# Deformation theory of morphisms by theoryman

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• pg 1
```									Deformation theory of morphisms
Bruno Vallette

Trends in Noncommutative Geometry

0-0
• 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
cc
c 
n       XX g Ô                n
f    g :=         ±
XX i ÔÔ =                 ±f ◦i g.
Ô
i=1             f             i=1

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 :

⊗2
c
c
µ : V           →V ,
c  c c
c  cc c
c  − c = 0
c                            in   Hom(V ⊗3 , V )

⇐⇒ µ          µ = 0 ⇐⇒ [µ, µ] = 0
In this case,
dµ (f ) := [µ, f ] veriﬁes dµ (f )2 = 0.
Explicitly, for f   ∈ Hom(V ⊗n , V )

cc         aa      }       ee
c        a } }}          ee ÑÑÑ
n     XX µ Ô         f b       X      f
XX i ÔÔ ±      bb ÔÔ ± XXX Ð
dµ (f ) =           ±       Ô           b Ô
Ô      ÐÐ
Ð
i=1             f                    µ                µ

∈ Hom(V ⊗n+1 , V )
dµ                   dµ                     dµ
Hom(K, V ) −→ Hom(V, V ) −→ Hom(V ⊗2 , V ) −→ · · ·
−             −                 −
Hochschild cohomology of the associative algebra V “with coefﬁcients into itself”
(C • (Ass, V ), dµ , [ , ]) 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

WW d Õ                   aa
n       WW i ÕÕ                  a ÑÑÑ
Õ                       f
D(f ) :=               f        − (−1)|f |
i=1
d
(EndV, D, ) is a dg preLie algebra and (EndV, D, [ ]) is a dg Lie algebra.
(V, d, µ) is a dg associative algebra ⇐⇒

1
Dµ + µ      µ = 0 ⇐⇒ Dµ +                [µ, µ] = 0 : Maurer-Cartan equation
2
General solutions :
µ ∈ {Hom(V ⊗n , V )}n≥1 ,              µn : V ⊗n → V    with   µ1 = d.
Dµ + µ µ = 0 ⇐⇒

d         WW   d  hh z
b ÕÕ            hzz
n=2           :        bÕ + WWÐ Ð =
bÕ      Ð
d
c  c c
c  cc c
n=3 :  c  − c  = D(µ3 )
c    
µ2 is associative up to the homotopy µ3

cc i 
cc 
n     :               ±  j  = D(µn )
cc 
i+j=n+1        
i,j≥2

Deﬁnition (Stasheff).
A Maurer-Cartan element µ is an associative algebra up to homotopy or
A∞ -algebra structure on (V, d, µ = {µn }n ).

Viewpoint : An associative algebra = very particular A∞ -algebra.
Once again, dµ (f ) := D(f ) + [µ, f ] veriﬁes d2 = 0. µ
(C • (Ass, V ), dµ , [ , ]) 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

i    /            s
(V, dV )         o              (W, dW )         h
p

p ◦ i = IdV          and         i ◦ p − IdW = dW ◦ h + h ◦ dW
V is a deformation retract of W .

If ν = {νn }n is an A∞ -algebra structure on W , than

i    dd            i ii
ii
i
~
i
dd      yy
yy
i
~~~
ν2                        ν3
pp
ph
yy
h
µn =                                         p            yy
planar trees with                       ν2
n leaves

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


dg Lie algebra


Maurer-Cartan elements   = (bi)algebra∞ structure on V   /   Homological perturbation lemma


twisted dg Lie : deformation complex (cohomology)
2 ......., ........., .....
d ~
d~
•                    d ~
d~
Operations                 •                                             •
no symmetry

d ~ d ~
d~     d~            d ~ d ~
d~     d~
•vv • r•              •vv • r•
•                  v rr                  v rr
Composition                                       •                      •
•
planar               non-planar

Monoidal category    (Vect, ⊗)               (gVect, ◦)           (S-Mod, ◦)
Monoid              A⊗A→A                   P ◦P →P              P ◦P →P
Non-symmetric
Modules                Modules                 algebras
Algebras
Lie, commutative,
Examples                                 associative algebras   Gerstenhaber algebras
Free monoid          (Tensor module)            trees
Trees

d ~
d~                 d ~
d~
Operations
~•d
~ d                  ~•d
~ d
d ~
d~                 d ~ d ~
d~      d~
•d ~
d •               •d ~
d • •
d
~                   d
~
Composition
~ d•d
~
• ~ d                ~ d•d •
~
• ~ d
~                   ~
Monoidal category   (S-biMod, c )          (S-biMod, )
Monoid              P cP→P                 P P→P
Modules               (Bial)gebras          (Bial)gebras
Lie, associative
Examples                 bialgebras
Connected
Free monoid                graphs
Graphs
d ~
d~
•                     d ~
d~
Operations                 •                                                •
no symmetry

d ~ d ~
d~     d              d ~ d ~
d~     d
•vvv• rr•~             •vvv• rr•~
Composition
•                      r •                    r  •
•
planar                non-planar

Monoidal category    (Vect, ⊗)                 (gVect, ◦)             (S-Mod, ◦)
Associative              Non-symmetric
Non-symmetric
Modules               Modules                    algebras
Algebras
Lie, commutative,
Examples                                   associative algebras    Gerstenhaber algebras
Free monoid         (Tensor module)                trees
Trees
d ~
d~                    d ~
d~
Operations
~•d
~ d                     ~•d
~ d
d ~
d~                    d ~ d ~
d~      d~
•d ~
d •                  •d ~
d • •
d
~                      d
~
Composition
~ d•d
~
• ~ d                   ~ d•d •
~
• ~ d
~                      ~
Monoidal category   (S-biMod,         c)      (S-biMod,        )
Modules              (Bial)gebras              (Bial)gebras
Lie, associative
Examples                 bialgebras
Connected
Free monoid                graphs
Graphs
• Recall for associative (co)algebras [Cartan, Eilenberg, MacLane, Moore, ...].

bar construction    B : {dg algebras}         {dg coalgebras} : Ω cobar construction
¯
B(A) := (T c (sA), δ), where
• T c : cofree connected coalgebra (tensor module)
• s homological suspension
¯
• A augmentation ideal
• δ unique coderivation which extends the product of A
−1
¯
T c (sA)         ¯    s
−→ ¯      ¯ sµ   ¯
(sA)⊗2 − − s(A ⊗ A) −→ sA
−

Explicitly,

δ(a1 ⊗ · · · an ) =          ±a1 ⊗ · · · ⊗ µ(ai , ai+1 ) ⊗ · · · ⊗ an .
i
          
          
    an    
          
                              an
          
          
     .    
     .    
     .                         .
                               .
                               .
          
    ai    
          
δ

=
             ± µ(ai , ai+1 )
   ai+1            i
          
          
                               .
                               .
     .                         .
     .    
     .    
          
                              a1
          
    a1    
          

`
Contracting internal edges : Graph homology a la Kontsevich

bar construction   B : {dg operads}           {dg cooperads} : Ω cobar construction
¯
B(P) := (F c (sP), 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)
−1
¯
c
F (sP)             ¯    s
−→ ¯      ¯ sγ
(sP)⊗2 − − s(P ⊗ P) −→ sP
−  ¯
Explicitely,

                                                                    
ff                 q

ff
ww
ww qqq ||
||       

 ||||
            p2           p3                              uuu               p
                hh
hh                                          uuu          qq 3
{{                                               qqq
                                      
δ
                      {{              =

±
                       p1                                     γ(p1 ⊗ p2 )
                                      

`
Contracting internal edges : Graph homology a la Kontsevich

• Where do these constructions come from conceptually ?

(C, ∆) coalgebra, (A, µ) algebra; f, g : C → A
g
C        /A
µ
f       g := C     ∆
/   ⊗            ⊗       /A
f
C        /A
(Hom(C, A), ) associative convolution algebra.

Theorem (Merkulov-V.).

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 (sP), 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

x
x   xxx             p
x
p2
x p γ p      x         ppp
p xxxx
p xx   xx       →
−    γ(p1 , p2 )
xxx
p1              pppp            x
x
x    p
p
Remark : The number of internal edges is not relevant.

Explicitly,

   p                                qq
p      xx                           qq               w
qq          www


p3
pp           
                           ww
                pp                               p3

                  p  x
x     

d                  p2       =        ±

              xxx p
xx    p     


            xxx
xx


γ(p1 ⊗ p2 )
iii
         p1                                yyy           iii
x p
x     p                           yyy
Recover particular cases : Associative algebras, operads.

Cobar construction Ω(C) is dual.

Theorem (Merkulov, V.).

Homdg prop(erad)s (Ω(C), P) ∼ Tw(C, P) ∼ Homdg coprop(erad)s (C, B(P))
=          =

Representation of Tw(−, −) and adjunction.

P ROOF.

∼
=                ¯
HomS (C, P) ⊂ HomS (C, P)
−1                 −1

S
Homdg prop(erad)s (Ω(C), P)         =     MC(Hom      (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

∂ is a derivation ←→ ∂|X : X → F (X)
d ~
d~
d ~
d~   •d ~
d •
: ~d →     d
~
∂|X
~
•d
~ d•d :
~
• ~ d
Vertex expansion

~
∼
Deﬁnition. A quadratic model is a minimal model               →
(F(X), ∂) − P such that

∂|X : X → F (X)(2)              :     graphs with 2 vertices

d ~
d~
d ~
d~     •dd
dd
~~•d →
d
~•d
~ d
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
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
f
→
P − Q 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).

→     Noncommutative geometry           [Nonlinear]

Theorem (Merkulov-V.).

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

P ROOF.
F : dg S-bimodules              dg prop(erad)s   : U
Deﬁnition (Deformation complex).

coﬁbrant resolution   : (R, ∂)
∼
/P
ii
ii
ii f
i" 
Q

C • (P, Q) := (Der(R, Q), ∂ ∗ )
• Well deﬁned :
Extension of Quillen theory of commutative rings to prop(erad)s
To                 O


I        /P          /Q

DerI (O, Q)         ∼
=    Homdg prop(erad)s /P (O,         P      Q            )
Eilenberg-MacLane space

∼
=    HomP−bimodules (P      O    ΩO/I       O    P , Q),
Cotangent complex

where ΩO/I is the module of Kahler differentials of the prop(erad) O .
¨
f
(Read the properties of the morphism P    →
− Q on the cotangent complex)
=⇒ 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) = HomS (P ¡ , Q) ⊂ HomS (P ¡ , Q)
•−1
Theorem (Merkulov-V.).
HomS (P ¡ , Q) is a non-symmetric dg prop(erad)
=⇒ it is a dg Lie algebra.
f
Proposition (Merkulov-V.).       →
P − Q is a morphism of dg prop(erad)s iff
¯           f
f : P ¡ → P − Q is a Maurer-Cartan element in HomS (P ¡ , Q)
→
In this case,

¯
(C • (P, Q), d) ⊂ (HomS (P ¡ , Q), [f , −])
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, Poisson) :
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∨ and C • (BiLie, EndV ) :
3
Ciccoli-Guerra cohomology of Lie bialgebras
P = BiAss, not Koszul and C • (BiAss, EndV ) :
Gerstenhaber-Shack bicomplex ???

1      2
1   c 2
c
•        t t utt
vt uttt t
t • t u•t
v u tt
ÕWW −
•
Õ
•    •
1       2      1      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 HomS (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 ...

(HomS (P ¡ , EndV ), [f, −]) : cohomology of P∞ -algebras
Interpretation of H• in terms of deformations
f
→
• Operations on the deformation complex of P − Q

When P is a Koszul operad,

Hom(P ¡ , Q) ← Hom(P ¡ , P) ∼ P ! ⊗ P ← P ! ◦ P
=
Manin complex    Manin products
tangent complex

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 )

P ROOF.
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)

d ~
d~
d ~
d~   •d ~
d •
: ~d →     d
~
∂|X
~
•d
~ d•d :
~
• ~ d

~
∂ 2 = 0 =⇒ X is a homotopy coprop(erad).
(coassociative of the coprop(erad) holds ‘up to homotopy’)

Theorem (Merkulov-V.).
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
1
Maurer-Cartan elements :   n≥0 n! ln (f, . . . , f ) = 0
f
→
P − Q is a morphism of dg prop(erad)s iff
¯           f
f : C → P − Q is a Maurer-Cartan element in HomS (P ¡ , Q)
→
In this case,

¯
(C • (P, Q), d) ⊂ ( HomS (P ¡ , Q), lf )
twisted homotopy Lie algebra
Theorem (Merkulov-V.). For every minimal model of BiAss, the deformation
complex

C • (BiAss, Q) ∼ Q
=
and
2` .
uuu ` . . m−ssm
1
Ò1
uuu ÒÒs
`` ss
`
u
s
Ò
ÒÒ uuu
sss ``` uu
d          •        =

sss Ò . .
Ò.
1   2         n 1
−      n

1       2                −
m 1     m

2` .
uuu ` . . m−ssm                                    ÔXX Ô•XX
...
ÔXX Ô•XX
Ô•X ÔÔ X                    Ô•X ÔÔ X
1
uuu ÒÒs
`` ss  Ò1                                    Ô                           Ô
n−2
ÙsS
ssÒuuuu
u
S
i+1          •S
(−1)                                             +

sssÙÙ•X S ..u
m                       m
i=0
s.. Ù Ô X                                         r r ... u v
ru u                      iiU... × y
U
i Ô X                                            •u                         ii × y
1       Ô       n                                      quu vvv
q
•q
UUy
v                     y
y
××UU i
yiii
•

yy×...
i+1           i+2                           •
×
1                      −
23 n 1n
1           2               −
m 1       m

ÔXX ÔÔXX                       ÔXX ÔÔXX
...
Ô•X Ô•X
Ô                              Ô•X Ô•X
Ô
n+1
+ (−1)
m                      m
iiU... × y
U                      rr sss s•vs
ii × y
Uiy                    rr ... w v v
s
××ii
y
U                       rw•w
yy UU i
•

yy×...
•
×
12          n 1
−                  n

1                   iiU × y
U   −
23 m 1m
...
ii × y
U×y
XX i+2                                              wr
w• r
wssssrrr                    i
U
×yii
yy UU i
i+1
X Ô m                                            •                          •
1u i S •Ô
uuuSS Ô Ù s..s
..                                                v•s ...
v
vs                         yy×...
×
n−2                uu`ss s
SsÙ
ÙÙ
uuu                                               n                       n
i+1
s•`
ÒÒ u
sss. . `` uu
+     (−1)                                                +
XX Ô XX Ô
XÔ XÔ                     XX Ô XX Ô
XÔ XÔ
ss ÒÒ
i=0
.                                               •Ô •Ô                      •Ô •Ô
1       2               n 1
−     n
...
1       2                 −
n 1    n

iiU...m−y
1U1
×
1                      m
ii × y
UUy                        q
•q
××ii
•y
i
yy UU i                    vv uq r
vvuuu u•r
•
yy×...
×                        v ... u r
m+1                    n                       n
+ (−1)
XX Ô XX Ô
XÔ XÔ                        XX Ô XX Ô
XÔ XÔ
•Ô •Ô                         •Ô •Ô
...
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.).
i
/         s
(V, dV )   o           (W, dW )   h
p

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

[3] B. Vallette, Manin products, Koszul duality, Loday algebras and Deligne
conjecture, to appear in The Journal fur 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.