theorem (Merkulov-V.).

•

The category of dg prop(erad)s is a cofibrantly generated model category structure

•

Quasi-free prop(erad)s are cofibrant PROOF.

F :

dg

S

-bimodules

­

dg prop(erad)s

: U

¤

Definition (Deformation complex).

cofibrant resolution

: (R, ∂ ) E E E

^∼

E E E E // "" ^P

²²

f

Q C

^•

(P , Q) := (Der(R, Q), ∂

^∗

)

•

Well defined :

Extension of Quillen theory of commutative rings to prop(erad)s

To

O

²² I // _P // _Q

Der

_I

(O, Q) ∼ = Hom

dg prop(erad)s/_P

(O, P n Q

| {z }

Eilenberg-MacLane space

)

∼ = Hom

_{P−bimodules}

(P £

_O

Ω

_O/I

£

_O

P

| {z }

Cotangent complex

, Q),

where

Ω

_O/I is the module of K ¨ahler differentials of the prop(erad)

O

. (Read the properties of the morphism

P − → Q

^f on the cotangent complex) Quillen adjunction

= ⇒

Total derived functor (non-additive)

= ⇒

well defined in the homotopy categories

•

Explicitly,

When

P

is Koszul :

R = Ω(P

^¡

) = F (s

⁻¹

P ¯

^¡

) −→ P

^∼ quadratic model In this case,

C

^•

(P , Q) = Der(F (s

⁻¹

P ¯

^¡

), Q) = Hom

^S_•−1

( ¯ P

^¡

, Q) ⊂ Hom

^S

(P

^¡

, Q)

Theorem (Merkulov-V.).

Hom

^S

(P

^¡

, Q)

is a non-symmetric dg prop(erad)

= ⇒

it is a dg Lie algebra.

Proposition (Merkulov-V.).

P − → Q

^f is a morphism of dg prop(erad)s iff

f ¯ : P

^¡

→ P − → Q

^f is a Maurer-Cartan element in

Hom

^S

(P

^¡

, Q)

In this case,

(C

^•

(P , Q), d) ⊂ (Hom

^S

(P

^¡

, Q), [ ¯ f , −]

| {z }

twisted dg Lie algebra

)

•

Examples

P = Ass

,

P

^¡

= Ass

^∨ and

C

^•

(Ass, EndV ) = EndV

: Hochschild cohomology of associative algebras

P = Ass

,

P

^¡

= Ass

^∨ and

C

^•

(Ass, P oisson)

: Invariant of knots [Vassiliev, Turchine]

P = Lie

,

P

^¡

= Com

^∨ and

C

^•

(Lie, EndV )

: Chevalley-Eilenberg cohomology of Lie algebras

P = Com

,

P

^¡

= Lie

^∨ and

C

^•

(Com, EndV )

: Harrison cohomology of commutative algebras

P = BiLie

,

P

^¡

= F rob

^∨₃ and

C

^•

(BiLie, EndV )

: Ciccoli-Guerra cohomology of Lie bialgebras

P = BiAss

, not Koszul and

C

^•

(BiAss, EndV )

: Gerstenhaber-Shack bicomplex ???

P

Koszul

= ⇒ P

quadratic but

BiAss

not quadratic

Ä Ä

?? •

¦¦ 9 • 9

2 1

1 2

− LLt t t • ^• u J u J J J uu J _• ^• J J J t t t

1

1

2

2

Interpretation of H⁰

,

H¹

,

H² in terms of formal deformations

Definition. A

P

_∞-gebra (or homotopy

P

-gebra structure) on

V

is a Maurer-Cartan element in

Hom

^S

(P

^¡

, EndV )

•

Examples

P = Ass

, homotopy associative algebra [Stasheff]

P = Lie

, homotopy Lie algebra [Stasheff, Hinich-Schetchman, Kontsevich]

P = Com

,

C

_∞-algebra [Stasheff]

P = Gerstenhaber

,

G

_∞-algebra [Getzler-Jones]

P = BiLie

, homotopy Lie bialgebra [Gan]

P = BiAss

...

(Hom

^S

(P

^¡

, EndV ), [f, −])

: cohomology of

P

_∞-algebras Interpretation of H^• in terms of deformations

•

Operations on the deformation complex of

P − → Q

^f

When

P

is a Koszul operad,

Hom(P

^¡

, Q) ← Hom(P

^¡

, P ) ∼ = P

^!

⊗ P

| {z }

Manin complex tangent complex

← P | {z }

^!

◦ P

Manin products

Example :

Ass ◦ Ass = Ass = ⇒

cup product

Theorem (V.).

P

finitely generated binary non-symmetric Koszul operad Little disk operad

←− • −→ C

^•

(P , Q)

Generalized Deligne conjecture (

P = Ass

,

Q = EndV

) PROOF.

. C

^•

(P , Q) = Hom(P

^¡

, Q)

non-symmetric operad

= ⇒

braces operations

. Ass → P

^!

◦ P = ⇒

cup product

∪ . ∂ = [∪, −]

.

(McClure-Smith)

¤

Examples :

4

infinite families of operads (Koszul by poset method)

5 Beyond the Koszul case

Minimal model for

P

:

(F (X ), ∂) −→ P

^∼

where

∂

_|X

: X → F (X)

∂

_|X

: @ @

• ~~

~~ @ @ 7→

@ @

• ~~

@ @

@ @ ^•

~~ ~~

• •

~~ @ @ ^:

not necessarily quadratic

∂

²

= 0 = ⇒ X

is a homotopy coprop(erad).

(coassociative of the coprop(erad) holds ‘up to homotopy’) Theorem (Merkulov-V.).

For

C

a homotopy coprop(erad) and

P

a dg prop(erad),

Hom(C , P )

is a homotopy prop(erad)

= ⇒ (Hom(C, P ), [ , ])

is a homotopy Lie algebra.

Proposition (Merkulov-V.).

In this case,

Hom(C, P )

is a filtered homotopy Lie algebra Maurer-Cartan elements :

P

n≥0 1

n!

l

_n

(f, . . . , f ) = 0 P − → Q

^f is a morphism of dg prop(erad)s iff

f ¯ : C → P − → Q

^f is a Maurer-Cartan element in

Hom

^S

(P

^¡

, Q)

In this case,

(C

^•

(P , Q), d) ⊂ ( Hom

^S

(P

^¡

, Q), l

^f¯

| {z }

twisted homotopy Lie algebra

)

Theorem (Merkulov-V.). For every minimal model of

BiAss

, the deformation complex

C

^•

(BiAss, Q) ∼ = Q

and

d •

KKKK << KK

^{. . .}

<< £ £ £ £ s s s s s s

1 2 m−1 m

sss sss

££ ££

. . .

< < < <

K K K K K K

1 2 n−1 n

=

n−2X

i=0

(−1)i+1 •

sss sss ªª ªª

.. •

¥¥¥ : : :

i+1 i+2

K K K K K K

5 5 5 5

..

1 i n

KKKK << KK

^{. . .}

<< £ £ £ £ s s s s s s

1 2 m−1 m

+

•

¨¨ ¨ yy yy 7 7 7

E E E E

...

2 3 n−1n

77 7 EEE E ¨ ¨ y ¨ y y y

z }| {...

m

• 1

HHH GGG

^•

v v v v v u u u u

•

u u

^{z }| {...}

m

•

: : :

¥¥¥

1

•

: : :

¥¥¥

2

¥¥¥ :

•

: :

m−1

¥¥¥ :

•

: :

m ...

+ (−1)n+1

•

7 7 7 E E E E

¨¨ ¨ yy yy

...

1 2 n−1

¨ ¨

¨ y y y 77 7 y EEE E

^...

z }| { m

• n

w w w

v v HHH IIII HH

z }| {...^•

II v

^•

m

•

: : :

¥¥¥

1

•

: : :

¥¥¥

2

¥¥¥ :

•

: :

m−1

¥¥¥ :

•

: :

m ...

+ n−2X

i=0

(−1)i+1 •

KKKK 555 KK 5

..

::: ¥¥

•

¥

i+1 i+2

s s s s s ª s ª ª ª

1 i .. m

sss sss

££ ££

. . .

< < < <

K K K K K K

1 2 n−1 n +

•

77 7 EEE E ¨ ¨ y ¨ y y y

2 3 m...−1m

¨¨ ¨ yy yy 7 7 7

E E E E

| {z }...

n

• 1

vvv www H H H H H

•

I I I I

•

I I

_...

| {z } n

•

¥ ¥ ¥ :::

1

•

¥ ¥ ¥ :::

2

::: ¥¥

•

¥

n−1

::: ¥¥

•

¥

... n

+ (−1)m+1

•

¨ ¨

¨ y y y 77 7 y EEE E

^...

1 1 m−1

7 7 7 E E E E

¨¨ ¨ yy yy

| {z }...

n

• m

G G G

H H vv vv v

_•

H

uuu u uu

•

| {z }...

n

•

¥ ¥ ¥ :::

1

•

¥ ¥ ¥ :::

2

::: ¥¥

•

¥

n−1

::: ¥¥

•

¥

n ...

Corollary (Merkulov-V.).

C

^•

(BiAss, EndV ) =

Gerstenhaber-Shack bicomplex

There is a homotopy Lie agebra structure on the Gerstenhaber-Shack bicomplex which measures the deformation of structures of bialgebras

6 Homological perturbation lemma

Theorem (V.).

(V, d

_V

)

i

//

(W, d

_W

)

oo

p

ss

^h

If

W

is a homotopy

P

-gebra, than there is an induced homotopy

P

-gebra structure on

V

.

•

Examples

P = Ass

(Kontsevich-Soibelman)

P = Lie

(Costello, Goncharov, Mnev, ...)

P = Com

(Cheng-Getzler)

P = Gerstenhaber

‘new’

P = BiLie

‘new’

P = BiAss

TBC ...

References

[1] B. Vallette, A Koszul duality for props, To appear in Trans. of Amer. Math.

Soc..

[2] S. Merkulov, B. Vallette, Deformation theory of representations of prop(erad)s, preprint.

[3] B. Vallette, Manin products, Koszul duality, Loday algebras and Deligne conjecture, to appear in The Journal f ¨ur die reine und angewandte Mathematik.

[4] B. Vallette, Homology of generalized partition posets, Journal of Pure and Applied Algebra, Volume 208, Issue 2, February 2007, 699-725.

[5] B. Vallette, Free monoid in monoidal abelian categories, preprint.

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