Rational homotopy of non-connected spacesRational homotopy of non-connected spaces Urtzi Buijs 5th GeToPhyMa CIMPA summer school on Rational Homotopy Theory and its interactions Celebrating

Rational homotopy of non-connected spaces

Urtzi Buijs

5th GeToPhyMaCIMPA summer school on Rational Homotopy Theory and its interactions

Celebrating Jim Stashe� and Dennis Sullivan

for their respective 80th and 75th anniversaries.

July 11-21, 2016

Rabat, Morocco

http://algtop.net/geto2016

These notes are based on the papers \Algebraic models of non-connected

spaces and homotopy theory of L∞ algebras" [5] and \Lie models of simplicial

sets and representability of the Quillen functor" [7] treated in the Sections 1

and 2 respectively.

There is also an introductory section on rational homotopy theory to present

the basic tools necessary for the rest of the notes. This introduction is mainly

based on [11, 21].

1 Introduction to rational homotopy theory

In 1967 Daniel Quillen published \Homotopical algebra" [22] where the author

presents a categorical framework in which make homotopy theory. Two years

later, in 1969, he applies these ideas in the fundamental paper \Rational homo-

topy theory" [23].

1.1 “Homotopical algebra”. Model categories

Definition 1.1. A model category is a category C together with three distin-

guished classes of maps

� weak equivalences∼ // ,

� �brations // // ,

1.1 “Homotopical algebra”. Model categories 2

� co�brations // // ,

satisfying the following axioms:

(1) (�nite) limits and colimits exist in C.

(2) Given morphisms Xf // Y

g // Z if two out of the maps f; g and

g ◦ f are weak equivalences, then so is the third.

(3) Suppose that f is a retract of g, i.e., there is a commutative diagram

A

f

��

// X

g

��

// A

f

��B // Y // B

;

with the two horizontal compositions being identities; Then, if g is a weak

equivalence, �bration or co�bration, then so is f .

(4) The lifting problem

A

i

��

// X

p

��B

>>

// Y

can be solved whenever i is a co�bration, p is a �bration, and at least one

of i and p is a weak equivalence.

(5) Every map f : X → Y can be factored in two ways

X

i

f // Y X

j

∼

f // Y

Z

p

∼?? ??

W

>> >>

In order to de�ne the notion of homotopy in a model category we need to

introduce some notation.

Let C be a model category. If f : A → C and g : B → C are given maps,

f + g : A ∪ B → C is the unique map de�ned by the universal property of the

coproduct. If h : C → A and k : C → B are given maps, (h; k) : C → A × B is

the unique map de�ned by the universal property of the product.

Definition 1.2. Let � be the initial object of C. An object X of C is co�brant if

�→ X is a co�bration.

Let e be the �nal object in C. An object X of C is �brant if X → e is a

�bration.

Definition 1.3. A cylinder (A× I; @0; @1; �) of an object A is given by an object

A× I called a cylinder object and maps

A@1

//@0 //

A× I � // A

such that �@0 = �@1 = idA, � is a weak equivalence and (@0+@1) is a co�bration.

1.1 “Homotopical algebra”. Model categories 3

Definition 1.4. A path structure (BI ; d0; d1; �) of an object B is given by an

object BI called a path object and maps

B� // BI

d1

//d0 //

B

such that d0� = @1� = idB , � is a weak equivalence and (d0; d1) is a co�bration.

Definition 1.5. Let f; g : A//// B be two maps in C. A left homotopy from f

to g is a map H : A×I → B such that H@0 = f , H@1 = g where (A×I; @0; @1; �)is a cylinder on A.

Af //g//

@1

��@0

��

B

A× IH

<<

A right homotopy from f to g is a map K : A → BI such that d0K = f ,

d1K = g where (BI ; d0; d1; �) is a path structure on B.

A

K

f //g// B

BI

d0

OO

d1

OO

If there exists a left (right) homotopy between f and g, we call f and g left

(right)homotopic .

Proposition 1.6. Let f; g : A//// B be two maps in C with A co�brant and B

�brant. If f and g are left (right) homotopic, a left (right) homotopy exists

for any cylinder (path structure). Moreover, f and g are left homotopic if

and only if they are right homotopic.

Definition 1.7. We call this relation homotopy and write f ' g. This is an

equivalence relation on the set of maps on C from A to B.

Definition 1.8. The cstegory hoC has

Objects: �brant and co�brant objects of C.

Maps: homotopy classes of maps of C.

Quillen also introduce the homotopy category of a model category C, de-

noted by HoC, obtained from C by formally inverting all weak equivalences:

Objects: Objects of C.

Maps: Maps of C plus formal inverses of any weak equivalence.

Theorem 1.9. The canonic functor C → HoC induce an equivalence of cate-

gories

hoC→ HoC:

1.2 Commutative differential graded algebras 4

Definition 1.10. A Quillen adjunction between two model categories C and D is

an adjunction

CF //

DGoo

(i.e., homC(FY;X) ∼= homD(Y;GY ) natural for all variables X in C and Y in

D.) such that the left adjoint F preserves co�brations and the right adjoint G

preserves �brations.

Every Quillen adjunction induces an adjunction between the associated ho-

motopy categories

HoCLF //

HoDRGoo

and the Quillen adjunction is called a Quillen equivalence if this induced ad-

junction is an equivalence of categories.

1.2 Commutative differential graded algebras

Definition 1.11. A graded algebra consists of a Z-graded vector space A =

⊕p∈ZAp together with a bilinear product

Ap ⊗ Aq → Ap+q

which is associative.

A di�erential graded algebra is a graded algebra A endowed with a linear

derivation d of degree +1 such that d2 = 0.

A commutative di�erential graded algebra (CDGA from now on) is a dif-

ferential graded algebra such

a · b = (−1)|a||b|b · a; for any a; b ∈ A:

Here |a| denotes the degree of a.An augmentation of CDGA's is a morphism of commutative di�erential

graded algebras " : (A; d) → (Q; 0). The augmentation ideal of (A; ") is the

kernel Ker", denoted by A.

A morphism f : (A; ") → (A′; "′) of augmented CDGA's is a degree 0 mor-

phim s of CDGA's such that " = "′ ◦ f .An augmented CDGA (A; d) is n-connected (cohomologically n-connected)

if Ap= 0 for p ≤ n. (H(A; d) = 0 for p ≤ n). We denote by CDGAn (CDGAcn)

the associated categories.

Example 1.12. (1) (Tensor product of graded algebras) If A and B are graded

algebras, then A⊗B is a graded algebra with multiplication

(a⊗ b)(a′ ⊗ b′) = (−1)|b||a′|aa′ ⊗ bb′:

Exercise 1. (i) Show that the above multiplication is associative,


(ii) Prove that with the di�erential

d(a⊗ b) = da⊗ b+ (−1)|a|

A⊗B is a di�erential graded algebra.

(iii) Show that if A and B are commutative then A⊗B is a CDGA.

(2) (Tensor algebra) For any graded vector space V , the tensor algebra TV is

de�ned by

TV =

∞⊕k=0

T kV; T kV = V ⊗ · · · ⊗ V︸︷︷︸k

;

where T 0V = Q. Multiplication is given by a · b = a⊗ b.Any linear map of degree zero from V to a graded algebra A extends to a

unique morphism of graded algebras, TV → A. Any degree k linear map

V → TV extends to a unique derivation of TV .

(3) (Free commutative graded algebras) Let V be graded vector space. The

elements v⊗w−(−1)|v||w|w⊗v, where v;w ∈ V generate an ideal I ⊂ TV .The quotient graded algebra

�V = TV=I

is called the free commutative graded algebra on V .

Exercise 2. Show that �(V ⊕W ) = �V ⊗ �W .

(4) (The simplicial de Rham algebra) De�ne ∗•, the simplicial de Rham al-

gebra, as the simplicial CDGA with n-simplices

∗n =�(t0; : : : ; tn; dt0; : : : ; dtn)(∑

i ti − 1;∑i dti

) ; |ti| = 0; |dti| = 1:

The face and degeneracy morphisms are the unique cochain algebra mor-

phisms

@i : ∗n+1 → ∗n and sj :

∗n → ∗n+1

satisfying:

@i : tk 7→

tk if k < i;

0 if k = i;

tk−1 if k > i;

and sj : tk 7→

tk if k < j;

tk + tk+1 if k = j;

tk+1 if k > j:

Theorem 1.13. [3] The category of commutative cochain algebras CDGA ad-

mits the structure of a model category where:


� Weak equivalences: quasi-isomorphisms, i.e., morphisms ' : (A; d)→(A′; d′) such that the induced map in cohomology

H∗ (') : H∗(A; d)∼= // H(A′; d′)

is an isomorphism.

� Fibrations: surjective morphisms,

� Co�brations: morphisms i : A→ X satisfying the extension problem

A

i

��

// B

' ��

X

�

>>

// C

In [13], Steve Halperin works with the category "-CDGAc0 of augmented

CDGA's cohomologically connected. The di�erence with the Bous�eld-Gugenheim

approach is that the co�brations can be explicitely determined by KS-extensions

[21, xII.3. (3)].

Definition 1.14. Let A be a commutative cochain algebra. The spatial realization

of A is the simplicial set

〈A〉 = HomCDGA(A;∗•):

There exists an adjunction

SimpSet∗(−) // CDGA〈−〉oo :

The contravariant functor ∗(−) : SimpSet→ CDGA takes rational homotopy

equivalences to quasi-isomorphisms, and it takes inclusions of simplicial sets to

surjections of cochain algebras.

The contravariant functor 〈−〉 : CDGA→ SimpSet takes co�brations of cochain

algebras to rational Kan �brations and it takes trivial co�brations to trivial �-

brations.

Theorem 1.15. The spatial realization functor induces an equivalence of cat-

egories

Ho(CDGAcof;f.t.1 )

' // Ho(SimpSetf.t.1 )

The left hand side denotes the homotopy category of co�brant, simply

connected, �nite type, commutative cochain algebras. The right hand side

denotes the homotopy category of simply connected rational Kan complexes

of �nite Q-type.

1.3 Differential graded Lie algebras 7

1.3 Differential graded Lie algebras

Throughout this notes we assume that Q is the base �eld.

Definition 1.16. A graded Lie algebra consists of a Z-graded vector space L =

⊕p∈ZLp together with a bilinear product called the Lie bracket denoted by

[−;−] such that [x; y] = −(−1)|x||y|[y; x] and

(−1)|x||z|[x; [y; z]

]+ (−1)|y||x|

[y; [z; x]

]+ (−1)|z||y|

[z; [x; y]

]= 0:

Here |x| denotes the degree of x.A di�erential graded Lie algebra is a graded Lie algebra L endowed with

a linear derivation @ of degree −1 such that @2 = 0.

A chain Lie algebra is a DGL such that L = L≥0, i.e. Ln = 0 for n < 0. A

chain Lie algebra is connected if L = L≥1.

The category of di�erential graded Lie algebras and its morphisms will be

denoted by DGL and the subcategory of connected DGL's by DGL1.

Example 1.17. (1) Every chain complex L can be seen as a DGL with [−;−] =0. We call these DGL's abelian.

(2) Every associative di�erential graded algebra A has a DGL structure ALie

with the commutator Lie bracket

[a; b] = ab− (−1)|a||b|ba:

(3) (Free Lie algebras) For any graded vector spave V , the free Lie algebra

L(V ) is characterized by the following universal property: there is a nat-

ural inclusion of graded vector spaces V ,→ L(V ) such that any morphism

of graded vector spaces V → L into a graded Lie algebra extends uniquely

to a morphism of graded Lie algebras L(V )→ L

V //

f !!

L(V )

f

��L:

The free Lie algebra L(V ) may be constructed as follows. Consider the

free associative tensor algebra on V

T (V ) =⊕k≥0

(V ⊗k); V ⊗0 = Q:

T (V ) becomes a Lie algebra with the commutator Lie bracket. Then

the free Lie algebra L(V ) is the smallest sub Lie algebra of T (V )Lie that

contains V ⊂ T (V ).

(4) (Products and coproducts) If L and M are DGL's, then the direct sum

L⊕M becomes a DGL with the coordinate-wise structure:

(L⊕M)n = Ln ⊕Mn;


d(x; y) = (dx; dy);[(x; y); (x′; y′)

]=([x; x′]L; [y; y

′]M

):

The direct sum together with the projections

L L⊕Moo // M

represent the product of L and M in the category DGL.

If L and M are DGL's the free product

L ∗M = L(L⊕M)=J

where J is the ideal generated by elements of the form

[x; y]L − [x; y]L; x; y ∈ L;

[z;w]L − [z;w]M ; x; y ∈M:

The free product together with the inclusions

L // L ∗M Moo

represent the coproduct of L and M in the category DGL.

Exercise 3. Show that if L = L(V ) and M = L(W ) are free as Lie algebras,

then

L ∗M = L(V ⊕W ):

Example 1.18. (Tensor product of a DGL and a commutative di�erential graded

algebra).

Let (L; @) be a DGL and (A; d) be a commutative di�erential graded algebra.

Exercise 4. Show that the tensor product L⊗A inherits a natural DGL struc-

ture with di�erential and Lie bracket given by:

D(x⊗ a) = @x⊗ a+ (−1)|x|x⊗ da;[x1 ⊗ a1; x2 ⊗ a2

]= (−1)|a1||x2|[x1; x2]⊗ a1a2:

Proposition 1.19. DGL1 is a model category.

Weak equivalences: quasi-isomorphisms, i.e., morphisms : (L; @)→(L′; @′) such that the induced map in homology H ( ) : H(L; @)

∼= // H(L′; @′)

is an isomorphism.

Fibrations: surjective morphisms

Co�brations: morphisms i : A→ X satisfying the extension problem

A

i

��

// B

' ��

X

�

>>

// C


Every object in DGL1 is �brant and the co�brant objects are the free Lie

algebras.

We can characterize the co�brations more explicitely:

Definition 1.20. A KQ-extension (Koszul-Quillen) is a sequence

L

##

� � // L ∗ L(V ) // //

∼=��

L(V )

L′

99

where the vertical map is an isomorphism of graded Lie algebras (no di�eren-

tial!).

Whenever L is a free Lie algebra L = L(W ) a KQ-extension is a sequence(L(W ); d

)� � //

(L(W ⊕ V ); D

)// //(L(V ); @

)where

Dw = dw; w ∈W; and Dv = @v +�; v ∈ V; � ∈ L+(W ) ∗ L(V ):

Definition 1.21. An object (L; @) of DGL1 is minimal if and only if:

(i) L is isomorphic as Lie algebras to a free Lie algebra L ∼= L(V ).

(ii) The di�erential of the generators of the free Lie algebra has no linear term

@(V ) ⊂ L≥2(V ).

Definition 1.22. Let (L′; @′) ∈ DGL1. If : (L; @)' // (L′; @′) is a quasi-

isomorphism we say that (L; @) is DGL-model of L′. If L′ = L(V ) we say

that it is a Quillen model and if it is minimal that it is a minimal Quillen

model.

Theorem 1.23. Every object in DGL1 has a minimal Quillen model

We can describe explicitely the homotopy in DGL1 by means of a path struc-

ture and a cylinder.

Path structure. Consider the free commutative di�erential graded algebra

�(t; dt) where |t| = 0, |dt| = −1 with d(t) = dt.

Let (L; @) be an object in DGL1. Consider the DGL

(M;d) = (L; @)⊗ �(t; dt)

product described in example 1.3.

Consider the projections

p0; p1 : (L; @)⊗ �(t; dt)→ (L; @);

characterized by p0(t) = 0; p1(t) = 1;


and the canonical injection

` : (L; @)→ (L; @)⊗ �(t; dt):

De�ne the DGL (L; @)I by

(LI)i =

Mi if i > 0;

Z(M0) if i = 0;

0 if i < 0;

Then, ((L; @)I ; p0; p1; `

)is a path structure on (L; @).

Then f; g : (L′; @′) //// (L; @) are homotopic if there exists such that

(L; @)⊗ �(t; dt)

p0

��p1

��(L′; @′)

77

g//

f //(L; @)

commutes.

Definition 1.24. Let f : (L; @)→ (L′; @′) be a DGL morphism,((L(V ); d);

)and(

(L(V ′); d′); ′)Quillen models of L and L′. Then,

f : (L(V ); d)→ (L(V ′); d′)

is a Quillen model of f (relative to and ′) if the following diagram is

commutative up to homotopy

(L; @)f // (L′; @′)

(L(V ); d)

'

OO

f

// (L(V ′); d′):

' ′

OO

If (L(V ); d) and (L(V ′); d′) are minimal then f is a minimal Quillen model of

f .

Proposition 1.25. Let f : (L; @)→ (L′; partial′) be a map in DGL1.((L(V ); d);

)and

((L(V ′); d′); ′

)two Quillen models of (L; @) and (L′; @′) respectively.

Then:

(i) There exists a Quillen model f of f relative to these models.

(ii) Two Quillen models f and g of f are homotopic.

(iii) If f; g : (L(A); @) //// (L′; @′) have as Quillen models f and g respec-

tively, then f ' g if and only if f ' g.


Definition 1.26. (L; @) and (L′; @′) have the same homotopy type if they have a

common Quillen model

(L; @) (L(V ); @)' oo ′

'// (L′; @′):

In [23] Quillen de�nes a sequence of adjunctions

SimpSet1

G //SGp1

W

ooQ• //

SCHA1G•

ooP• //

SLA1

U•

ooN //

DGL1

N∗oo

We brie y explain the elements of this composition following the main source

[23]:

(i)

SimpSet1

G //SGp1

W

oo

If K is a reduced simplicial set, GK is the simplicial group constructed by

Kan [14] playing the role of the loop space of K. If G is a simplicial group

WG is the simplicial set which acts as its \classifying space". [14, 1, 8]

(ii)

SGp1

Q• //SCHA1

G•

oo

If G is a group then QG is the complete Hopf algebra obtained by complet-

ing the group ring QG by the powers of its augmentation ideal. If R is a

complete Hopf algebra, then GR is its group of group-like elements. These

functors are extended dimension-wise to simplicial groups and simplicial

CHA's and denoted by the same letters.

(iii)

SCHA1

P• //SLA1

U•

oo

If G is a Lie algebra over Q, UG is the CHA obtained by completing the

universal enveloping algebra Ug by powers of its augmentation ideal. If R

is a CHA, then PR is its Lie algebra of primitive elements. These functors

are applied dimension-wise to simplicial objects.

(iv)

SLA1

N //DGL1

N∗oo

If L is a simplicial Lie algebra, its complex of normalized chains NL is a

DGL with bracket de�ned by means of the Eilenberg-Zilber map ⊗. N∗is the left adjoint of N .

1.4 The bridge between DGL’s and CDGA’s 12

The composition NP•Q•G is called the � Quillen functor.

Theorem 1.27. The above sequence of adjunctions induce an equivalence of

categories

Ho(DGLcof1 )' // Ho(SimpSetQ;1)

The left hand side denotes the homotopy category of co�brant, 1-connected

di�erential graded Lie algebras. The right hand side denotes the homotopy

category of simply connected rational Kan complexes.

1.4 The bridge between DGL’s and CDGA’s

In the fundational paper, Quillen gave another adjunction between di�eren-

tial graded Lie algebras and cocommutative di�erential graded coalgebras. Let

de�ne �rst the necessary objects.

A graded coalgebra C is a graded vector space C together with two linear

maps of degree 0: A comultiplication �: C −→ C ⊗ C and an augmentation

� : C −→ Q such that (� ⊗ id)� = (id ⊗�)� y (id ⊗ �)� = (� ⊗ id)� = idC ,

i.e., the following diagrams are commutative

C� //

�

��

C ⊗ C

id⊗��

C ⊗ C�⊗id// C ⊗ C ⊗ C

C ⊗ C

id⊗��

C

idC

��

� //�oo C ⊗ C

�⊗id��

C ⊗Q // C Q⊗ Coo

A morphism ' : C −→ C ′ of graded coalgebras is a linear map of degree 0

such that ('⊗ ')� = �′' and � = �′'. A graded coalgebra is cocommutative

if �� = �, where � : C ⊗C −→ C ⊗C is the involution a⊗ b 7→ (−1)|a||b|b⊗ a.A graded coalgebra is coaugmented by the choice of an element 1 ∈ C0 such

that �(1) = 1 and �(1) = 1⊗ 1. Given such a choice, the above relations imply

that for a ∈ Ker �,

�a− (a⊗ 1 + 1⊗ a) ∈ Ker �⊗Ker �:

An element, a, in a coaugmented coalgebra is called primitive if a ∈ Ker �

and �a = a⊗1+1⊗a. The primitive elements constitute a graded subspace of

Ker �, and a morphism of augmented graded coalgebras send primitive elements

to primitive elements.

For a coaugmented graded coalgebra we write C = Ker � and therefore

C = Q⊕ C.A coderivaci�on of degree k in a graded coalgebra C is a linear map � : C −→

C of degree k such that �� = (� ⊗ id+ id⊗ �)� y �� = 0.

1.4 The bridge between DGL’s and CDGA’s 13

A di�erntial graded coalgebra (CDGC from now on) is a graded coalgebra

C together with a di�erential which is a coderivation on C.

If C is a graded coalgebra, then C∨ = Hom(C;Q) is a graded algebra with

multiplication de�ned by

(f · g)(c) = (f ⊗ g)(�c); f; g ∈ Hom(C;Q); c ∈ C

and with identity given by the map � : C −→ Q. If (C; d) is a CDGC , then

C∨ = Hom(C;Q) es un graded di�erential algebra.

Example 1.28. (1) (primitively cogenerated coalgebras) The reduced co-

multiplication �: C −→ C ⊗ C is de�ned by �c = �c− (c⊗ 1 + 1⊗ c).Its kernel is the graded subspace generated by the primitive elements.

Write now �(0)

= idC , �(1)

= � and de�ne the n-th reduced diagonal

�(n)

= (�⊗ id⊗ · · · ⊗ id) ◦�(n−1): C −→ C ⊗ · · ·C (n+ 1 factors). We

say that C is primitively cogenerated if C = ∪nKer �(n)

.

(2) The main example for us of a cocommutative graded coalgebra primitively

cogenerated is �V which comultiplication � id the unique morphism of

algebras such that �(v) = v ⊗ 1 + 1 ⊗ v, v ∈ V . It is augmented by

� : �+V −→ 0, 1 7→ 1 and coaugmented by Q = �0V . It is trivially

cocommutative.

Exercise 5. Show that �V is primitively cogenerated

(3) Among the cocommutative graded coalgebras primitively cogenerated, �V

has an important universal property. Let � : �+V −→ V be the surjective

linear map de�ned by a− �a ∈ �≥2V .

Lemma 1.29. If C = Q⊕C is a cocommutative graded coalgebra prim-

itively cogenerated, then for any linear map of degree 0: f : C −→ V

lifts to a unique morphism of grade coalgebras ' : C −→ �V such that

�'|C = f .

Theorem 1.30. [23] There is an adjunction

DGLC∗ //

CDGCLoo

Let (C;�; �) be a cocommutative di�erential graded coalgebra. L is de�ned

by

L(C) = (L(s−1C); @ = @1 + @2);

where

@1(s−1c) = −s−1�c;

@2(s−1c) =

1

2

∑i

(−1)|ai|[s−1ai; s−1bi];

where c ∈ C and �c =∑i ai ⊗ bi.

2 L∞-algebras and the cochain functor 14

Let (L; [−;−]; @) be a di�erential graded Lie algebra. C∗, the Cartan-

Eilenberg-Chevalley construction, is de�ned by

C∗(L) = (�sL; � = �1 + �2);

where

�1(sx1 ∧ · · · ∧ sxk) = −k∑i=1

(1)nisx1 ∧ · · · ∧ s@(xi) ∧ · · · ∧ sxk

�2(sx1 ∧ · · · ∧ sxk) =∑i<j

(1)|sxi|+nijs[xi; xj ] ∧ sx1 ∧ : : : sxi : : : sxj · · · ∧ sxk

where the signs are given by ni =∑j<i |sxj |, and

sx1 ∧ · · · ∧ sxk = (−1)nijsxi ∧ sxj ∧ sx1 ∧ : : : sxi : : : sxj · · · ∧ sxk:

In fact, with the appropriate restrictions we have an equivalence between

the corresponding homotopy categories of DGL's and CDGC's. Moreover, if we

consider �nite type DGL's and �nite type CDGC's we can connect DGL's with

CDGA's

DGL f.t.

C∗

&&C∗ // CDGCf.t. (−)∨ //Loo CDGAf.t.

(−)∨oo

where the composition C∗ = (−)∨ ◦ C∗ is called the cochain functor.

The whole picture can be summarized in the following diagram:

Necessary: 1-connected. Non necessary: �nite type.

SimpSet1

G //SGp1

W

ooK• //

SCHA1G•

ooP• //

SLA1

U•

ooN //

DGL1

N∗oo

C∗

��CDGC1

L

OO

(−)∨

��SimpSet1

APL //CDGA1

(−)∨OO

〈−〉oo

;

Necessary: �nite type. Non necessary: 1-connected (nilpotent is good enough).

2 L∞-algebras and the cochain functor

The main goal of the present section is to extend the de�nition of the cochain

functor of di�erential graded Lie algebras to the more general setting of (non-

bounded) L∞-algebras. Then, using the Sullivan realization functor de�ne a

geometrical realization for L∞-algebras.


Recall from the previous section that there is a functor

DGLf.t.1C∗ // CDGAf.t.

1 :

If (L; @) ∈ DGLf.t.1 , then

C∗(L) = Hom(C∗(L);Q) = C∗(L)∨ (1)

This cochain CDGA turns out to be in fact a Sullivan algebra

Lemma 2.1. If (L; @) is a connected chain Lie algebra and each Li is �nite

dimensional, then

� : �(sL)∨∼= // C∗(L)

is an isomorphism of graded algebras which exhibits C∗(L) as a Sullivan

algebra

Proof.

Exercise 6. check all the details of the proof given in ref

In fact, we can describe the di�erential of the cochain algebra C∗(L) in terms

of the di�erential and the Lie bracket of the DGL (L; @).

Proposition 2.2. If (L; @) is a connected chain Lie algebra of �nite type then:

(1) C∗(L) = (�V; d) with V and sL dual graded vector spaces.

(2) d = d1 + d2 is the sum of its linear and quadratic parts where

〈d1v; sx〉 = (−1)|v|〈v; s@x〉; x ∈ L; v ∈ V

〈d2v; sx ∧ sy〉 = (−1)|y|+1〈v; s[x; y]〉; x; y ∈ L; v ∈ V:

Proof.

Exercise 7. Check carefully all the details of the proof given in

The main goal of the present section is to extend the de�nition of the cochain

functor to the setting of L∞-algebras.

An L∞-algebra on a graded vector space L is a collection of degree k − 2

linear maps `k : L⊗k → L, for k ≥ 1, satisfying the following two conditions:

(i) For any permutation � of k elements,

`k(x�(1); : : : ; x�(k)) = ��`k(x1; : : : ; xk);

where �� is the signature of the permutation and � is the sign given by

the Koszul convention.


(ii) The generalized Jacobi identity holds, that is∑i+j=n+1

∑�∈S(i;n−i)

��(−1)i(n−i)`n−i(ì(x�(1); : : : ; x�(i)); x�(i+1); : : : ; x�(n)) = 0;

where S(i; n− i) denotes the set of (i; n− i)-shu�es.

Equivalently, and L∞-algebra structure on the graded vector space L can be

seen as a di�erential graded coalgebra structure on �+sL, the cofree graded

cocommutative coalgebra generated by the suspension of L.

Exercise 8. [11, Lemma 22.2]

Suppose g : �V → V is a linear map of some arbitrary degree, k ≥ 1.

De�ne �g : �V → �V by

�g(v1∧· · ·∧vn) =∑

1≤i1<···<ik≤n

±g(vi1 ∧· · ·∧vik)∧v1∧ : : : i1 : : : ik · · ·∧vn: (2)

where − means delete and the sign ± is given by v1 ∧ · · · ∧ vn = ±vi1 ∧ · · · ∧vik ∧ v1 ∧ : : : i1 : : : ik · · · ∧ vn.

Prove the following assertions:

(1) �g decreases wordlength by k − 1.

(2) �g is a coderivation in �V .

(3) �g is the unique coderivation that extends g and decreases wordlength

by k − 1.

Exercise 9. Let � : �sL→ �sL be a coderivation. Write �� =∑k≥1 hk, where

hk : �ksL→ sL, k ≥ 1.

Prove the following:

(1) The \in�nite" sum �� =∑k≥1 hk is well de�ned.

(2) The collection of degree k − 2 linear maps

`k = s−1 ◦ hk ◦ s⊗k : L⊗k → L;

de�nes an L∞-algebra structure on the graded vector space L.

(3) Reciprocally, if (L; {`k}) is an L∞-algebra. Then, the collection of

linear maps

hk = (−1)k(k−1)

2 s ◦ `k ◦ (s−1)⊗k : �ksL→ sL

de�nes a coderivation � in the cofree coalgebra �sL.

If (L; {`k}) is an L∞ algebra, we denote the di�erential coalgebra structure

on �sL as C∗(L) by analogy with the Cartan-Eilenberg-Chevaley construction.

Given two L∞-algebras L and L′, a morphism of L∞-algebras is a cocom-

mutative di�erential graded coalgebra morphism

f : C∗(L)→ C∗(L′):


f is determined by �f : �sL → sL′ which can be written as∑k≥1(�f)

(k).

Note that, as before, the collection of linear maps {(�f)(k)}k≥1 is in one to one

correspondence with a system {f (k)} of skew-symmetric maps of degree 1− k,where f (k) : L⊗k → L′. Indeed, each f (k) is uniquely determined by (�f)(k) as

follows:

f (k) = s−1 ◦ (�f)(k) ◦ s⊗k;

(�f)(k) = (−1)k(k−1)

2 s ◦ f (k) ◦ (s−1)⊗k:

Exercise 10. Write explicitely the equations that the linear maps f (1) and

f (2) must satisfy.

For which L∞-algebras (L; {`k}k≥1) can we give an analogue of Proposition

2.2?

Definition 2.3. An L∞-algebra (L; {`k}k≥1) is mild if every bracket is locally

�nite, i.e. for any a ∈ L there are �nite dimensional subspaces Sk ⊂ L⊗k, k ≥ 1

which vanish for k� 0 and such that `−1k 〈a〉 ⊂ Ker`k ⊕ Sk.

Definition 2.4. Given a mild L∞-algebra (L; {`k}k≥1), choose a homogeneous

basis {zi} of L and denote by V ⊂ (sL)∨, V = 〈{vi}〉, where vi(szr) = �ri .

Then de�ne

C∗(L) = (�V; d); where the di�erential satis�es

d =∑k≥1

dk; dk(V ) ⊂ �kV

〈dkv; sx1 ∧ · · · ∧ sxk〉 = ±〈v; s`k(x1; : : : ; xk)〉: (3)

Exercise 11. (1) Show that the free commutative di�erential graded algebra

C∗(L) is well de�ned when L is a mild L∞-algebra.

(2) Prove that (L; {`k}k≥1) is a mild L∞-algebra if and only if for each vi(�xed)

〈vi; s`k(zj1 ; : : : ; zjk)〉 = 0

for almost all zj1 ⊗ · · · ⊗ zjk ∈ L⊗k, k ≥ 1.

(3) We say that a free commutative graded algebra (�V; d) is mild if for

each zj1 ⊗ · · · ⊗ zjk ∈ L⊗k (�xed) we have

〈dkvi; zj1 ∧ · · · ∧ zjk〉 = 0

for almost all vi ∈ V .Show that the L∞-algebra (L; {`k}k≥1) is mild if and only if the free

CDGA C∗(L) is mild.

(4) Give an example of a non-mild free CDGA (�V; d) and show that it

can not be of the form C∗(L) for an L∞-algebra L.


C∗(−) does not de�ne a functor unless we also restircts the class of L∞-

morphisms.

Definition 2.5. An L∞-morphism

� : (�sL; �)→ (�sM; �′)

is mild if every �(k) : �ksL→ sM is locally �nite, i.e. for any a ∈M there is a

�nite dimensional subspace Sk ⊂ L⊗k, k ≥ 1 with Sk = 0, k� 0, such that

(�(k))−1〈a〉 = Ker�(k) ⊕ Sk:

Definition 2.6. If � is a mild L∞-morphism, de�ne

C∗(�) : C∗(M) = (�W;d)→ (�V; d) = C∗(L);

with C∗(�) =∑k≥1 C

∗(�)k where C∗(�)k : W → �kV is given by

〈C∗(�)k; sx1 ∧ · · · ∧ sxk〉 = ±〈w; s�(k)(x1 ⊗ · · · ⊗ xk)〉: (4)

Exercise 12. (1) Rewrite the de�nition of a mild L∞-morphism in an ana-

logue way as in the previous exercise.

(2) Show that if C∗(M) = (�W;d) and C∗(L) = (�V; d) are mild CDGA's

not all CDGA morphism ' : (�W;d)→ (�V; d) can be written as C∗(�)

for a mild L∞-morphism �.

(3) De�ne properly the concept of a mild CDGA morphism verifying that

� is a mild L∞-morphism if and only if ' = C∗(�) is a mild CDGA

morphism.

We denote by Lmild∞ the category of mild L∞-algebras and mild L∞-morphisms.

The next remarks are a discussion with examples of the implications between

mildness and other conditions classically used for L∞-algebras

Remark 2.7.

FINITE TYPE + BOUNDED < MILD

Exercise 13. (1) ;. Find a non-mild, �nite type and bounded L∞-algebra

L.

(2) Find a mild, non-�nite type and non-bounded L∞-algebra L.

Remark 2.8.

WHY NOT DEFINE C∗(L) = C∗(L)∨ ?

In general

(�sL)∨ � �(sL)∨

unless very strict restrictions are assumed. Recall by Lemma 2.1 that if V is a

graded vector space of �nite type, bounded and with V0 = 0 then (�V )∨ ∼= �V ∨.

Exercise* 14. Show that if V is a graded vector space of �nite type, bounded

and with V0 6= 0, then (�V )∨ � �V ∨.

2.1 The connected components in the CDGA setting 19

Remark 2.9.

NILPOTENT 6= MILD.

The lower central �ltration on an L∞-algebra is de�ned inductively by

F 1L = L; F iL =∑

i1+···+ik=i

`k(Fi1L; : : : ; F ikL); i > 1:

L is nilpotent if F iL = 0 for i� 0.

Exercise 15. (1) Compute (�V; d), the Sullivan minimal model of S3 ∨S3(until generators of degree 11).

(2) Show that the associated L∞-algebra L such that C∗(L) = (�V; d) is

not nilpotent.

Finally de�ne the realization functor 〈−〉 : Lmild∞ → SimpSet as the composi-

tion:

Lmild∞

C∗ $$

〈−〉 // SimpSet

CDGA

〈−〉S

::

2.1 The connected components in the CDGA setting

Let K be a simplicial set. Given a 0-simplex x0 ∈ K0, we de�ne the connected

component of K containing x0 as the simplicial subset Kx0 of K where

(Kx0)q = {x ∈ Kq | @q−i@ix = x0 for any i }:

Denote also by x0 the point in |K| identi�ed with the 0-simplex x0, and by

|K|x0 the component which contains the point x0. We have then

Lemma 2.10. (1) If K is a Kan complex, there exists an homotopical equiv-

alence Kx0' // S∗(|K|x0) which makes the following diagram com-

mutative

Kx0� _

��

' // S∗(|K|x0)

��K '

// S∗|K|:

In particular |K|x0 ' |Kx0 |.

(2) Given a homotopical equivalence h : K' // L between simplicial

sets, and x0 ∈ K0, then the restriction Kx0// Lhx0 is a homo-

topical equivalence whose geometric realization corresponds with the

equivalence |K|x0' // |L|hx0 .


�Let (�V; d) be a CDGA where V = ⊕p∈ZVp. Its realization 〈�V; d〉S ' X is

a non-necessarily connected simplicial set.

Consider a 0-simplex of this simplicial set ' ∈ 〈�V; d〉S = HomCDGA

((�V; d); (APL)0

)=

HomCDGA

((�V; d);Q

). This 0-simplex represents a map

∗ ,→ X;

◦

•

◦• ,,

Figure 1

or equivalently a connected component.

We can describe algebraically in terms of Sullivan algebras, the di�erent

connected components of the (non-connected) simplicial set X.

Associated to the CDGA morphism ' : (�V; d)→ Q we have in (�V; d) the

ideal K' generated by A1 ∪ A2 ∪ A3 where

A1 = (�V )<0; A2 = d(�V )0; A3 = {�− '(�) | � ∈ (�V )0}:

Lemma 2.11. The ideal K' agrees with the ideal K ′' generated by A′1∪A′2∪A′3,where

A′1 = V <0; A′2 = d(V 0); A′3 = {v − '(v) | v ∈ V 0}:

Proof. The inclusion K ′' ⊂ K' is trivial since A′i ⊂ Ai, i = 1; 2; 3. Let's check

the inclusion K' ⊂ K ′'.If � ∈ A1, then � ∈ �+(V <0) · �V ⊂ K ′'.Consider � = � − '(�) ∈ A3, with � ∈ (�V )0. Write � = a + b where

a ∈ �+(V <0) · (�V ) and b ∈ �W 0. Then, � − '(�) = a + b − '(a) − '(b) =a+b−'(b). Since a ∈ A1 ⊂ K ′' it only remains to show that b−'(b) ∈ K ′'. We

will suppose that b ∈ �nV 0 and proceed by induction. If n = 1, then b ∈ V 0

and b− '(b) ∈ A′3 ⊂ K ′'. Suppose that b = b1 · · · bn with bi ∈ V 0, i = 1; : : : ; n.

Then

b− '(b) = b1

((b2 · · · bn)− '(b2 · · · bn)

)+ '(b2 · · · bn)

(b1 − '(b1)

);

which obviously belongs to K ′' by inductive hypothesis.

Finally, consider an element � = d� ∈ A3 with � ∈ (�V )0, and write

� = a + b + c where a ∈ �+(V <−1) · �V , b ∈ �(V 0) and c ∈ V −1 · �V .


Clearly, da; db ∈ K ′'. We can write c as a sum of terms of the form c1 · c2with |c1| = −1. Then dc =

∑(dc1 · c2 − c1 · dc2). Thus, on the one hand

c1 · dc2 ∈ K ′' trivially. On the other hand '(dc1) = d('c1) = 0 and therefore

dc1 · c2 =(dc1 − '(dc1)

)· c2 ∈ K ′'

Another important property of K' is the following

Lemma 2.12. K' is a di�erential ideal of (�V; d).

Proof. By Lemma 2.11, in order to prove that dK' ⊂ K', it su�ces to show

that dA′i ⊂ K ′' = K', i = 1; 2; 3.

Let v ∈ A′1 = V <0. If v ∈ V −1, then dv ∈ (�V )<0 ⊂ K'. If w ∈ V −1, since'(dv) = d'(v) = 0, we have dv = dv − '(dv) ∈ K'.

If v ∈ A′3, then v = x− '(x) for some x ∈ V 0 and dv = dx− d'(x) = dx ∈K'. Finally, dA

′2 = 0.

It turns out that (�V; d)=K' is isomorphic to a free CDGA. To prove that

consider the ideal K' generated only by A1∪A3. Following the proof of Lemma

2.11 it is easy to see that this ideal agrees with the one generated by A′1 ∪ A′3.

Lemma 2.13. The map

� : V 1 �� // (�V )1 // (�V=K')

1;

is an isomorphism of vector spaces.

Proof. The map is clearly injecitve by de�nition of K'. Let [�] ∈ (�V=K')1

and write � = �0 +�1 +�2 where

�0 ∈ �+V <0 · (�V ); �1 ∈ (�+V 0) · V 1; �2 ∈ V 1:

In order to simplify the notation we will write the elemt �1 as �1 = � · where

� ∈ (�+V 0) and ∈ V 1. Then '(�) + �1 ∈ V 1 and

�('(�) + �1

)= ['(�) + �1] = [�]:

Consider now the linear map

@ : V 0 d // (�V )1 // // (�V=K')1 �−1 // V 1;

and denote by V1a complement of the image of @, i.e. V 1 = @V 0 ⊕ V 1

.

In what follows, if we have an element of the form � = � · ∈ �+V 0 ·�V we

will denote by �=' the element '(�)·. With this notation if dv = �0+�1+�2

where

�0 ∈ �+V <0 · (�V ); �1 ∈ (�+V 0) · V 1; �2 ∈ V 1;

then @(v) = �1='+�2.

Then we have:


Proposition 2.14. There exists a di�erential d' such that we have an isomor-

phism of CDGA's

(�V=K'; d) ∼= (�(V1 ⊕ V ≥2); d'):

Proof. We �rst de�ne a morphism of graded algebras : �V → �(V1 ⊕ V ≥2)

by

(v) =

0 if v ∈ V <0 ⊕ @V 0;

'(v) if v ∈ V 0;

v otherwise.

This morphism is clearly surjective and we will show that its kernel is precisely

K':

In order to show that (K') = 0 it is enough to see (A′i) = 0, i = 1; 2; 3.

It is immediate that (A′1) = 0. For an element v − '(v) ∈ A′3, we have

v ∈ V 0 and then

(v − '(v)

)= (v)− '(v)

= '(v)− '(v) (1)= '(v)− '(v) = 0:

Finally, if |v| = 0, then the element dv ∈ A′2 can be written as

dv = �0 +�1 +�2; �0 ∈ �+V <0 · (�V ); �1 ∈ (�+V 0) · V 1; �2 ∈ V 1;

or equivalently,

dw = �0 +�1 − �1='+�1='+�2

= �0 +�1 − �1='+ @v;

and we have (sw) = (�0) + (�1 − �1=') + (@v) = 0. Indeed, (�0) =

(@v) = 0 by de�nition.

Now, if we write �1 = � · w with � ∈ �+V 0 and w ∈ V 1, then

(�1 − �1=') = (� · w − '(�) · w

)=

(�− '(�)

)· w = 0:

On the other hand we will show that any element � ∈ ker belongs to K':

Write

� = �0 +�1 +�2

where �0 ∈ �+(V <0 ⊕ @V 0) · �V , �1 ∈ �+V 0 · �(V 1 ⊕ V ≥2) and �2 ∈ �(V1 ⊕

V ≥2). Then,

0 = (�) = (�0 +�1 +�2) = (�1) + �2 = �1='+�2:

Take a basis {�i} in �(V1 ⊕ V ≥2) and write

�1 =∑

�i�i; �i ∈ �+V 0;

�2 =∑

�i�i; �i ∈ Q:

2.2 Points, augmentations and Maurer-Cartan elements 23

Thus,

0 = �1='+�2 =∑

'(�i)�i +∑

�i�i =∑(

'(�i)− �i)�i;

so we conclude that '(�i) = �i and then �1 + �2 =∑(

�i − '(�i))�i ∈ K'.

Since �0 ∈ K' by de�nition, we obtain = 0 +1 +2 ∈ K'.

Therefore, induces an isomorphism of graded algebras making the fol-

lowing diagram commutative:

�V

%%

// // �V=K'

∼=��

�(V1 ⊕ V ≥2)

Finally, we endow �(V1 ⊕ V ≥2) with the di�erential d' = ◦ d ◦ −1.

Exercise 16. Write an explicit formula for the di�erential d' = ◦ d ◦ −1.

Theorem 2.15. The projection (�V; d)→ (�V=K'; d) induces an homotopical

equivalence

〈�V=K'〉' // 〈�V 〉'

making the following diagram commutative

〈�V 〉' �� // 〈�V 〉

〈�V=K'〉

'

OO

// 〈�V 〉;

where 〈−〉 stands for the Sullivan realization 〈−〉S.

�In conclusion, we have that the Sullivan algebra

(�V1 ⊕ V ≥2; d')

is a model of the component of X = 〈�V 〉S which contains the 0-simplex

' : �V → Q.

2.2 Points, augmentations and Maurer-Cartan elements

We have seen that augmentations ' : (�V; d)→ Q represent maps as in Figure

1. The problem to \model" this �gure with DGL's (or L∞-algebras) is that the

morphisms represent base-point preserving maps and then the unique inclusion

would be


◦

•

◦•

44

Figure 2

So if we want to describe algebraically in the DGL/L∞-algebra setting the

map of Figure 1 we need to factor the map through the inclusion of the singleton

in S0

◦

•

◦• ,,

WW

Figure 3

◦ •

))

��

Then if we want to \model" this �gure, we need �rst a model for S0.

Definition 2.16. A Maurer-Cartan element of an L∞-algebra L is an element

z ∈ L−1 for which `k(k

z; : : : ; z) = 0 for k su�ciently large and∑k≥1

1

k!`k(

kz; : : : ; z) = 0:

Observe that, whenever (L; @) is a DGL, i.e., an L∞-algebra such that `k = 0

for k ≥ 3, then z ∈ L−1 is a Maurer-Cartan element if

@z = −12[z; z]:

We will denote the set of Maurer-Cartan elements in L by MC(L).

Exercise 17. (The DGL model of S0). Consider the free diferential graded

Lie algebra L(u) with |u| = −1 and with a di�erential that makes u a

Maurer-Cartan element @u = − 12 [u; u].

(1) Show that the cochain functor on this DGL is

C∗(L(u); @

)∼=(�(x; y); d

);


where x and y are generators of degrees 0 and −1 respectively, dx = 0

and dy = 12 (x

2 − x).

(2) Show that the geometric realization of(�(x; y); d

)has the homotopy

type of S0.

In the CDGA setting we can model the map

•

WW

Figure 4

◦ •

By the CDGA morphism

� :(�(x; y); d

)→ Q; �(x) = 1; �(y) = 0:

Then, if we de�ne a based augmentation of a given CDGA A by a morphism

A →(�(x; y); d

)it is clear that the composition of any based augmentation

with � :(�(x; y); d

)→ Q gives rise to a classical augmentation A → Q. Con-

versely, we have the following.

Lemma 2.17. Let (�V; d) be a free CDGA and let � ∈ �+x such that �(�) =

1. Then, any augmentation f : (�V; d) → Q has a unique lifting f� to(�(x; y); d

)such that, for any v ∈ V 0,

f�(v) = f(v)�:

◦

•

◦• ,,

WW

Figure 5

◦ •

))

��

(�(x; y); d

)�

��Q (�V; d)

foo

f�ee

Proof. For degree reasons we set f� to be zero in V ≥1 and V ≤−2. Let w ∈ V −1and write dw = � + �, where � ∈ �+V 0 and � ∈ �+V 6=0 · (�V ). Then,


f(dw) = f(�). Write � = p(v1; : : : ; vn) as a polynomial without constant term

in the generators of V 0, and set �i = f(vi), for i = 1; : : : ; n. Then,

p(�1; : : : ; �n) = fd(w) = df(w) = 0:

On the other hand,

f�(dw) = p(�1�; : : : ; �n�

)= P (x);

which is a polynomial in x without constant term, and it satis�es P (1) =

p(�1; : : : ; �n) = 0. Hence P (x) = x(x − 1)r(x) and we de�ne f�(w) = 2yr(x)

so that df�(w) = f�(dw).

Finally, we check that, for any generator u ∈ V −2, f�(du) = 0. Indeed,

write f�(du) = yQ(x) whose di�erential 12 (x

2 − x)Q(x) has to vanish. Thus

Q(x) = 0 and the lemma holds.

Following the same spirit of algebraically describe Figure 3 in the DGL/L∞setting we have the following.

Lemma 2.18. Let L be an L∞-algebra. Then, for any z ∈ L−1, there exists aunique L∞ morphism � : (L(u); @)→ L such that

�(1)(u) = z;

�(k)(u⊗ : : :⊗ u) = 0; k ≥ 2:

Moreover, z ∈MC(L) if and only if �(k)([u; u]⊗ u⊗ : : :⊗ u) = 0 for k large

enough.

Proof. Since (L(u); @) is the vector space spanned only by u and [u; u], with

@u = − 12 [u; u], an L∞ morphism � : L(u)→ L is simply a CDGC morphism,

� : (�(su; s[u; u]); �) −→ (�sL; �);

which is completely determined by the elements

�(k)(u⊗ : : :⊗ u); �(k)([u; u]⊗ u⊗ : : :⊗ u); k ≥ 1;

satisfying the system referred in Exercise 10. In this particular case, if we set

�(1)(u) = z; �(k)(u⊗ : : :⊗ u) = 0; k ≥ 2;

and since ì = 0, for i ≥ 3 in (L(u); @), a direct computation shows that � is

indeed an L∞ morphism if the following identities hold for any k ≥ 1,

`k(z; : : : ; z) =

(k

2

)�(k−1)([u; u]⊗ u⊗ · · · ⊗ u)− k

2�(k)([u; u]⊗ u⊗ · · · ⊗ u);

k∑j=1

(k − 1

j − 1

)`j

(�(k−j+1)([u; u]⊗ u⊗ · · · ⊗ u); z; j−1: : : ; z

)= 0: (5)


We will show that �(k)([u; u]⊗ u⊗ : : :⊗ u), satisfying the above identities,

are uniquely determined by the formula

�(k)([u; u]⊗ u⊗ · · · ⊗ u) = −2(k − 1)!

k∑i=1

1

i!ì(z; : : :; z): (6)

First of all, for k = 1, the �rst identity in (5) is simply

`1z = −1

2�(1)[u; u]:

Thus, we are forced to de�ne

�(1)[u; u] = −2`1(z);

as in (6). The second identity in (5) for k = 1 reads `1�(1)[u; u] = 0 which is

trivially satis�ed:

`1�(1)[u; u] = −2`21(z) = 0:

Assume the identities in (5) are satis�ed for k − 1 by setting formula (6) for

integers smaller than k.

Again, from the �rst identity in (5) for k, we are forced to de�ne

�(k)([u; u]⊗u⊗ :::⊗u) = (k− 1)�(k−1)([u; u]⊗u⊗ :::⊗u)− 2

k`k(z; :::; z): (7)

Now, by the inductive hypothesis for k − 1, this expression becomes

− 2(k − 1)(k − 2)!

k−1∑i=1

1

i!ì(z; : : : ; z)−

2

k`k(z; : : : ; z)

= −2(k − 1)!

k∑i=1

1

i!ì(z; : : : ; z);

which is precisely the equation (6) for k. To �nish, we must check that the

second identity in (5) for k,∑kj=1

(k−1j−1)`j

(�(k−j+1)([u; u]⊗ u⊗ · · · ⊗ u); z; j−1: : : ; z

)= 0

holds.

For it, replace in this equation �(k−j+1)([u; u]⊗ u⊗ · · · ⊗ u) by its value on

equation (7) above for k − j + 1. This yields the following, in which we have

avoid the ⊗ sign for simplicity:∑kj=1

(k−1j−1)`j

(((k − j)�(k−j)([u; u]u:::u)− 2

k−j+1`k−j+1(z; :::; z)); z; j−1::: ; z

):

Then, this expression splits as

(k − 1)

k−1∑j=1

(k − 2

j − 1

)`j

(�(k−j)([u; u]u· · ·u); z; j−1: : : ; z

)


−2k

k∑j=1

(k

j − 1

)`j(`k−j+1(z; : : : ; z); z;

j−1: : : ; z)):

By induction hypothesis the �rst summand is zero as it is the second identity

in (5) for (k − 1). The second summand is also zero by the kth higher Jacobi

identity on L.

Now we prove the second assertion. If z ∈ MC(L), then there is an integer

N such that `k(z; : : : ; z) = 0 for k ≥ N . Therefore, via equation (6), and for

k ≥ N ,

�(k)([u; u]⊗ u⊗ · · · ⊗ u) = −2(k − 1)!

∞∑i=1

1

i!ì(z; i: : :; z) = 0:

The converse is also trivially satis�ed in light of (6).

Remark 2.19. Note that by the previous Lemma, any element z of degree −1of a given L∞-algebra L can be written as

∑i≥1

1i!�

(i)(u ⊗ · · · ⊗ u) and thus,

independently of any �nitness or mildness assumption, Maurer-Cartan elements

are not preserved in the standard fashion by L∞ morphisms. Note also that,

even for � mild, the condition �(k)([u; u]⊗ u⊗ · · ·⊗) = 0 for k large enough is

not automatically satis�ed.

Lemmas 2.17 and 2.18 are related by the following diagram:

C∗(L(u)

)L(u)oo

�

##

z ∈ L−1

(�(x; y); d

)�

��

L

��Q (�V; d)

foo

f�ee

C∗(L);

C∗(�)

dd

where the lower left corner is Lemma 2.17 and the upper right corner is Lemma

2.18. The hypothesis required to � and f� just translate into the fact that

C∗(�) = f�.

In order to detect Maurer-Cartan elements at the cochain level, let L be a

mild L∞-algebra and let {zj}j∈J and {vj}j∈J be basis of L−1 and V 0 respec-

tively (see De�nition 2.4). Then, any z ∈ L−1, written as z =∑j �jzj , is

obviously identi�ed with the linear map V 0 → Q sending vj to �j for all j ∈ J .However, Maurer-Cartan elements of L are not, in general, those z for which

this map can be extended as an augmentation of the cochains, i.e., as a CDGA

morphism C∗(L)→ Q. The following exercises corroborates this assertion.

Exercise 18. Show that if L is an in�nite-dimensional abelian L∞-algebra

concentrated in degree −1, the sets MC(L) and

Aug(C∗(L)

)= HomCDGA(C

∗(L);Q)

can not be one-to-one correspondant.


Exercise 19. Let L be the mild L∞ algebra generated by B = {!i; �}i≥2, with|!i| = −2, |�| = −1, and where the only non zero brackets on generators

are:

`1(�) = −!2; `k(�; : : : ; �) = k!(!k − !k+1); k ≥ 2:

(1) Show that C∗(L) = (�V; d) in which V is generated by {v; ui}i≥2, with|v| = 0, |ui| = −1, dv = 0 and dui = vi − vi−1 for i ≥ 2.

(2) Show that the morphism C∗(L) → Q sending ui to 0 for all i and v

to 1 is a well de�ned augmentation, but � is not a Maurer-Cartan

element.

Remark 2.20. In light of previous exercises, it is important to note that, if

one considers non-�nite type mild L∞ algebras, the Maurer-Cartan set can not

be identi�ed with the set of augmentations from the cochain algebra and very

special and technical restrictions are needed to have this identi�cation. In the

same way, in view of Lemma 2.18 and Remark 2.19, Maurer-Cartan elements

are not preserved by mild L∞ morphisms unless either �nite type is assumed,

or again, special restrictions are applied. Thus, hereafter, and again for the

sake of clearness, we restrict the categroy L∞ to the class of mild, �nite type

L∞-algebras, denoted by Lf.t.∞ .

Corollary 2.21. Let L be an L∞-algebra of �nite type. Then, an element

z ∈ L−1 is Maurer-Cartan if and only if there exists a mild L∞ morphism

� : (L(u); @)→ L such that �(1)(u) = z and �(k)(u⊗ : : :⊗ u) = 0 for k ≥ 2.

Proof. If z ∈ MC(L) the morphism � of Lemma 2.18 is obviously mild as

�(k)([u; u] ⊗ u ⊗ : : : ⊗ u) = 0 for k large enough. Conversely, if � is a mild

L∞ morphism and L is of �nite type, then �(k)([u; u]⊗ u⊗ : : :⊗ u) necessarilyvanishes for k large.

Definition 2.22. Let g : L → L′ be a morphism in Lf.t.∞ and z ∈ MC(L). De�ne

the map MC(g) : MC(L) −→ MC(L′) by

MC(g)(z) =∑k≥1

1

k!g(k)(z ⊗ · · · ⊗ z):

In the next result we see that MC(g) is well de�ned. Moreover, with the

�niteness type assumptions in the above remark, we identify the Maurer-Cartan

elements of L ∈ Lf.t.∞ in a functorial way with the set AugC∗(L) of augmentations

of C∗(L). We stress here that, to our knowledge, the following result and the

Corollary 2.24 that follows are not straightforward and do not follow at once

by simply generalizing their classical DGL counterpart of [6, Remark 16] or [12,

Proposition 1.1] (compare to [2, Lemma 2.3] or [10, Proposition 2.2]).

Proposition 2.23. Let g : L→ L′ be a morphism in Lf.t.∞ and z ∈ MC(L). Then,

MC(g)(z) is indeed a Maurer-Cartan element in L′. That is,∑k≥1

1

k!g(k)(z ⊗ · · · ⊗ z) ∈ MC(L′):


Moreover, the functor

MC: Lf.t.∞ → Set

is naturally equivalent to the functor

Aug : Lf.t.∞ → Set

which assigns to g : L → L′ the map Aug(g) : AugC∗(L) → AugC∗(L′) given

by composition, Aug(g)(") = "C∗(g).

Proof. We �rst show that there is a natural bijection

MC(L) ∼= AugC∗(L):

Choose a basis {zj}mj=1 of L−1, set C∗(L) = (�V; d) with V = (sL)], and for

each j denote by vj the element (szj)] of V 0.

Given z ∈ MC(L), write z =∑mj=1 �jzj and apply Corollary 2.21 (recall that

L is assumed to be of �nite type) to obtain the mild L∞ morphism � : L(u)→ L

for which �(1)(u) = z, �(n)(u ⊗ · · · ⊗ u) = 0 for n ≥ 2. Then, since � is mild,

we can construct the based augmentation

C∗(�) : (�V; d)→ (�(x; y); d)

which sends each vj to �jx. Therefore, the composition �C∗(�) : C∗(L) → Q is

an augmentation denoted by "z.

Conversely, consider any augmentation " : (�V; d) → Q and set "(vj) = �j .

Lift " via Lemma 2.17 to a based augmentation "x : (�V; d) → (�(x; y); d).

Then, observe that "x = C∗(�) for a mild L∞ morphism � : L(u)→ L in which

�(1)(u) =∑mj=1 �jzj and �

(n)(u ⊗ · · · ⊗ u) = 0 for n ≥ 2. Since L is of �nite

type, again by Corollary 2.21, the element z =∑mj=1 �jzj is a Maurer-Cartan

element of L.

Thus, the correspondence z ↔ "z establishes the asserted bijection.

Next, we prove the �rst assertion of the proposition by showing that, given

g : L→ L′ a morphism in Lf.t.∞ , then

MC(g) : MC(L)→ MC(L′)

is identi�ed with

Aug(g) : AugC∗(L)→ AugC∗(L′):

For it, let z ∈ MC(L). By the bijection MC(L) ∼= AugC∗(L), the Maurer-

Cartan element z corresponds to the augmentation in AugC∗(L) given by

�C∗(�)

where � : L(u) → L is the mild L∞ morphism, obtained via Corollary 2.21,

corresponding to z ∈ MC(L). Applying Aug(g) to this augmentation we obtain,

Aug(g)(�C∗(�)

)= �C∗(�)C∗(g) = �C∗(g�) ∈ Aug(L′):


We will prove that this augmentation corresponds, via again the bijection

MC(L′) ∼= AugC∗(L′), with the element 1k!

∑k g

(k)(z ⊗ · · · ⊗ z) ∈ L′−1 which

must be then a Maurer-Cartan element in L′ as stated.

For it, we need to lift this augmentation �C∗(�g), via Lemma ??, to a based

augmentation "x : C∗(L′)→ (�(x; y); d). Observe that "x is, in general, far from

being C∗(g�) = C∗(�)C∗(g). Indeed, although the image of C∗(�) on degree zero

elements is linear on x, the image of C∗(g) may not be linear on degree zero

elements. Let us then describe explicitly "x.

Choose �nite basis {zj}j∈J , {z′i}i∈I of L−1 and L′−1 respectively and write

C∗(L) = �W , C∗(L′) = �V . Observe thatW 0 and V 0 are generated by {wj}j∈Jand {vi}i∈I where wj = (szj)

] and vi = (sz′i)] for each i ∈ I and j ∈ J .

If z =∑j �jzj , then C∗(�) : C∗(L) → (�(x; y); d) is de�ned on W 0 by

C∗(�)wj = �jx.

On the other hand, write C∗(g) =∑k≥1 C

∗(g)k with C∗(g)kV ⊂ �kW and

set

C∗(g)k(vi) = Pik +Qik; with Pik ∈ �kW 0 and Qik ∈ �+W 6=0 · �W:

Then,

�C∗(�)C∗(g)k(vi) = �C∗(�)(Pik) = Pik(�j);

where Pik(�j) is the scalar obtained by evaluating the \polynomial" Pik on the

�j 's. Thus, "x is de�ned on V 0 as,

"x(vi) =∑k≥1

Pik(�j)x;

being this a �nite sum due to the mildness assumption.

Now that we have explicitly precised the lifting "x of the augmentation

�C∗(g�), we need to identify the Maurer-Cartan element z′ that it represents.

By the �rst part of the present proof, this element is precisely,

z′ =∑i

(∑k

Pik(�j))z′i:

On the other hand, an easy computation shows that

〈C∗(g)kvi; sz; : : : ; sz〉 = k!Pik(�j)

which, in light of (3) of Section 1, let us conclude that

Pik(�j) =1

k!〈vi; sg(k)(z ⊗ · · · ⊗ z)〉:

Therefore,

z′ =∑i;k

Pik(�j)z′i =

∑i;k

1

k!〈vi; sg(k)(z ⊗ · · · ⊗ z)〉z′i =

∑k

1

k!g(k)(z ⊗ · · · ⊗ z)

and the proposition is proved.

2.3 The connected components in the DGL/L∞-algebra setting 32

Exercise 20. Show that, given an L∞ algebra L and a commutative di�er-

ential graded algebra A, the tensor product L ⊗ A inherits a natural L∞structure with brackets:

`1(x⊗ a) = @x⊗ a+ (−1)|x|x⊗ da;`k(x1 ⊗ a1; : : : ; xk ⊗ ak) = "`k(x1; : : : ; xk)⊗ a1 : : : ak; k ≥ 2;

where " = (−1)∑

i>j |xi||aj | is the sign provided by the Koszul convention.

Corollary 2.24. Let L ∈ Lf.t.∞ and A ∈ CDGA such that L ⊗ A is of �nite type.

Then, there is a bijection

MC(L⊗ A) = HomCDGA (C∗(L); A):

Proof. Since L⊗A is mild and of �nite type, apply Proposition 2.23, to identify

a given Maurer-Cartan element z of L⊗ A with an augmentation

"z : C∗(L⊗ A) ∼= �(sL⊗ A)∨ → Q:

This produces a degree zero linear map (sL)∨ → A which is extended to an

algebra morphism C∗(L) → A. A straightforward computation shows that it

commutes with di�erential since "z does. Conversely, any CDGA morphism

C∗(L)→ A gives rise, by the procedure above, to an augmentation C∗(L⊗A)→Q.

It is important also to observe that if L ⊗ A fails to be of �nite type, and

even if L and A are, MC(L⊗A) is no longer identi�ed with the set of morphisms

HomCDGA(C∗(L); A) as shown in the following exercise. In the general case, as

in Remark 2.20, it is necessary to impose technical �niteness restrictions in the

class of morphisms.

Exercise 21. Let L =∑n<0 L2n+1 be an abelian L∞-algebra (i.e., all brackets

are zero) concentrated in odd negative degrees, with L2n+1 of dimension 1

for all n, and let A = (�x; 0) be the polynomial algebra on a single generator

of degree 2, without constant terms.

(1) Show that MC(L⊗A) = (L⊗A)−1 and is of in�nite countable dimen-

sion.

(2) Show that, C∗(L) = (�(y0; y2; y4; : : :); 0) and that HomCDGA(C∗(L); A)

is of in�nite, uncountable dimension.

2.3 The connected components in the DGL/L∞-algebra setting

Definition 2.25. Given an L∞-algebra L and z ∈ MC(L), de�ne the perturbation

of `k by z as

`zk(x1; : : : ; xk) = [x1; : : : ; xk]z =

∞∑i=0

1

i!ì+k(z; i: : : ; z; x1; : : : ; xk):

3 Complete DGL’s and the representable realization functor 33

Exercise 22. Show that whenever the above sum is always �nite, (L; {`zi }) isagain an L∞ algebra which will be denoted by Lz.

We can truncate Lz to produce a non-negatively graded L∞ algebra L(z)

whose underlying graded vector space is

L(z)i =

Lzi = Li if i > 0;

Ker`z1 if i = 0;

0 if i < 0;

and with brackets induced by `zk for any k ≥ 1.

Theorem 2.26. [2, Corollary 1.2] [?, Theorem 1.1] Let ' : C∗(L) → Q be the

augmentation corresponding to the Maurer-Cartan element z of a given

mild L∞-algebra. Then 〈L〉' and 〈L(z)〉 are homotopy equivalent simplicial

sets.

Proof. First, observe that, for a given augmentation f : (�V; d) → Q of a free

CDGA, the quotient (�V; d)=Kf is again a free CDGA (�(V1 ⊕ V ≥2); df ) in

which V1is the coker of the map d : V 0 → V 1 resulting by applying the di�er-

ential d and then projecting over the ideal generated by V <0 and {v−f(v); v ∈V 0}.

In particular, if (�V; d) = C∗(L), we write,

C∗(L)=K' = (�(V1 ⊕ V ≥2); d'):

A straightforward computation shows that (�(V1⊕V ≥2); d') is precisely C∗(L(z)).

Then,

〈L(z)〉 = 〈C∗(L(z))〉 = 〈C∗(L)=K'〉 ' 〈C∗(L)〉' = 〈L〉':

Exercise 23. Check carefully the step

(�(V1 ⊕ V ≥2); d') ∼= C∗(L(z))

in the previous proof.

3 Complete DGL’s and the representable realization functor

The second part of this course deals also with realization functors. Recall that

Quillen's adjoint pair is the composition of a sequence of adjoint pairs:

{Arc-connected, simply

connected spaces

}G //

SGp1

W

ooK• //

SCHA1G•

ooP• //

SLA1

U•

ooN //

DGL1

N∗oo ;

which induce an equivalence between the corresponding homotopical categories

[18].

3.1 The gauge action and the Baker-Campbell-Hausdorff product 34

This situation contrasts with the simplicity of Sullivan realization functor{Arc-connected, nilpotent,

�nite type spaces

} APL//CDGA

〈−〉Soo :

which is (co)representable by •, the simplicial de Rham algebra, i.e., the

simplicial CDGA with n-simplices

n =�(t0; : : : ; tn; dt0; : : : ; dtn)(∑

i t1 − 1;∑i dti

) ; |ti| = 0; |dti| = 1:

To any commutative di�erential graded algebra A on the category of the

right we associate the simplicial set 〈A〉S , where

(〈A〉S)n = HomCDGA(A;n):

The idea in the origin of [7] was trying to describe a new realization functor

for di�erential graded Lie algebras representable by some cosimplicial DGL.

3.1 The gauge action and the Baker-Campbell-Hausdorff product

Recall that a di�erential graded Lie algebra is called free if L is free as a Lie

algebra, L = L(V ) for some graded vector space V .

The completion L of a graded Lie algebra L is the limit

L = lim−→n

L=Ln

where L1 = L and for n ≥ 2, Ln = [L;Ln−1]. A Lie algebra L is called complete

if L is isomorphic to its completion. From now on, and unless explicitly stated

otherwise, by a cDGL we mean a complete di�erential graded Lie algebra.

Recall that in a di�rential graded Lie algebra L, a Maurer-Cartan element

is an element a ∈ L−1 such that @a+ 12 [a; a] = 0. Maurer-Cartan elements are

trivially preserved by DGL morphisms.

Exercise 24. Let L = (L(V ); @) be a complete free DGL, and � a derivation

satisfying �(V ) ⊂ L≥2(V ) and [�; @] = 0.

Prove that e� =∑n≥0

�n

n! is an automorphism of L and so, if a ∈ MC(L),

then e�(a) is also a Maurer-Cartan element.

Given (L(V ); @) a complete free DGL and v ∈ V , we will often write @v =∑n≥1 @nv where @nv ∈ Ln(V ).Let (L; @) be a DGL and a ∈ MC(L). Then, as we have seen in the �rst part

of the course, the derivation @a = @ + ada is again a di�erential on L.

Given L a complete DGL, the gauge action G of L0 on MC(L) determines

an equivalence relation among Maurer-Cartan elements de�ned as follows (see

for instance [?, x4]): given x ∈ L0 and a ∈ MC(L),

xG a = eadx(a)− eadx − 1

adx(@x):

3.1 The gauge action and the Baker-Campbell-Hausdorff product 35

Here and from now on, 1 inside an operator will denote the identity. Ex-

plicitly,

xG a =∑i≥0

adix(a)

i!−∑i≥0

adix(@x)

(i+ 1)!:

We denote the quotient set by MC(L) = MC(L)=G. Geometrically [15, 16],

interpreting Maurer-Cartan elements as points in a space, one thinks of x as

a ow taking xG a to a in unit time. In topological terms [5], the points a

and xG a are in the same path component. Let L be a complete Lie algebra

concentrated in degree 0. We denote by UL its enveloping algebra, by IL its

augmentation ideal and by UL and IL the completions of UL and IL with

respect to the powers of IL,

UL = lim−→n

UL=InL ; IL = lim−→n

IL=InL :

Denote �nally by GL = {x ∈ UL |�(x) = x⊗x} the group of grouplike elements

in UL. Moreover, the injection of L in the set of primitive elements in UL is an

isomorphism and the functions exp and log give inverse bijections between L

and GL. This induces a product on L, called the Baker-Campbell-Hausdor�

product, BCH product henceforth, de�ned by

a ∗ b = log(exp(a) · exp(b)):

Note that a ∗ (−a) = 0. Therefore, −a is the inverse of a for the BCH

product and we also use the notation −a = a−1.

As the law in GL is associative, the BCH product is also associative. An

explicit form of the product is given by the Baker-Campbell-Hausdor� formula

a ∗ b = a+ b+1

2[a; b] +

1

12

[a; [a; b]

]− 1

12

[b; [a; b]

]+ · · ·

It follows from the Jacobi identity that in the Lie algebra of derivations of L we

have ada∗b = ada ∗ adb. Hence eada∗b = eada ◦ eadb .Note that the BCH product is compatible with the gauge action on MC(L);

i.e., if y ∈ L0 and a ∈ MC(L), we have

(x ∗ y)Ga = xG(yGa):

We also need the following property.

Proposition 3.1. Let L be a complete DGL and let x; y ∈ L0. Then,

x ∗ y ∗ (−x) = eadx(y):

With the previous convention, the formula also reads

x ∗ y ∗ x−1 = eadx(y):

3.2 The Lawrence-Sullivan model of the interval 36

Proof. First, we note that, in UL,

eadx(y) = exye−x:

Indeed,

eadx(y) =

∞∑n=0

adnx(y)

n!=

∞∑n=0

1

n!

n∑i=0

(−1)i(n

i

)xn−iyxi

=

∞∑i=0

∞∑n=i

xn−i

(n− i)!y(−x)i

i!= exye−x:

Replacing y by ey, we deduce

eadx(ey) = exeye−x: (8)

In a second step, we prove the equality

(eadx)(yn) =(eadx(y)

)n: (9)

On the left hand side, the term of length k, k ≥ 0, in x equals

adkx(yn)

k!:

In the right hand side, this term is∑k1+···+kn=k

adk1x (y)

k1!· · · ad

knx (y)

kn!:

As adx is a derivation, both terms coincide and the equality (9) is proved.

Therefore,

eadx(ey) =∑n≥0

(eadx)(yn)

n!=∑n≥0

(eadx(y)

)nn!

= eeadx (y): (10)

Finally, the proposition follows from

x ∗ y ∗ (−x) = log(exeye−x) =(8) log(eadx(ey)

)=(10) log(e

eadx (y)) = eadx(y):

3.2 The Lawrence-Sullivan model of the interval

In the �rst part of the course we have seen that in some sense the points be-

longing to a particular path-component were described using Maurer-Cartan

elements.

In order to determine if two Maurer-Cartan elements are in the same path-

component we need an algebraic abstraction of an interval.

Consider the standard 1-simplex �1,

�10 = {(0); (1)}; �1

1 = {(0; 1)};


and �1p = ∅ if p > 1. Let (L(s−1�1); d) be the complete free DGL on the

desuspended rational simplicial chain complex on �1,

da0 = da1 = 0; da01 = a1 − a0

Here, ai0:::ip denotes the generator of degree p− 1 represented by the p-simplex

(i0; : : : ; ip) ∈ �np .

If we replace this di�erential by a new one, @, such that the linear part

agrees with d and with the extra condition that 0-simplexes are Maurer-Cartan

elements @a0 = − 12 [a0; a0] and @a1 = − 1

2 [a1; a1], then @a01 = a1 − a0 is no

longer a di�erential. In order to recover a di�erential we need to add more

terms

@a01 = a1 − a0 +�; � ∈ L≥2(s−1�1):

The construction of a model for the interval was �rst introduced in [16]:

Definition 3.2. [16] The Lawrence-Sullivan interval is the complete free DGL

L = (L(a; b; x); @);

in which a and b are Maurer-Cartan elements, x is of degree 0 and

@x = adxb+

∞∑n=0

Bnn!

adnx(b− a) = adxb+adx

eadx − 1(b− a) (11)

where Bn are the Bernoulli numbers.

Exercise 25. Bernoulli numbers are de�ned by the series

x

ex − 1=

∞∑n=0

Bnn!xn:

(1) Since

ex − 1

x=

∞∑n=0

1

n+ 1!xn;

Deduce the �rst Bernouilli numbers from the equationx

ex − 1· e

x − 1

x= 1.

(2) Use the formula (−x

e−x − 1

)= x+

(x

ex − 1

); (12)

to deduce that Bn = 0 if n is odd and n ≥ 3.

In the following theorem we give an alternative proof of the fact that the

di�erential of the Lawrence-Sullivan interval is indeed a di�erential (@2 = 0),

interpreting it as the Tanr�e cylinder of a free DGL with a unique generator in

degree −1 which is a Maurer-Cartan element.

Consider (L(b); @), @(b) = − 12 [b; b] the model of S0. Consider the Tanr�e

cylinder

(L(b; u; v); @); @(u) = v);


where |u| = 0, |v| = −1.Let L(a; b; x) be a complete free Lie algebra with |a| = |b| = −1, |x| = 0.

We can de�ne an isomorphism of Lie algebras

' : L(a; b; x)→ L(b; u; v);

'(a) = e�(b); '(b) = b; '(x) = −u;

where � = @i+ i@ and i is the derivation of degree +1 de�ned by i(v) = u and

i(other) = 0.

This isomorphism de�nes a di�erential in L(a; b; x) by d = '−1@'.

Theorem 3.3. (L(a; b; x); d) is the Lawrence-Sullivan interval.

Proof. We will decompose the proof in the following exercises:

Exercise 26. Check that a and b are Maurer-Cartan elements

Exercise 27. Prove inductively the following equation

�k+1(b) = (−1)kadku(v) + (−1)k+1adk+1u (b): (13)

Exercise 28. Deduce from equation (13) the following formula:

e�(b) =e−adu − id

−adu(v) + e−adu(b): (14)

Now we can compute dx:

dx = '−1@'x = '−1@(−u) = −'−1v;

So the di�erential of the top cell will hold the equation

dx = −'−1v: (15)

We can write this equation in a more explicit way by using formula (14).

Indeed:

'(a) = e�(b)

(14) =e−adu − id

−adu(v) + e−adu(b):

We can isolate v from this equation:

v =−adu

e−adu − id('(a)− b) + adu(b);

and apply '−1 obtaining:

'−1v =adx

eadx − id(a− b)− [x; b]

yielding

dx = [x; b] +adx

eadx − id(b− a):

3.3 The cDGL model for the triangle and the tetrahedra 39

Moreover, the di�erential @ in L is the only one for which a and b are Maurer-

Cartan elements and either, its linear part @1 sati�es @1x = b−a [7, Thm. 1.4],

or else xG b = a [16, Thm. 1], [17].

Exercise 29 (change of orientation). Show that the automorphism

: L∼=−→ L; (a) = b; (b) = a; (x) = −x:

commutes with the di�erential.

We can use also the LS-interval to give a short proof of the fact that the

gauge action preserves Maurer-Cartan elements.

Proposition 3.4. Let (L; d) be a complete DGL, z ∈ MC(L) and x ∈ L0. Then,xG z is a Maurer-Cartan element.

Proof. Write c = xG z. The formula for the gauge action can be written

dx = adxz +adx

eadx − 1(z − c):

Then d2x = 0 implies that w = dc + 12 [c; c] belongs to the ideal generated by

x. Suppose w 6= 0 then w = wr + w′r with wr a non zero linear combination of

Lie brackets containing r times x, and w′r a linear combination of Lie brackets

containing at least r+1 times x. If we denote by @ the usual di�erential on the

LS-interval on L(c; z; x), then dx = @x, dz = @z. From d2 = 0 and @2 = 0 we

deduce

0 = −w +1

2[x;w]−

∑n≥2

Bnn!

adnx(w):

This implies that wr = 0. Therefore w = 0 and c = xG z is a Maurer-Cartan

element.

3.3 The cDGL model for the triangle and the tetrahedra

Consider the standard 2-simplex �2,

�20 = {(0); (1); (2)}; �2

1 = {(0; 1); (0; 2); (1; 2)};�22 = {(0; 1; 2)}



da0 = da1 = da2 = 0; da01 = a1 − a0; da02 = a2 − a0; da12 = a2 − a1;

da012 = a12 − a02 + a01:

Here, ai0:::ip denotes the generator of degree p − 1 represented by the p-

simplex (i0; : : : ; ip) ∈ �np .

If we replace d by a di�erential whose linear term is precisely d and such

that 0-simplexes are MC-elements and 1-simplexes are LS-intervals, how should

we extend da012 = a12 − a02 + a01 in order to still having a di�erential @?


We �rst recall the Lawrence-Sullivan model of the subdivision of the interval.

Let (L(a0; a1; a2; x1; x2); @) be two glued LS-intervals. That is, a0; a1 and a2 areMaurer-Cartan elements, @x1 = adx1a1 +

adx1eadx1−1

(a1 − a0) and @x2 = adx2a2 +adx2

eadx2−1(a2 − a1).

Theorem 3.5. [16, Thm. 2] The map

: (L(a; b; x); @) −→ (L(a0; a1; a2; x1; x2); @);

(a) = a0, (b) = a2 and p(x) = x1 ∗ x2, is a DGL morphism.

Here, ∗ denotes the Baker-Campbell-Hausdor� product.

Exercise 30. Let L be a complete di�erential graded Lie algebra.

(1) Show that two Maurer-Cartan elements z; z′ ∈ L are gauge equivalent

if and only if there is a morphism of cDGL's

� : (L(a; b; x); @)→ L

such that �(a) = z and �(b) = z′.

(2) Show that Theorem 3.5 translates to the statement

(x1 ∗ x2)G a2 = a0 = x1 G (x2 G a2);

which reproves that the gauge action is in fact a group action of (L0; ∗)on MC(L).

Lemma 3.6. Given a complete free di�erential graded Lie algebra (L(V ); @)where V = 〈ai; xi, i = 1; : : : ; n〉 are generators such that (L(ai−1; ai; xi); @)are Lawrence-Sullivan intervals for any i (with a0 = an) describing a cycle,

then

@(x1 ∗ · · · ∗ xn) = adx1∗···∗xn(a0);

•a0

x1•a1

x2

•a2xi

•aixn−1•

an−1

xn

Figure 6

where ∗ stands for the Baker-Campbell-Hausdor� product.

Proof. Recall that the iterated subdivision of the Lawrence-Sullivan interval

gives a morphism

: L(a; b; x)→ L(a0; a1; : : : an; x1; x2; : : : ; xn)


given by (a) = a0, (b) = an and (z) = x1 ∗ · · · ∗xn. Since commutes with

di�erentials we have

(@z) = (adz(b) +

∑k≥0

Bkk!

adkz(b− a))

= adx1∗···xn(an) +∑k≥0

Bkk!

adx1∗···∗xn(an − a0)

= @(x1 ∗ · · · ∗ xn) = @ (z):

Now identifying a = b and a0 = an we have

@(x1 ∗ · · · ∗ xn) = adx1∗···∗xn(a0):

Remark 3.7. In what follows for a an element of degree −1 we will use the

notation @a = @ + ada, which is a di�erential if and only a is a Maurer-Cartan

element.

The above lemma allows us to construct a model for the triangle as a par-

ticular case:

(L(a; b; c; x; y; z; e); @) where |a| = |b| = |c| = −1, |x| = |y| = |z| = 0, e = 1

and

@a = −12[a; a]; @b = −1

2[b; b]; @c = −1

2[c; c];

@x = adx(b) +∑i≥0

Bii!adix(b− a);

@y = ady(c) +∑i≥0

Bii!adiy(c− b);

@z = adz(a) +∑i≥0

Bii!adiz(a− c);

@e = x ∗ y ∗ z − ade(a):

y z

x

e

b •

Figure 7

• a

•c

Equivalently we will write simply

@a(e) = xyz:

Exercise 31. Prove that @e = x ∗ y ∗ z − ade(a) gives, indeed, a di�erential


For the next step we consider the standard 3-simplex �3,

�30 = {(0); (1); (2); (3)};

�31 = {(0; 1); (0; 2); (0; 3); (1; 2); (1; 3); (2; 3)};�32 = {(0; 1; 2); (0; 1; 3); (0; 2; 3); (1; 2; 3)};

�33 = {(0; 1; 2; 3)};



dai0:::ip =

p∑j=0

(−1)jai0:::ij :::ip

; 0 ≤ p ≤ 3: (16)

Here as usual, ai0:::ip denotes the generator of degree p − 1 represented by

the p-simplex (i0; : : : ; ip) ∈ �3p.

If we replace d by a di�erential whose linear term is precisely d and such that

0-simplexes are MC-elements and 1-simplexes are LS-intervals and 2-simplexes

are BCH-triangles, how should we extend

da0123 = a123 − a023 + a013 − a012

in order to still having a di�erential @?

To answer to that question we �rst give a subdivision for triangles

Lemma 3.8. Consider a complete free graded di�erential Lie algebra for two

glued triangles (L(a; b; c; d; x1; x2; x3; y1; y2; e1; e2); @)

x1 x3

y1 y2

e1 e2

b • • c

•a

•d

x2

Figure 8

@ae1 = x1y1x−12 ; @ae2 = x2y2x

−13 ;

There is an element Be1e2 = e1+e2+� of degree 1 where � ∈ L≥2(v1; v2; e1; e2)with v1 = x1y1x

−12 and v2 = x2y2x

−13 such that

@a(Be1e2) = x1y1y2x−13 :

Proof. Consider the product v1 ∗ v2 = v1 + v2 +12 [v1; v2] + · · · . To de�ne

Be1e2 substitute in each bracket one (and only one!) of the elements vi by the

corresponding ei. Then Be1e2 = e1 + e2 + � where � ∈ L≥2(v1; v2; e1; e2) andits clear that

@a(Be1e2) = v1v2 = (x1y1x−12 )(x2y2x

−13 ) = x1y1y2x

−13 :


It is also clear that we can extend this Lemma to any set of adjacent triangles

with a common point or to polygons instead of triangles.

Proposition 3.9. Let L be a complete DGL which contains a LS-interval

(L(a; b; x); @). Then, for any v ∈ L,

@a eadx(v) = eadx(@bv):

In other words, the map

eadx : (L; @b)→ (L; @a)

is an isomorphism of DGL's.

Proof. In fact,

@b e−adx(v) = e−adx(@bv) + (−1)|v|e−adxadv

eadx − 1

adx(@bx)

= e−adx(@bv + (−1)|v|adv(b− a)

)= e−adx(@av);

where the �rst equality comes from [16, Lemma 1] and the second is a direct

computation using (11).

Consider now the subdivision of the triangle into three triangles:

x1 x2 x3 x4

y2y1 y3

e1 e3e2

b • • c

•a

Figure 9

That is, we have three triangles with

@a(e1) = x1y1x−12 = v1; @a(e2) = x2y2x

−13 = v2; @a(e3) = x3y3x

−14 = v3;

and we de�ne an element

Be1e2e3 = e1 + e2 + e3 +�; where � ∈ L≥2(vi; ei); i = 1; 2; 3;

satisfying

@a(Be1e2e3) = x1y1y2y3x−14 :

Make x1 = x4 = x and b = c so we have

@a(Be1e2e3) = xy1y2y3x−1:

3.4 The cosimplicial DGL 44

•b

x

•a

y1 y2

y3

e1 e2

e3

Figure 10

We can now attach a new triangle e4 with @b(e4) = y−13 y−12 y−11 = −y1y2y3to the base of the above picture:

•b

x

•a

y1 y2

y3

e1 e2

e3

e4

Figure 11

Finally we have that the element

Be1e2e3 + eadx(e4)

is a cycle, since

@a(Be1e2e3 + eadx(e4)) = xy1y2y3x−1 + eadx(@be4)

= xy1y2y3x−1 − eadx(y1y2y3) = 0;

where in the last equality we have applied Proposition 3.1.

We can attach a new generator o of degree 2 with

@a(o) = Be1e2e3 + eadx(e4) = e1 + e2 + e3 + e4 +;

where ∈ L≥2(xi; yi; ej), i = 1; 2; 3; j = 1; 2; 3; 4.

3.4 The cosimplicial DGL

In the general case, for each n ≥ 0, we consider the standard n-simplex �n,

�np = {(i0; : : : ; ip) | 0 ≤ i0 < · · · < ip ≤ n}; if p ≤ n;

and �np = ∅ if p > n. Let (L(s−1�n); d) be the complete free DGL on the

desuspended rational simplicial chain complex on �n,

dai0:::ip =

p∑j=0

(−1)jai0:::ij :::ip

: (17)

3.4 The cosimplicial DGL 45

Here, ai0:::ip denotes the generator of degree p− 1 represented by the p-simplex

(i0; : : : ; ip) ∈ �np . Henceforth, unless explicitly needed, we drop the desuspen-

sion sign to avoid unnecessary notation and write simply (L(�n); d).

For each 0 ≤ i ≤ n consider the i-th coface a�ne map �j : �n−1 → �n;

de�ned on the vertices by,

�i(j) =

{j; if j < i,

j + 1; if j ≥ i:

We use the same notation for the induced DGL morphism,

�i : (L(�n−1); d) −→ (L(�n); d); (18)

de�ned by

�i(aj0:::jp) = a`0:::`p with `k =

{jk; if jk < i;

jk + 1; if jk ≥ i:

Finally, we denote by _�n and �ni the boundary of �n and the i-horn obtaining

by removing the i-th coface from _�n.

Definition 3.10. A sequence of compatible models of ∆ is a family {(Ln; @) =(L(�n); @)}n≥0 of DGL's satisfying the following properties:

(1) For each i = 0; : : : ; n, the generator ai ∈ �n0 is a Maurer-Cartan element,

@ai = − 12 [ai; ai].

(2) The linear part @1 of @ is precisely d as in (17).

(3) For each i = 0; : : : ; n, the coface maps, �i : (L(�n−1); @) → (L(�n); @),

are DGL morphisms.

Each element (L(�n); @) of this sequence is called a model of �n, which is

thus implicitly endowed with models of �q, q < n, satisfying the compatibility

condition (3) above.

Definition 3.11. A sequence of models {(L(�n); @)}n≥0 is called inductive if, for

n ≥ 2, we have

@a0a0:::n ∈ L( _�n): (19)

Theorem 3.12. [7, Theorem 2.3] There exists sequences of compatible inductive

models of ∆.

In fact, these sequences of compatible inductive models are unique up to

isomorphism:

Theorem 3.13. [7, Theorem 2.8] Two sequences {(L(�n); @)}n≥0 and {(L(�n); @′)}n≥0of compatible models of ∆ are isomorphic: for n ≥ 0, there are DGL iso-

morphisms,

'n : (L(�n); @)∼=−→ (L(�n); @′);

3.5 The realization functor and its adjoint 46

which commute with the coface maps �i, for i = 0; : : : ; n,

(L(�n); @)'n

∼=// (L(�n); @′)

(L(�n−1); @)

� ?

�i

OO

'n−1

∼=// (L(�n−1); @′);

� ?

�i

OO(20)

and such that Im('n − id) ⊂ L≥2(�n).

Theorem 3.14. [7, Theorem 3.5] Any sequence {Ln}n≥0 of models of ∆ admits

a cosimplicial DGL structure for which the cofaces are the usual ones.

3.5 The realization functor and its adjoint

Based on a sequence L• of compatible models of ∆ with the cosimplicial struc-

ture given by Theorem 3.14, we de�ne a pair of adjoint functors,

SimpSet cDGL〈−〉oo

L //

between the categories of simplicial sets and complete DGL's.

Definition 3.15. Let L ∈ cDGL. The realization of L is the simplicial set,

〈L〉• = cDGL(L•; L):

On the other hand, let ∆ be the category whose objects are the sets [n] =

{0; : : : ; n}, n ≥ 0, and whose morphisms are monotone maps. Now, let I : ∆→SimpSet the functor that associates to [n] the simplicial set �n whose p-simplices

are the sequences 0 ≤ i0 ≤ · · · ≤ ip ≤ n. Observe that, by construction, L• is a

functor from ∆ to cDGL.

Definition 3.16. The functor model L : SimpSet→ cDGL is de�ned as the left Kan

extension of L• along I,

∆I //

L•��

SimpSet

L=LanIL•zzcDGL

The DGL L(K) is thus the colimit of L• over the comma category I ↓ K,

L(K) = LanIL•(K) = lim−→f : �n→K

Ln:

For simplicity, we write

L(K) = lim−→K

L•:

and refer to it as the L-model of the simplicial set K.

In the case K is a �nite simplicial complex, then K ⊂ �n for some n, and

L(K) is trivially isomorphic to the complete sub DGL (L(V ); @) ⊂ Ln where

(V; @1) is the desuspension of the chain complex of K.

3.6 The connected components in the cDGL setting 47

Theorem 3.17. The functors L and 〈−〉 are adjoint. More precisely, for any

simplicial set K and any complete di�erential graded Lie algebra L, there

is a bijection,

SimpSet(K; 〈L〉) ∼= cDGL(L(K); L):

Proof. The result follows from classical properties of commutation of limits

with hom functors, i.e.,

cDGL(L(K); L) = cDGL(lim−→K

Ln; L) = lim←−K

cDGL(Ln; L)

= lim←−K

〈L〉n = lim←−K

SimpSet(�n; 〈L〉)

= SimpSet(lim−→K

�n; 〈L〉) = SimpSet(K; 〈L〉):

3.6 The connected components in the cDGL setting

We now interpret the homotopy groups of the realization of a cDGL and its

path component.

Proposition 3.18. For any cDGL, (L; @), there is a natural bijection �0〈L〉 ∼=MC(L).

Proof. By [4, Proposition 3.1], two Maurer-Cartan elements z0; z1 ∈ MC(L) are

gauge equivalent if there is a map ' : L1 = (L(a; b; x); @) → L with '(a) = z0and '(b) = z1. By De�nition 3.15, 〈L〉0 is the set of Maurer-Cartan elements

of L, and 〈L〉1 is the set of DGL morphisms from the LS-interval L1 to L. This

implies the result.

Proposition 3.19. Let (L; @) be a non-negatively graded cDGL. Then, 〈L〉 is aconnected simplicial set and there are natural bijections

�n〈L〉 ∼= Hn−1(L; d); n ≥ 1;

which are group isomorphisms for n ≥ 2.

Proof. By Proposition 3.18, 〈L〉 is connected. The coface maps, �j : Ln−1 → Ln,

induce the face maps

di = cDGL(�j ; L) : 〈L〉n → 〈L〉n−1:

We denote ker dj = {f : (Ln; @)→ (L; @) | djf = 0}. Recall that

�n〈L〉 = ∩n−1i=0 ker di= ∼

where f ∼ g if dnf = dng and there is h ∈ 〈L〉n+1 such that dnh = f , dn+1h = g.

We denote by f the element of �n〈L〉 represented by f . De�ne,

' : �n〈L〉∼=−→ Hn−1(L); '(f) = [f(a0:::n)];

REFERENCES 48

and observe that, for any f ∈ �n〈L〉, the morphism f vanishes in any p-simplex

of �n, with 0 ≤ p < n. Hence, it is uniquely determined by f(a0:::n). Straight-

forward computations show that ' is a well de�ned isomorphism for n ≥ 2 and

a bijection for n = 1.

For any di�erential graded Lie algebra L and any Maurer-Cartan element

z ∈ MC(L) consider the localization of L at z which is the cDGL,

L(z) = (L; @z)=(L<0 ⊕M)

where M is a complement of ker @z in L0.

Proposition 3.20. 〈L〉 ' _∪z∈MC(L)

〈L(z)〉.

Proof. As we know, the components of 〈L〉 are identi�ed with MC(L). Via

this identi�cation, the component of a given z ∈ MC(L) is of the same ho-

motopy type as the reduced simplicial set which we denote by 〈L〉z whose n-

simplices are the DGL morphisms f : Ln → L such that f(ai) = z for any

0-simplex ai, i = 0; : : : ; n. Perturbing both DGL's, these are the DGL mor-

phisms f : (L(�n); @a0)→ (L; @z) such that f(ai) = z.

But, composing with the projection (L; @z)→ L(z), this is isomorphic to the

simplicial set whose n-simplices are the DGL morphisms from (L(�n); @a0) to

L(z). This is immediate for n ≥ 2. For n = 1, let f : L1 = (L(a; b; x); @a) →(L; @z) be a morphism such that f(a) = f(b) = z. Then, as f commutes with

di�erentials, f(x) lies in ker @z = L(z)0 . In other words, 〈L〉z ∼= 〈L(z)〉.

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