Homotopy of Operads & Grothendieck-Teichmüller Groups
Book project, by Benoit Fresse (Université de Lille 1)
Chapter II.13
* Date of completion: 10 May 2015 (with minor writing corrections
after the overall proofreading of the manuscript on 13 December 2015
and minor updates on 31 December 2015)
* First draft on 17 February 2014, comprehensively revised
and augmented on 10 May 2015
* This chapter may not be in final form: I might change and simplify
some of the constructions of this chapter
CHAPTER 13
Complete Lie Algebras and Rational Models of
Classifying Spaces
We tackle the applications of the rational homotopy theory to En -operads in
this part. In a preliminary step, we explain general constructions on Lie algebras
which we use in our definition of models for the rational homotopy of En -operads
in the category of Hopf cochain dg-cooperads.
Let us give an overview of the definition of these models first. We can forget
about Λ-structure for the moment. We consider a graded version, defined for any
dimension n ≥ 2, of the Drinfeld–Kohno Lie algebras of §I.11.2. We use the notation
p̂n (r), r > 0, for this collection of Lie algebras in graded modules. We consider the
Chevalley–Eilenberg cochain complexes of these Lie algebras C∗CE (p̂n (r)) which form
a Hopf cochain dg-cooperad C∗CE (p̂n ). We are precisely going to check that C∗CE (p̂n )
forms a cofibrant object of the category of Hopf cochain dg-cooperads whose image
under the functor G• : dg ∗ Hopf Op c∅1 → sSet Op op
∅1 of §10.2 is an En -operad up to
rational weak-equivalence of operads.
In the case n = 2, where the Lie algebras p̂(r) = p̂2 (r) are the classical Drinfeld–
Kohno Lie algebras, we will more explicitly prove that the image of the Hopf dgcooperad C∗CE (p̂) under the functor G• : dg ∗ Hopf Op c∅1 → sSet Op op is weaklyequivalent to the classifying space of the chord diagram operad of §I.11.2. In this
case, we can use the operadic interpretation of Drinfeld associators in order to
identify the obtained operad with a rationalization of the operad of little 2-discs.
Recall that the chord diagram operad CDb is defined aritywise by the set of
group-like elements of the (completed) enveloping algebra of the Drinfeld–Kohno
Lie algebras Û(p(r)), r > 0. Besides the Hopf cochain dg-cooperad C∗CE (p̂), we
have a simple model of E2 -operad in the category of cosimplicial Hopf cooperads
B c ∈ c Hopf Op c∅1 which is defined, in each arity r > 0, by the cobar construction
B c (r) = Bc (Û(p(r))∨ ), where we consider the dual coalgebra of the enveloping
algebra of the rth Drinfeld–Kohno Lie algebra Û(p(r))∨ . We actually have the
identity N∗] (B c ) = C∗CE (p̂) when we apply the commutative algebra upgrade of the
conormalized cochain complex functor of §5.2 to this cosimplicial Hopf cooperad
B c ∈ c Hopf Op c∅1 .
We study the cobar construction of the dual coalgebras of enveloping algebras and we recall the general definition of the Chevalley–Eilenberg complex of Lie
algebras in this chapter. We therefore forget about cooperad structures for the moment. We are precisely going to prove that the cobar complex of the dual coalgebra
of a complete enveloping algebra Bc (Û(g)∨ ) forms a cofibrant object in the category of cosimplicial unitary commutative algebras and that the space G• Bc (Û(g)∨ )
associated to this cosimplicial algebra, where we consider our functor from cosimplicial unitary commutative algebras to spaces G• : c Com + → sSet op (see §7.2),
395
396
13. COMPLETE LIE ALGEBRAS AND CLASSIFYING SPACES
is identified with the classifying space of the Malcev complete group G = G Û(g)
associated to the Lie algebra g ∈ fˆ Lie (see §I.8.2). We obtain a similar result
for the Chevalley–Eilenberg cochain complex C∗CE (g). We also check that C∗CE (g)
represents the image of the cobar construction Bc (Û(g)∨ ) under the commutative
algebra upgrade of the conormalized cochain complex N∗] : c Com + → dg ∗ Com + in
general. We address these topics in §§13.1-13.2.
We revisit the definition of a complete Lie algebra and of a complete enveloping
algebra before tackling this main subject. We devote a preliminary section §13.0
to this revision. We take a characteristic zero field k as a ground ring all through
this chapter.
13.0. Reminders on Lie algebras and enveloping algebras
In §I.7, we explained the definition of a Lie algebra and of the enveloping
algebra of a Lie algebra in the setting of a Q-additive symmetric monoidal category
M. We also used this general approach in order to define complete Lie algebras
and complete enveloping algebras. We then worked in the category of complete
modules M = fˆ Mod .
We actually deal with complete Lie algebras equipped with an additional simplicial (respectively, differential graded) structure in this chapter. We therefore
consider a simplicial (respectively, differential graded) analogue of our category of
complete filtered modules as symmetric monoidal category M. We mainly check
in this section that the definitions and results of §I.7 remain valid in this setting.
We give a brief survey of these applications of our constructions after reviewing the
definition of a complete filtered module.
13.0.1. Complete objects in simplicial modules and in chain graded dg-modules.
Recall that the category of complete modules M = fˆ Mod consists of modules
M ∈ Mod equipped with a decreasing filtration
(1)
M = F0 M ⊃ · · · ⊃ Fs M ⊃ . . .
such that M = lims M/ Fs M . The morphisms of this category are the module
morphisms f : M → N which satisfy the relation f (Fs M ) ⊂ Fs N , for all s ∈ N.
In this chapter, we precisely deal with simplicial modules (respectively, chain
graded dg-modules) M equipped with a filtration as in (1), but where each Fs M
forms a sub-object of M in the category of simplicial modules (respectively, of
chain graded dg-modules) and such that the identity M = lims M/ Fs M holds in
this category s Mod (respectively, dg ∗ Mod ). In what follows, we generally use
the notation M = s fˆ Mod (respectively, M = dg ∗ fˆ Mod ), where we still use the
prefix fˆ to mark the consideration of complete filtered objects, for this category of
complete simplicial modules (respectively, of complete chain graded dg-modules ).
In principle, we should rather use the expression M = fˆ s Mod to refer to our
category of complete simplicial modules (defined as filtered objects in the category
of simplicial modules), whereas the notation M = s fˆ Mod refers to the category
of simplicial objects in the category of complete modules fˆ Mod . But we have an
obvious category identity fˆ s Mod = s fˆ Mod . We can therefore use both prefix
orders in the notation of the category of complete simplicial modules. We should
similarly use the notation M = fˆ dg ∗ Mod (rather than M = dg ∗ fˆ Mod ) for the
category of complete chain graded dg-modules since we define of a complete chain
graded dg-module as a filtered object in the category of chain graded dg-modules.
13.0. REMINDERS ON LIE ALGEBRAS AND ENVELOPING ALGEBRAS
397
But we can also identify an object of our category of complete chain graded dgmodules
M ∈ fˆ dg ∗ Mod with a module M equipped with a decomposition M =
L
d∈N Md such that Md is a (plain) complete module (in the sense of §I.7.3.1)
together with a differential δ : M → M consisting of morphisms δ : M∗ → M∗−1
in the category of complete modules fˆ Mod such that δ 2 = 0. We adopt the
commutation rule fˆ dg ∗ Mod = dg ∗ fˆ Mod to symbolize this identity of categories.
Besides complete simplicial modules, we consider a category of weight graded
simplicial modules s w Mod whose L
objects are simplicial modules M ∈ s Mod
∞
equipped with a decomposition M = s=0 M(s) in s Mod . We may equivalently assume that M is a simplicial object of the category of weight graded modules w Mod .
We symbolize this relationship by the category identity w s Mod = s w Mod . We
similarly consider a category of weight graded chain graded dg-modules dg ∗ w Mod
whose objects
L∞chain graded dg-modules M ∈ dg ∗ Mod equipped with a decomposition M = s=0 M(s) in dg ∗ Mod . We can also give an equivalent definition of this
notion, by swapping the order of the structures which we attach to our objects,
and we symbolize this relation by the category identity w dg ∗ Mod = dg ∗ w Mod .
In both cases, we refer to the summand M(s) as the homogeneous component of
weight s of our weight graded object M ∈ s w Mod (respectively, M ∈ dg ∗ w Mod ).
We then consider the natural functor E0 : s fˆ Mod → s w Mod (respectively,
0
E : dg ∗ fˆ Mod → dg ∗ w Mod ) which extends the functor E0 : fˆ Mod → w Mod
of §I.7.3.6 and maps any complete simplicial module (respectively,
complete chain
L∞
graded dg-module) M to the weight graded object E0 M = s=0 E0s M such that
E0s M = Fs M/ Fs+1 M,
for any s ∈ N.
13.0.2. Completed tensor products. We provide the category M = s fˆ Mod (respectively, M = fˆ dg Mod ) with a symmetrical monoidal structure. We adapt the
construction of §I.7.3.12. We now start with the tensor product of plain simplicial modules (respectively, plain chain graded dg-modules) M ⊗ N such as defined in §6.1. We
P provide this tensor product M ⊗ N with the filtration such that
Fr (M ⊗ N ) = p+q=r Fp (M ) ⊗ Fq (N ), for any r ∈ N, and we perform the comˆ = limr M ⊗ N/ Fr (M ⊗ N ) to get a
pletion with respect to this filtration M ⊗N
complete simplicial module (respectively, chain graded dg-module) associated to M
and N .
In the simplicial setting, we have an identity:
ˆ )n = Mn ⊗N
ˆ n,
(M ⊗N
for each dimension n ∈ N, where on the right-hand side, we form the tensor product
of the n-dimensional components of the objects M, N ∈ s fˆ Mod in the category of
complete modules fˆ Mod . The action of simplicial category on this tensor product
is also identified with the obvious diagonal action. Therefore, our tensor product
on the category of complete simplicial modules M = fˆ s Mod = s fˆ Mod is actually
identified with the usual diagonal extension, to simplicial objects, of the tensor
ˆ : fˆ Mod ×fˆ Mod → fˆ Mod . In the
product in the category of complete modules ⊗
differential graded setting, we similarly have an identity:
M
ˆ )n =
ˆ q,
(M ⊗N
Mp ⊗N
p+q=n
398
13. COMPLETE LIE ALGEBRAS AND CLASSIFYING SPACES
for each degree n ∈ N, where we consider the tensor products of the homogeneous
components of M and N in the category of complete modules fˆ Mod on the right
hand side.
Thus, in both cases M = s fˆ Mod , dg ∗ fˆ Mod , we can also identify our comˆ with a natural extension of the classical tenplete tensor product operation ⊗
sor product of simplicial modules (respectively, chain graded dg-modules) to complete objects. The completed tensor product of simplicial modules (respectively,
chain graded dg-modules) accordingly inherits an obvious associativity isomorˆ ⊗M
ˆ ' K ⊗(L
ˆ ⊗M
ˆ ) from the completed tensor product in fˆ Mod . The
phism (K ⊗L)
ground ring k, equipped with the trivial filtration such that F0 k = 0, F1 k = k, and
regarded as a constant simplicial module (respectively, as a dg-module concentrated
in degree zero) still defines a unit object in s fˆ Mod (respectively, in dg ∗ fˆ Mod ),
ˆ ' N ⊗M
ˆ , which is also identified with
and we have a symmetry isomorphism M ⊗N
a natural extension of the symmetry isomorphism of §5.3.2 (respectively, §5.3.1)
to complete objects. In the differential graded case, we still assume that this
ˆ
ˆ with
symmetry isomorphism involves a sign, so that we have c(x⊗y)
= ±y ⊗x,
deg(x) deg(y)
± = (−1)
, for any pair of homogeneous elements x ∈ M , y ∈ N .
We also have an obvious extension to the category of weight graded objects
s w Mod (respectively, dg ∗ w Mod ) of the tensor product of simplicial modules (respectively, of chain graded dg-modules). We readily deduce from our relations and
the observations of §I.7.3.13 that the functor E0 : s fˆ Mod → s w Mod (respectively,
E0 : dg ∗ fˆ Mod → dg ∗ w Mod ) of §13.0.1 is symmetric monoidal.
13.0.3. Lie algebras and enveloping algebras. The definitions and results of
§§I.7.2-7.3 extend to the simplicial module (respectively, chain graded dg-module)
setting. First, by applying the general concepts of §I.7.2 to the symmetric monoidal
category of complete simplicial modules M = s fˆ Mod (respectively, chain graded
dg-module M = dg ∗ fˆ Mod ), such as defined in §§13.0.1-13.0.2, we get the definition of the notion of a Lie algebra, of an associative algebra, and of a Hopf
algebra in M = s fˆ Mod (respectively, M = dg ∗ fˆ Mod ). For our purpose, we still
take the assumption that a complete Lie algebra g satisfies the connectedness condition E00 g = 0 (see §I.7.3.20), and we adopt the notation s fˆ Lie (respectively,
dg ∗ fˆ Lie) to refer to the subcategory of these Lie algebras in M = s fˆ Mod (respectively, M = dg ∗ fˆ Mod ). We similarly consider the category s fˆ As + (respectively,
dg ∗ fˆ As + ) whose objects A are the associative algebras in M = s fˆ Mod (respectively, M = dg ∗ fˆ Mod ) such that E00 A = k, and the category s fˆ Hopf Alg (respectively, dg ∗ fˆ Hopf Alg) whose objects A are the Hopf algebras in M = s fˆ Mod
(respectively, M = dg ∗ fˆ Mod ) which satisfy the same connectedness assumption
E00 A = k.
The complete tensor algebra, the complete symmetric algebra, and the enveloping algebra construction (see §§I.7.3.22-7.3.24) have an obvious extension to
simplicial modules (respectively, chain graded dg-modules). The Structure Theorem of §I.7.3 (see Theorem I.7.3.25), the Poincaré-Birkhoff-Witt Theorem, and
the Milnor-Moore Theorem (see Theorem I.7.3.26) also hold in this setting. The
validity of this extension follows from the functoriality of our methods in §I.7.
In the simplicial setting, we still have an identity between the Lie algebras, the
associative algebras, and the Hopf algebras in s fˆ Mod and the simplicial objects
in the category of Lie algebras, associative algebras, and Hopf algebras in fˆ Mod .
13.0. REMINDERS ON LIE ALGEBRAS AND ENVELOPING ALGEBRAS
399
(We just extend observations of §6.1.3 to complete modules.) Furthermore, the
complete enveloping algebra of a complete simplicial Lie algebra g ∈ s fˆ Lie can
be defined by an obvious dimensionwise application of the complete enveloping
algebra functor of §I.7.3.24. We accordingly have Û(g)n = Û(gn ), for all n ∈ N. We
have similar observations for complete tensor algebras and for symmetric algebras.
Furthermore, the isomorphisms that give the Structure Theorem, the PoincaréBirkhoff-Witt Theorem, and the Milnor-Moore Theorem are formed dimensionwise
in the category of complete modules.
Let g = ĝ be an object of our category of complete Lie algebras s fˆ Lie (respectively, dg ∗ fˆ Lie). Recall that the Poincaré-Birkhoff-Witt Theorem asserts that we
have an isomorphism of counitary cocommutative coalgebras induced by the natural
symmetrization map from the symmetric algebra to the tensor algebra:
'
e : Ŝ(g) −
→ Û(g).
In the dg-setting, we simply have to add a sign, arising from the permutation of
tensors, when we form this mapping. We therefore get an expression of the form
X
e(x1 · · · xr ) = (1/r!) ·
±(xσ(1) · · · xσ(r) ),
σ∈Σr
for a homogeneous monomial x1 · · · xr ∈ Sr (g).
13.0.4. Duality. Recall that we use the notation D(M ) = HomMod (M, k) for the
dual of a module M ∈ Mod . In the context of complete modules, we replace this
ordinary duality functor by the continuous duality functor such that:
D(M ) = colim D(M/ Fs+1 M ),
s
for any M ∈ fˆ Mod . In what follows we also use the notation M ∨ = D(M ) for
this continuous dual of a complete filtered module M ∈ fˆ Mod (especially when
we deal with objects equipped with comultiplicative or multiplicative structures).
This duality functor D : M 7→ M ∨ from the category of complete filtered modules
fˆ Mod to the category of ordinary modules Mod is unit-preserving (in the sense
that k∨ = k) and inherits a symmetric comonoidal transformation β : M ∨ ⊗ N ∨ →
ˆ )∨ (like the ordinary duality functor on plain modules).
(M ⊗N
This comonoidal transformation β is an isomorphism as soon as the weight
graded modules E0 M, E0 N ∈ w Mod associated to the complete modules M, N ∈
fˆ Mod are finitely generated in each weight dim E0s M, dim E0s N < ∞, ∀s ∈ N. (But
we do not need to assume that M or N are globally finitely generated as k-modules
in order to get this result.) In this situation, we say that the weight graded modules E0 M, E0 N ∈ w Mod are locally finitely generated. (Recall that we also assume
that the ground ring is a field throughout this chapter.) From this observation, we
deduce that the continuous dual of a complete unitary commutative (respectively,
associative) algebra A inherits a counitary cocommutative (respectively, coassociative) coalgebra structure as soon as the weight graded module E0 A underlying A
satisfies our local finiteness condition dim E0s A < ∞, ∀s ∈ N. The single existence of the comonoidal transformation implies, on the other hand, that the dual
of a counitary cocommutative (respectively, coassociative) coalgebra in complete
modules always forms a unitary commutative (respectively, associative) algebra.
For the complete symmetric algebra Ŝ(M ) of a complete weight graded module
M satisfying E00 M = 0 and such that E0 M is locally finitely generated in our sense,
400
13. COMPLETE LIE ALGEBRAS AND CLASSIFYING SPACES
we have an identity Ŝ(M )∨ = S(M ∨ ), where we consider the ordinary symmetric
algebra of the module M ∨ on the right-hand side. In this relation, we also use
the characteristic zero assumption to get an identity between the coinvariant modules (−)Σr , which define the summands of the symmetric algebra, and invariants
modules which naturally arise in the duality process.
In the case of a complete Lie algebra g ∈ fˆ Lie such that E00 g = 0, the identity E0 Û(g) = E0 Ŝ(g) = S(E0 g) implies that the complete enveloping algebra Û(g)
satisfies the local finiteness condition dim E0s Û(g) < ∞ as soon as g does. In this situation, the Poincaré-Birkhoff-Witt Theorem implies that we have an isomorphism
of unitary commutative algebras Û(g)∨ ' S(g)∨ ' S(g∨ ).
Recall that we define the dual of a simplicial module K ∈ s Mod as the cosimplicial object D(K) ∈ c Mod such that D(K)n = HomMod (Kn , k), for any n ∈ N
(see §5.0.9). The dual of a chain graded dg-module C ∈ dg ∗ Mod is a cochain
graded dg-module D(C) ∈ dg ∗ Mod satisfying D(C)n = HomMod (Cn , k), as in the
simplicial module case, and equipped with the differential δ : D(C) → D(C) such
that δ(u)(ξ) = ±u(δξ), for any homogeneous element u ∈ D(C), and any element
ξ ∈ C (see also §5.0.9). The sign ± in this expression is produced by the commutation of the cochain u ∈ D(C) with the differential symbol δ. In the case of complete
objects, we replace the ordinary duality functor considered in these constructions
by the continuous duality functor (−)∨ : fˆ Mod → Mod to get a continuous duality functor on simplicial modules D : s fˆ Mod → c Mod , and a continuous duality
functor on chain graded dg-modules D : dg ∗ fˆ Mod → dg ∗ Mod .
The above observations, about the structures of the continuous dual of algebras and coalgebras, extend to objects in simplicial modules and chain graded
dg-modules.
13.1. The cobar construction of complete enveloping algebras
In §7.1, we explain the definition of (a cosimplicial variant of) the Sullivan model
for the rational homotopy of spaces. We mainly prove that the rationalization of
a space X ∈ sSet is determined by the (cosimplicial) unitary commutative algebra
A(X) = QX naturally associated to X. The correspondence between complete Lie
algebras and classifying spaces which we examine in this chapter is related to an
anterior approach of rational homotopy theory, set up by Quillen, where the model
of a space X is given a counitary cocommutative chain dg-coalgebra which we
obtain from a simplicial Lie algebra associated to X.
To the enveloping algebra Û(g) of any complete simplicial algebra g ∈ s fˆ Lie,
we associate a bar complex B̂(Û(g)) which forms an object of the category of complete simplicial modules s fˆ Mod . The bar construction inherits the structure of a
simplicial counitary cocommutative coalgebra, and represents a simplicial counterpart of the counitary cocommutative chain dg-coalgebras considered by Quillen.
The cobar construction Bc (Û(g)∨ ), which we consider in the introduction of
this chapter, is the dual of this complete simplicial module B̂(Û(g)) ∈ s fˆ Mod .
We deal with the cobar construction Bc (Û(g)∨ ) in our constructions in order to
get a cosimplicial unitary commutative algebra rather than a simplicial counitary
cocommutative coalgebra and hence to retrieve an object of the model category
of §7.1. We can drop the connectivity assumptions occurring in the definition of
the Quillen model when we deal with cosimplicial unitary commutative algebras,
13.1. THE COBAR CONSTRUCTION OF COMPLETE ENVELOPING ALGEBRAS
401
but we have to restrict ourselves to complete Lie algebras which satisfy the local
finiteness condition of §13.0.4, because our constructions involve a duality process.
The definition of the cosimplicial algebra Bc (Û(g)∨ ) makes sense for any complete simplicial Lie algebra, but for our purpose, we can assume that g is a discrete
complete Lie algebra, with no simplicial structure, and we only examine this particular case in what follows.
We review the definition of the simplicial bar construction and of the cosimplicial cobar construction in the first part of this section. We prove afterwards that the
cobar construction Bc (Û(g)∨ ) forms a cofibrant object of the category of cosimplicial unitary commutative algebras and we check that the space G• Bc (Û(g)∨ ) ∈ sSet
associated to this cosimplicial algebra Bc (Û(g)∨ ) ∈ c Com + is identified with the
classifying space of the Malcev complete group G = G Û(g) of the Lie algebra
g ∈ fˆ Lie.
13.1.1. The simplicial bar and cosimplicial cobar constructions. We already
used simplicial bar complexes in §6.3. We then dealt with simplicial bar complexes
with coefficients B(R, A, S) = B• (R, A, S), and our purpose was to define simplicial
resolutions of pushouts in the category of cosimplicial commutative algebras.
The simplicial bar constructions which we consider in this chapter are formed
in the category of complete modules, and are associated to complete unitary associative algebras A ∈ fˆ As + equipped with an augmentation : A → k. Moreover,
we only take the ground field as coefficients R = S = k. We adopt the notation
B̂(A) = B̂(k, A, k) for this complete simplicial bar construction with trivial coefficients. For our purpose, we just consider the case of discrete algebras A (with no
simplicial structure). We also write fˆ As + / k for the category of complete unitary
associative algebras equipped with an augmentation over the ground field.
The simplicial object B̂(A) = B̂(A)• is defined, in any simplicial dimension
ˆ
n ∈ N, by the n-fold completed tensor product of our algebra B̂(A)n = A⊗n
. The
face operators di : B̂(A)n → B̂(A)n−1 are defined by:

ˆ . . . ⊗a
ˆ n,

for i = 0,
(a1 )a2 ⊗
ˆ . . . ⊗a
ˆ n ) = a1 ⊗
ˆ . . . ⊗a
ˆ i ai+1 ⊗
ˆ . . . ⊗a
ˆ n , for i = 1, . . . , n − 1,
di (a1 ⊗


ˆ . . . ⊗a
ˆ n−1 ,
(an )a1 ⊗
for i = n.
The degeneracy operators sj : B̂(A)n → B̂(A)n+1 are defined by:
ˆ . . . ⊗a
ˆ n ) = a1 ⊗
ˆ . . . ⊗a
ˆ j ⊗1
ˆ ⊗a
ˆ j+1 ⊗
ˆ . . . ⊗a
ˆ n,
sj (a1 ⊗
for all j = 0, . . . , n.
This complete bar complex B̂(A) is naturally identified with the completion of a
plain bar construction B(A) = B(k, A, k), defined by using the ordinary tensor
product Bn (A) = A⊗n instead of the completed one.
Besides the bar construction, we consider a cobar construction, which is a
cosimplicial object Bc (C) = Bc (C)• associated to any coaugmented counitary coassociative coalgebra C ∈ k / As c+ and defined, in any cosimplicial dimension n ∈ N,
by the n-fold tensor product of our coalgebra Bc (C)n = C ⊗n . The coface operators
of this cosimplicial object di : Bc (C)n−1 → Bc (C)n are defined by the formulas:


for i = 0,
1 ⊗ c1 ⊗ · · · ⊗ cn−1 ,
di (c1 ⊗ · · · ⊗ cn−1 ) = c1 ⊗ · · · ⊗ ∆(ci ) ⊗ · · · ⊗ cn−1 , for i = 1, . . . , n − 1,


c1 ⊗ · · · ⊗ cn−1 ⊗ 1,
for i = n,
402
13. COMPLETE LIE ALGEBRAS AND CLASSIFYING SPACES
where we set 1 = η(1) ∈ C for the unit element which determines the coaugmentation of C, and we use the notation ∆(c) ∈ C ⊗ C for the coproduct of an element
of C. The codegeneracy operators sj : Bc (C)n+1 → Bc (C)n are defined by:
sj (c1 ⊗ · · · ⊗ cn+1 ) = (cj )c1 ⊗ · · · ⊗ cbj ⊗ · · · ⊗ cn+1 ,
for all j = 0, . . . , n,
where we write : C → k for the counit morphism of our coalgebra C.
If a complete augmented algebra A ∈ fˆ As + / k satisfies the local finiteness
condition of §13.0.4 (explicitly, if the homogeneous components of the weight graded
module E0 A are finitely generated as k-modules), then we have an identity of
cosimplicial modules
D(B̂(A)) = Bc (A∨ ),
where D : s fˆ Mod → c Mod refers to the duality functor on complete simplicial
modules (see §13.0.4), and we consider the cobar construction of the counitary
coassociative coalgebra A∨ naturally associated to A on the right-hand side.
13.1.2. The simplicial bar construction of complete Hopf algebras. We readily
see that the complete simplicial bar construction of §13.1.1 satisfies B̂(k) = k,
and when we consider a tensor product of unitary associative algebras in complete
modules (by using the general symmetric monoidal structure of §I.7.3.11), we also
ˆ
ˆ B̂(B). The mapping B̂ : A 7→ B̂(A) therefore
get an identity B̂(A⊗B)
= B̂(A)⊗
defines a symmetric monoidal functor from the category of augmented unitary
associative algebras in complete modules fˆ As + / k towards the category of complete
simplicial modules s fˆ Mod . We deduce from this observation that the complete bar
construction B̂(H) of a complete Hopf algebra H, which we identify with a counitary
cocommutative coalgebra object in the symmetric monoidal category of complete
unitary associative algebras, forms a counitary cocommutative coalgebra in the
category of complete simplicial modules.
In the dual case of the cobar construction, we similarly have Bc (k) = k and
c
B (C ⊗ D) = Bc (C) ⊗ Bc (D). The mapping Bc : C 7→ Bc (C) therefore defines a
symmetric monoidal functor from the category of coaugmented counitary coassociative coalgebras k / As c+ towards the category of cosimplicial modules c Mod and
preserves unitary commutative algebra structures. We consider a complete Hopf
algebra H such that the weight graded object E0 H satisfies the local finiteness
condition of §13.0.4. We then get that the cobar complex Bc (H ∨ ), where we consider the dual coalgebra of H, inherits a unitary commutative algebra structure,
and the identity D(B̂(H)) = Bc (H ∨ ) in §13.1.1 also holds in the category of unitary
commutative algebras in cosimplicial modules.
We now assume that H = Û(g) is the enveloping algebra of a complete Lie
algebra g ∈ fˆ Lie + . We have the following theorem:
Theorem 13.1.3. Let g ∈ fˆ Lie be a complete Lie algebra. We assume that
this complete Lie algebra is connected as usual g = F1 g, and that the associated
weight graded object E0 g satisfies the local finiteness condition of §13.0.4 (explicitly,
the homogeneous components of this weight graded object form finite dimensional
modules).
(a) The cobar complex Bc (Û(g)∨ ), associated to the (continuous) dual of the
complete enveloping algebra of g, defines a cofibrant object of the category of cosimplicial unitary commutative algebras.
13.1. THE COBAR CONSTRUCTION OF COMPLETE ENVELOPING ALGEBRAS
403
(b) We have an identity of simplicial sets
G• Bc (Û(g)∨ ) = B(G Û(g)),
when we take the image of this cosimplicial algebra under the functor G• : c Com + →
sSet op of §7.2, and where, on the right-hand side of the relation, we consider the
classifying space B(G) of the group of group-like elements
ˆ
G = G Û(g) = {u ∈ Û(g)|(u) = 1, ∆(u) = u⊗u}
in our complete enveloping algebra Û(g) (see §I.8.1.2).
Proof. We establish assertion (a) first. We divide our proof in several steps.
Preliminaries. We consider the quotient modules ps g = g / Fs g, which we
identify with complete Lie algebras such that Fm (g / Fs g) = Fm g / Fs g for m < s,
and Fm (g / Fs g) = 0 for s ≥ m. The canonical projection qs : g → g / Fs g induces
'
an isomorphism Û(g)/ Fs Û(g) −
→ Û(g / Fs g)/ Fm Û(g / Fs g) for any m ≥ s, and
we readily deduce from this observation that the continuous dual of the complete
enveloping algebra Û(g) satisfies Û(g)∨ = colims Û(g / Fs+1 g)∨ . We then get
(1)
Bc (Û(g)∨ ) = colim Bc (Û(g / Fs+1 g)∨ ),
s
because the cobar construction clearly preserves sequential colimits.
We check that the morphisms q ∗ : Û(g / Fs g)∨ → Û(g / Fs+1 g)∨ induce cofibrations of cosimplicial unitary commutative algebras on the cobar construction. We
more precisely prove that these morphisms fit in pushout diagrams
(2)
S(Γ• (E0s (g)∨ ⊗ B2 ))
S(Γ• (E0s (g)∨ ⊗ E2 ))
φ
ψ
/ Bc (Û(g / Fs g)∨ ) ,
/ Bc (Û(g / Fs+1 g)∨ )
for all s ≥ 0, where we consider the symmetric algebras on the image of the cochain
graded dg-modules E0s (g)∨ ⊗ B2 , E0s (g)∨ ⊗ E2 ∈ dg ∗ Mod under the cosimplicial
Dold–Kan functor Γ• : dg ∗ Mod → c Mod .
Recall that E2 denotes the cochain graded dg-module generated by an element
1
e of degree 1 and an element b2 of degree 2 such that δ(e1 ) = b2 . We also
set B2 = k b2 ⊂ E2 (see §5.1.2). We identify (E0s g)∨ with a dg-module concentrated
in degree 0 when we form the tensor products E0s (g)∨ ⊗ B2 , E0s (g)∨ ⊗ E2 ∈ dg ∗ Mod .
In what follows, we also consider the dual of the dg-modules B2 , E2 ∈ dg ∗ Mod , and
we use the notation B2 = D(B2 ), E2 = D(E2 ), for these objects. We can also identify
E2 with the chain graded dg-module generated by an element e1 of degree 1 and an
element b2 of degree 2 such that δ(b2 ) = e1 . We also have an identity B2 = k b2
and the map E2 → B2 , dual to the embedding B2 ,→ E2 , is given by the obvious
projection of the module k e1 ⊕ k b2 onto the summand k b2 . We moreover have
D(E0s (g)∨ ⊗ B2 ) = E0s (g) ⊗ B2 and D(E0s (g)∨ ⊗ E2 ) = E0s (g) ⊗ E2 when the module
E0s (g) fulfills the finiteness assumption of our theorem.
The conclusion of the theorem will follow from the fact that the commutative
algebra morphisms S(Γ• (E0s (g)∨ ⊗ B2 )) → S(Γ• (E0s (g)∨ ⊗ E2 )), which we associate
to the cosimplicial module embeddings Γ• (E0s (g)∨ ⊗ B2 ) ,→ Γ• (E0s (g)∨ ⊗ E2 ), are
404
13. COMPLETE LIE ALGEBRAS AND CLASSIFYING SPACES
cofibrations (by definition of the model structure of cosimplicial unitary commutative algebras in §6.2), and from the fact that the class of cofibrations in a model
category is stable under pushouts and compositions (see Proposition 1.1.6).
The definition of our pushout (2) depends on the choice of a section s :
g / Fs (g) → g / Fs+1 (g) equivalent to a splitting
g / Fs+1 (g) = g / Fs (g) ⊕ E0s (g)
(3)
in the category of k-modules. Our general connectedness assumption on the Lie algebra E00 g = 0, which is equivalent to the relation g = F1 g, implies that we have the
inclusion relation [g, Fs g] ⊂ Fs+1 g for each s > 0. We accordingly have [E0s g, −] = 0
in the quotient Lie algebra g / Fs+1 g so that the module E0s g = Fs g / Fs+1 g is central in Û(g / Fs+1 g). We moreover have an identity of complete associative algebras
Û(g / Fs g) = Û(g / Fs+1 g)/hE0s (g)i
(4)
where we form the quotient of the complete algebra Û(g / Fs+1 g) over the (closure
of the) ideal generated by this central module E0s g = Fs g / Fs+1 g ⊂ Û(g / Fs+1 g).
We now tackle the construction of this pushout (2). We proceed as follows.
Step 1. The construction of the square. We first work with the chain graded
dg-modules E0s g ⊗ B2 , E0s g ⊗ E2 , dual to E0s (g)∨ ⊗ B2 , E0s (g)∨ ⊗ E2 ∈ dg ∗ Mod , and
with the normalized complexes N∗ (−) = N∗ B̂(Û(g / Fs+1 g)). We form a diagram:
(5)
···
δ
/ N3 (−)
ψ]
···
···
δ
/ N2 (−)
δ
ψ]
/0
/ E0s g ⊗ b2
/0
/ E0s g ⊗ b2
/ N1 (−)
δ
/ N0 (−) ,
ψ]
'
ψ]
/ E0s g ⊗ e1
/0
/0
/0
where we consider the chain complexes equivalent to these dg-modules E0s g ⊗ B2 ,
E0s g ⊗ E2 , and N∗ (−) = N∗ B̂(Û(g / Fs+1 g)).
We adopt the conventions of §I.7.3 for the structures attached to complete Hopf
algebras. In particular, we write I Û(−) for the augmentation ideal of any complete
enveloping algebra Û(−), and we use similar notation I Ŝ(−) for the augmentation
ideal of the complete symmetric algebra Ŝ(−). This augmentation ideal I Û(−)
occurs in the normalized complex of the bar construction of the enveloping algebra Û(−). Indeed, from the definition of the degeneracies in §13.1.1, and from the
expression of the normalized bar complex as a quotient over the image of degenˆ
eracies, we immediately get the identity Nn (B̂(Û(g))) = I Û(g)⊗n
, for each n ∈ N.
We consider the morphism of chain complexes ψ] : N∗ (−) → E0s g ⊗ E2 defined, in
degree 1, by the map
(6)
I Û(g / Fs+1 g) o
'
I Ŝ(g / Fs+1 g)
/ g / Fs+1 g
/ E0 g o
s
=
E0s g ⊗ e1 ,
where we take the isomorphism of the Poincaré-Birkhoff-Witt Theorem (see Theorem I.7.3.26), the projection onto the component Ŝ1 (g / Fs+1 g) = g / Fs+1 g of
the complete symmetric algebra Ŝ(g / Fs+1 g), and the projection onto the summand E0s g in our splitting (3). We use the preservation of differential to determine
13.1. THE COBAR CONSTRUCTION OF COMPLETE ENVELOPING ALGEBRAS
405
the component of degree 2 of this morphism ψ] . We go back to this construction
soon.
We also consider the obvious morphism E0s g ⊗ E2 → E0s g ⊗ B2 to complete our
diagram (5). We see that the map
(7)
/ E0 g ⊗ b2 ,
ˆ I Û(g / Fs+1 g)
I Û(g / Fs+1 g)⊗
s
=
δ
I Û(g / Fs+1 g) o
'
I Ŝ(g / Fs+1 g)
/ g / Fs+1 g
/ E0 g
s
which defines the morphism of chain complexes N∗ (−) → E0s g ⊗ E2 in degree 2,
ˆ I Û(g / Fs+1 g) such that u or v
vanishes over the tensors u ⊗ v ∈ I Û(g / Fs+1 g)⊗
0
belongs to the ideal generated by Es+1 g in Û(g / Fs+1 g). We basically use our
observation that E0s+1 g is central in Û(g / Fs+1 g) to establish this claim. Indeed, if
0
we have a product u · P
v = w · x such that
P w ∈ I Û(g / Fs+1 g), and x ∈ Es+1 g, then
we write w = e(ξ) = k ak · e(ξk ) + kl bkl · e(ξk · ξl ) + · · · , for a formal sum of
P
P
'
monomials ξ ∈ k ak · ξk + kl bkl · ξk · ξl + · · · ∈ I Ŝ(g), and where e : Ŝ(g) −
→ Û(g)
denotes the isomorphism of the Poincaré-Birkhoff-Witt Theorem (see our reminder
in §13.0.3). We useP
that x ∈ E0s g commutes
P with all factors of this expansion to get
an identity w · x = k ak /2 · e(ξk · x) + kl bkl /3 · e(ξk · ξl · x) + · · · , from which we
deduce that the image of the product w · x ∈ I Û(g / Fs+1 g) under our projection is
trivial.
We deduce from this vanishing result and identity (4) that the composite morphism N∗ B̂(Û(g / Fs+1 g)) → E0s g ⊗ E2 → E0s g ⊗ B2 in (5) factors through the normalized complex of the complete bar construction on Û(g / Fs g). We therefore have
a commutative square of dg-module morphisms
(8)
N∗ B̂(Û(g / Fs+1 g))
N∗ B̂(Û(g / Fs g))
ψ]
φ]
/ E0s g ⊗ E2
/ E0s g ⊗ B2
equivalent to our morphisms of chain complexes. We dualize this square and we
apply the cosimplicial Dold–Kan equivalence of §5.2 to get a commutative square
in the category of cosimplicial modules:
(9)
Γ• (E0s (g)∨ ⊗ B2 )
Γ• (E0s (g)∨ ⊗ E2 )
φ
ψ
/ Bc (Û(g / Fs g)∨ ) ,
/ Bc (Û(g / Fs+1 g)∨ )
from which we deduce the commutative square (2) in the category of cosimplicial
counitary commutative algebras. We are left to check that this commutative square
forms a pushout.
Step 2. The pushout property. If we forget about the action of coface operators in the cobar construction, then the Poincaré-Birkhoff-Witt Theorem (see
406
13. COMPLETE LIE ALGEBRAS AND CLASSIFYING SPACES
Theorem I.7.3.26) implies that we have an identity:
(10)
Bc (Û(g / Fs+1 g)∨ )n = S((g / Fs+1 g)∨ ⊕ · · · ⊕ (g / Fs+1 g)∨ )
|
{z
}
n
∨ n
= B (Û(g / Fs g) ) ⊗ S(E0s (g)∨ ⊕ · · · ⊕ E0s (g)∨ ),
|
{z
}
c
n
for each cosimplicial dimension n ∈ N. The image of an element of the form
(11)
di1 · · · din−1 (u ⊗ e1 ) ∈ Γn (E0s (g)∨ ⊗ e1 ),
under our morphism ψ : Γ• (E0s (g)∨ ⊗ E2 ) → Bc (Û(g / Fs+1 g)∨ ), where we assume
n ≥ i1 > · · · > in−1 ≥ 1, is given by a linear term in E0s (g)∨ ⊕ · · · ⊕ E0s (g)∨
plus higher order terms in the tensor product decomposition (10) of the algebra Bc (Û(g / Fs+1 g)∨ )n . We readily see that this mapping is one-to-one on linear
terms, and we deduce from this observation that our diagram of unitary commutative algebras (2) is a pushout in each cosimplicial dimension n ∈ N. We conclude
that (2) defines a pushout in the category of cosimplicial unitary commutative algebras and this verification finishes the proof of the first assertion (a) of our theorem.
The space associated to the cobar construction. We now determine the image
of the cosimplicial unitary commutative algebra Bc (Û(g)∨ ) = D(B̂(Û(g))) under
the functor G• : c Com + → sSet of §7.2. We have by definition G• Bc (Û(g)∨ ) =
Morc Com + (Bc (Û(g)∨ ), A(∆• )), where we set A(X) = kX = D(k[X]), for any X ∈
sSet.
For every simplicial dimension n ∈ N, we have a one-to-one duality correspondence between the morphisms of cosimplicial algebras u : D(B̂(Û(g))) → A(∆n ) and
the morphisms of simplicial coalgebras u] : k[∆n ] → B̂(Û(g)) such that u(c)(σ) =
c(u] ([σ])), for every simplex σ ∈ (∆n )m , and any continuous form c ∈ B̂(Û(g))∨
m.
To have this relation, we again use the local finiteness assumption of the theorem.
Recall that we have an identity Mors Mod (k[∆n ], L) = MorsSet (∆n , L) = Ln , for
any simplicial module L ∈ s Mod (see §5.0.7) because ∆n is defined by a representable functor on the simplicial category ∆ (see §0.3). Hence, the morphism u]
arising from our correspondence is uniquely determined as a morphism of simplicial
modules by an element g ∈ B̂(Û(g))n such that we have the relation g = u] ([ιn ]),
where ιn denotes the fundamental simplex of the n-simplex ιn ∈ (∆n )n (see §0.3).
We readily see that u] defines a morphism of coalgebras if and only if this element
g ∈ B̂(Û(g))n is group-like (see §I.7.1.14). We therefore get bijections
(12)
Gn (Bc (Û(g)∨ )) = Morc Com + (Bc (Û(g)∨ ), A(∆n )) ' G(B̂(Û(g))n )
which clearly intertwine the face and the degeneracy operators of our objects by
functoriality of our construction. We then use that the group-like element functor
is symmetric monoidal (see Proposition I.8.1.8) to get the relation
(13)
ˆ
G(B̂(Û(g))n ) = G(Û(g)⊗n ) = G(Û(g))×n ,
in each simplicial dimension n ∈ N, and we easily check that the face and degeneracy
operators on B̂(Û(g)) correspond to the face and degeneracy morphisms of the
classifying space of the group G = G Û(g). We therefore get that G• Bc (Û(g)∨ ) is
identified with this classifying space B = B(G Û(g)) as a simplicial object and this
observation completes the proof of the second assertion (b) of our theorem.
13.1. THE COBAR CONSTRUCTION OF COMPLETE ENVELOPING ALGEBRAS
407
From our proof of Theorem 13.1.3(a), we also deduce that the following refinement of our statement:
Proposition 13.1.4. The cobar construction Bc (Û(g)∨ ) on the dual of our
complete enveloping algebra Û(g) in Theorem 13.1.3 is identified with the colimit
of a sequence of cofibrations
k = Bc (Û(g / F1 g)∨ ) · · ·
· · · Bc (Û(g / Fs g)∨ ) Bc (Û(g / Fs+1 g)∨ ) · · ·
· · · colim Bc (Û(g / Fs+1 g)∨ ) = Bc (Û(g)∨ )
s
which fit in pushouts of the form:
S(Γ• (E0s (g)∨ ⊗ B2 ))
S(Γ• (E0s (g)∨ ⊗ E2 ))
/ Bc (Û(g / Fs g)∨ ) ,
/ Bc (Û(g / Fs+1 g)∨ )
for all s > 0.
13.1.5. The tower decomposition of the classifying space of a Malcev complete
group. In §I.8.2, we observe that the group G = G(H) associated to any complete
Hopf algebra H is equipped with a descending filtration G = F1 G ⊃ · · · ⊃ Fs G ⊃
· · · by subgroups Fs G ⊂ G such that (Fs G, Ft G) ⊂ Fs+t G for all s, t ≥ 1, and G =
lims Fs G (see Proposition I.8.2.3). Recall that we refer to this object G = G(H) as
the Malcev complete group associated to the Hopf algebra H.
By the complete Milnor-Moore Theorem (see Theorem I.7.3.26), we automatically have H = Û(g), for a complete Lie algebra such that g = P(H). If we assume
H = Û(g), then we have a group isomorphism Fs G/ Fs+1 G ' E0s g for all s ≥ 1,
where we consider the abelian group underlying the summand E0s g of the weight
graded object E0 g (see Proposition I.8.2.3), and we moreover have the relation
G/ Fs G ' G Û(g / Fs g), for all s ≥ 1 (see Proposition I.8.2.5).
We can apply the result of Theorem 13.1.3(b) to the quotients g / Fs g of our
complete Lie algebra g = lims g / Fs g. We then get an isomorphism of simplicial
sets G• Bc (Û(g / Fs g)∨ ) ' B(G/ Fs G), for every s ≥ 1, and we accordingly get
that the functor G• : c Com + → sSet op carries the sequence of cofibrations of
Proposition 13.1.4 to the tower of fibrations
(1)
B(G) = lim B(G/ Fs G) · · ·
s
· · · B(G/ Fs+1 G) B(G/ Fs G) · · ·
· · · B(G/ F1 G) = pt,
where we consider the classifying spaces B(G/ Fs G) associated to our tower of quotient groups G/ Fs G, s ≥ 1.
In §7.5.8, we also observed that the functor G• : c Com + → sSet op satisfies the relation G• (S(Γ• (M ))) = Γ• (D(M )), for any cochain graded dg-module
M ∈ dg ∗ Mod , where D(M ) ∈ dg ∗ Mod denotes the dual dg-module of M . We accordingly have G• (S(Γ• (E0s (g)∨ ⊗B2 ))) = Γ• (E0s g ⊗ B2 ) and G• (S(Γ• (E0s (g)∨ ⊗E2 ))) =
Γ• (E0s g ⊗ E2 ) for the symmetric algebras occurring in the pushout square of Proposition 13.1.4. The dg-module E0s g ⊗ B2 obviously reduces to the module E0s g in
408
13. COMPLETE LIE ALGEBRAS AND CLASSIFYING SPACES
degree 2. In §7.5.1, we observe that for a module of this form M = E[n], which
has a single component E ∈ Mod put in some degree n ∈ N, we can identify the
object Γ• (E[n]) with an Eilenberg–MacLane space K(E, n). We therefore have an
identity Γ• (E0s g ⊗ B2 ) = K(E0s g, 2) in our case. We can also identify the simplicial
module Γ• (E0s g ⊗ E2 ) with a contractible space L(E0s g, 2) of the form considered in
our models of principal fibrations in §7.5.4(b). We accordingly get that the functor
G• : c Com + → sSet op carries the pushout of Proposition 13.1.4 to the pullback:
(2)
B(G/ Fs+1 G)
/ L(E0s g, 2)
B(G/ Fs G)
/ K(E0s g, 2)
which is classically associated to the map of classifying space B(G/ Fs+1 G) →
B(G/ Fs G) when we identify G/ Fs G with the quotient of the group G/ Fs+1 G by
the subgroup E0s g ' Fs G/ Fs+1 G. Note simply that the relation (G, Fs G) ⊂ Fs+1 G
implies that Fs G/ Fs+1 G forms a central subgroup of G/ Fs+1 G.
Thus, the sequence of cofibrations and the pushouts of Proposition 13.1.4 represent counterparts of the tower decomposition (1) and of the pullback diagrams (2)
which we associate to the classifying space of our Malcev complete group G.
In the case where the filtration of our Lie algebra in Theorem 13.1.3 is bounded,
we can also use the equivalence of rational homotopy theory to determine the model
of the classifying space B(G Û(g)) from the result of our theorem. To be explicit,
we have the following statement:
Proposition 13.1.6. If the Lie algebra g of Theorem 13.1.3 satisfies g =
g / Fm g for some m ≥ 1, then we have a weak-equivalence of cosimplicial unitary
commutative algebras
∼
Bc (Û(g)∨ ) −
→ A(B(G Û(g))),
where we consider the cosimplicial algebra A(−) associated to the classifying space
of the group G = G Û(g) on the right-hand side.
Proof. The assumption g = g / Fm g implies, by the observations of §13.1.5,
that the classifying space B(G Û(g)) forms a Q-nilpotent space of finite Q-type
(see §7.5.3), while the cosimplicial unitary commutative algebra Bc (Û(g)∨ ) forms a
nilpotent cell complex of finite type by the result of Proposition 13.1.4 (see §7.5.7).
In this situation, we can use the result of Theorem 7.5.10 to conclude that the
'
adjoint morphism Bc (Û(g)∨ ) → A(B(G Û(g))) of the isomorphism B(G Û(g)) −
→
G• Bc (Û(g)∨ ) of Theorem 13.1.3 defines a weak-equivalence in the category of cosimplicial unitary commutative algebras.
13.1.7. Remarks: The rationalization of the classifying space of nilpotent groups.
We now assume that G is any nilpotent group. We equip this group with the lower
central series filtration G = Γ1 G ⊃ · · · ⊃ Γs G ⊃ · · · (see §I.8.2.1), which satisfies Γm G = 0, for some m ≥ 1. We have an analogue of the decomposition
of §13.1.5(1-2) for the tower of classifying spaces B(G/ Γs G), s ≥ 1. We just need
to replace the weight
graded Lie algebra E0 g in this construction §13.1.5(1-2) by the
L
0
object E G = s Γs G/ Γs+1 G which we determine from the filtration of our group
(see §I.8.2.2). We immediately see, by using this decomposition, that the space
13.2. THE CHEVALLEY–EILENBERG COMPLEX OF COMPLETE LIE ALGEBRAS
409
B(G) associated to our nilpotent group G is nilpotent of finite Q-type (see §7.5.3)
when E0s (G) ⊗Z Q is finitely generated as a Q-module for each weight s ≥ 1.
We briefly mention in §I.8.3.5 that the Malcev completion of a nilpotent group
Gb= G Q[G]bis equipped with a filtration Gb= F1 Gb⊃ · · · ⊃ Fs Gb⊃ · · · such that
Gb/ Fs Gb= (G/ Γs G)bfor every s ≥ 1, and is associated to a Lie algebra g = P Q[G]b
such that E0s g = E0s (G) ⊗Z Q, for each s ≥ 1. We immediately see, from this
correspondence, that the classifying space B(Gb) is identified with the rationalization
of the nilpotent space B(G), since we retrieve the tower decomposition considered in
the proof of Theorem 7.5.5 for this object. We accordingly have a weak-equivalence
∼
L G• A(B(G)) −
→ B(Gb), where we consider the image of the cosimplicial algebra
A(B(G)) ∈ c Com + under the left derived functor of G• : c Com + → sSet op . We
may again use the equivalence claim of Theorem 7.5.10 to deduce that we have
∼
a weak-equivalence A(B(Gb)) −
→ A(B(G)) from this statement. This observation
implies that the space B(G) is Q-good in the sense of Bousfield-Kan (see [20, §I.5]
and [19, §12]). To be explicit, we get that the map from B(G) to the rationalization
B(G)b= L G• A(B(G)) ∼ B(Gb) induces an isomorphism in rational homology.
Let g = P Q[G]b. We also have a weak-equivalence:
(1)
∼
Bc (Û(g)∨ ) −
→ A(B(G)),
∼
which we can obtain by lifting the weak-equivalence Bc (Û(g)∨ ) −
→ A(B(G)) of Propo∼
sition 13.1.6 through the morphism A(B(Gb)) −
→ A(B(G)) (observe that this morphism is trivially surjective, and hence, forms an acyclic fibration in the category
of cosimplicial unitary commutative algebras). We accordingly get that Bc (Û(g)∨ )
defines a cofibrant resolution of the model of the space B(G). We use a similar
result for the pure braid groups G = Pr in our study of E2 -operads. We then have
an analogue of the weak-equivalence (1) but we can not ensure that our spaces are
Q-good in this case. Let us mention that for certain examples of classifying spaces
B(G), where G is not a nilpotent group, the rationalization B(G)b= L G• A(B(G)) is
not equivalent to a classifying space, though such spaces may be good in certain
cases (see [20, §VII.3] for examples of such phenomena). We also refer to [48, §§7-8]
for a general study of the Sullivan model of classifying spaces B(G).
13.2. The Chevalley–Eilenberg complex of complete Lie algebras
We recall the definition of Chevalley–Eilenberg cochain complex in this section. We also recall the definition of a complete Chevalley–Eilenberg chain complex,
which is dual to the Chevalley–Eilenberg cochain complex. We aim to prove an
analogue of the statement of Theorem 13.1.3 for the Chevalley–Eilenberg cochain
complex C∗CE (g) of a complete Lie algebra g = ĝ. We also check that, in the situation of Theorem 13.1.3, the image of the cosimplicial bar construction of the dual
coalgebra of the enveloping algebra Û(g) under the commutative algebra upgrade
of the conormalized cochain complex functor N∗] : c Com + → dg ∗ Com + is identified
with the Chevalley–Eilenberg cochain complex of the Lie algebra g.
We first explain the definition of the Chevalley–Eilenberg chain and cochain
complexes in the general case of a complete Lie algebra in graded modules. We
use this setting in §14, when we explain the definition of a cofibrant model of the
operad of little n-discs.
In the definition of the Chevalley–Eilenberg complexes, we deal with a suspension functor Σ : M 7→ Σ M , defined on the category of dg-modules dg Mod , and
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13. COMPLETE LIE ALGEBRAS AND CLASSIFYING SPACES
which is given by the tensor product formula Σ M = M ⊗k e1 , for any M ∈ dg Mod ,
where e1 denotes a homogeneous element of lower degree 1. We also consider an
inverse desuspension functor Σ−1 : M 7→ Σ−1 M , which we identify with the tensor
product Σ−1 M = M ⊗ k e1 , where e1 now denotes a homogeneous element of upper
degree 1 (equivalent to a homogeneous element of lower degree −1). We see that
the suspension functor preserves the subcategory of chain graded dg-modules inside the category of dg-modules while the desuspension preserves the subcategory
of cochain graded dg-modules. We also consider the obvious extension of these
functors to complete chain graded dg-modules. We then have the duality relation
D(Σ M ) = Σ−1 D(M ) for every object M in this category dg ∗ fˆ Mod .
13.2.1. Complete Lie algebras in graded modules. In this section, we more
precisely consider complete Lie algebras g in the category of chain graded modules gr ∗ Mod . We use the same conventions as in the definition of a complete chain
graded dg-module in §13.0.1 in order to get the notion of a complete chain graded
module. We just forget about differentials. We can also identify our complete Lie
algebra g with a complete Lie algebra in chain graded dg-modules equipped with a
trivial differential. We still assume E00 g = 0, and we use the notation gr ∗ fˆ Lie for
the category of complete chain graded Lie algebras that satisfy this connectedness
condition. The weight graded Lie algebra E0 g associated to a complete chain graded
Lie algebra g forms an object of the category of chain graded modules in the sense
that each component of homogeneous weight of this weight graded Lie algebra E0s g
naturally defines an object of this category gr ∗ Mod . In our study, we assume that
this weight graded object in chain graded modules E0 g is locally finite in the sense
that each homogeneous component E0s g, s > 0, forms a chain graded module which
is finitely generated degreewise. The observations of §13.0.4 are valid under this
refined local finiteness requirement. We just consider the internal-hom of graded
modules in order to perform our duality constructions within the category gr ∗ Mod .
The results which we explain in this section have a straightforward generalization for complete Lie algebras in chain graded dg-modules, when we assume
that our Lie algebra g is equipped with a (possibly non-trivial) internal differential
δ : g → g. In this context, we just need to require that the differential increases the
filtration of our Lie algebra δ(Fs g) ⊂ Fs+1 g, for all s ≥ 1, in order to ensure that
our subsequent cofibration statements (see Theorem 13.2.5) remain valid. Now the
complete Lie algebras which we consider in the applications of this book are defined within the category of chain graded modules and are equipped with a trivial
internal differential δ = 0 in general. We therefore focus on this framework in what
follows, and we only give a survey of the applications of complete Lie algebras in
chain graded dg-modules in the concluding paragraph of this section.
13.2.2. The Chevalley–Eilenberg chain complex. We define the Chevalley-Eilenˆ
berg complex ĈCE
∗ (g) of a complete chain graded Lie algebra g ∈ gr ∗ f Lie as the
twisted complete dg-module:
(1)
ĈCE
∗ (g) = (Ŝ(Σ g), ∂),
equipped the twisting differential ∂ : Ŝ(Σ g) → Ŝ(Σ g) such that:
X
(2)
∂(ξ1 · · · ξn ) =
±[ξi , ξj ] · ξ1 · · · ξbi . . . · ξbj · · · ξn ,
i<j
for any ξ1 · · · ξn ∈ Ŝ(Σ g).
13.2. THE CHEVALLEY–EILENBERG COMPLEX OF COMPLETE LIE ALGEBRAS
411
In this expression, we consider the complete symmetric algebra on the suspension Σ g of the underlying graded module of our Lie algebra g, and we form the
composite
(3)
'
[−,−]
Σg⊗Σg −
→ Σ2 (g ⊗ g) −−−→ Σ g
to extend the Lie bracket of g to Σ g. We accordingly get a homomorphism [−, −] :
Σ g ⊗ Σ g → Σ g of degree −1. The sign in our formula is produced by the tensor
permutation
ξ1 ⊗ · · · ⊗ ξi ⊗ · · · ⊗ ξj ⊗ · · · ⊗ ξn 7→ ξi ⊗ ξj ⊗ ξ1 ⊗ · · · ⊗ ξbi ⊗ · · · ⊗ ξbj ⊗ · · · ⊗ ξn
involved in the expression of our differential. We also have a sign yielded by the
isomorphism Σ g ⊗ Σ g ' Σ2 (g ⊗ g) (which is equivalent to a tensor permutation) in
the extension of the bracket to the suspension (3). If g is a plain complete Lie
algebra (equivalent to a graded object concentrated in degree 0), then this extra
sign vanishes and we have ± = (−1)i+j .
We take the obvious extension of the map (2) to formal sums of monomials
in order to determine our differential on the complete symmetric algebra Ŝ(Σ g).
The relation of differentials ∂ 2 = 0 follows from an obvious verification. We also
see that we have the inclusion relation [Fp g, Fq g] ⊂ Fp+q g ⇒ ∂(Fs Ŝ(g)) ⊂ Fs Ŝ(g)
for the filtration of the complete symmetric algebra Ŝ(Σ g). We accordingly get
that ĈCE
∗ (g) = (Ŝ(Σ g), ∂) forms a complete chain graded dg-module in the sense
of §13.0.1.
13.2.3. The coalgebra structure of the Chevalley–Eilenberg chain complex. In
the chain complex ĈCE
∗ (g) = (Ŝ(Σ g), ∂), we actually regard the complete symmetric
algebra Ŝ(Σ g) as a counitary cocommutative coalgebra (not as unitary commutative
algebra).
The augmentation of this counitary cocommutative coalgebra structure :
Ŝ(Σ M ) → k is defined by the obvious projection, and we provide Ŝ(Σ g) with the
coproduct such that
X
ˆ j 1 · · · ξj q
(1)
∆(ξ1 · · · ξn ) =
±ξi1 · · · ξip ⊗ξ
{i1 <···<ip }q{j1 <···<jq }
={1<···<n}
for any monomial ξ1 · · · ξn ∈ Ŝ(Σ g) (see §I.7.2.6 and §I.7.3.22).
We readily see that our differential satisfies the counit relation ∂ = 0 as well as
the coderivation relation ∆∂ = (∂⊗id + id ⊗∂)·∆ with respect to the coproduct (1).
Thus, the object ĈCE
∗ (g) = (Ŝ(Σ g), ∂) actually forms a counitary cocommutative
coalgebra in the symmetric monoidal category of complete chain graded dg-modules.
13.2.4. The Chevalley–Eilenberg cochain complex. We now consider a complete
chain graded Lie algebra g ∈ gr ∗ fˆ Lie which satisfies the local finiteness assumption
of §13.2.1 in addition to our standard connectedness condition E00 g = 0.
We define the Chevalley–Eilenberg cochain complex of this chain graded Lie
algebra g as the (continuous) dual C∗CE (g) = D ĈCE
∗ (g) of the complete Chevalley–
Eilenberg chain complex of §13.2.2. We deduce from the general duality identities
D(Σ M ) = Σ−1 D(M ) and Ŝ(−)∨ = S(D(−)) (see §13.0.4) that this cochain complex
is identified with a twisted symmetric algebra such that:
(1)
C∗CE (g) = (S(Σ−1 g∨ ), ∂).
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13. COMPLETE LIE ALGEBRAS AND CLASSIFYING SPACES
In §13.0.4, we observed that the dual of a counitary cocommutative coalgebra
in complete dg-modules inherits a unitary commutative algebra structure. We can
also readily check that the dual of the complete symmetric coalgebra, which we
use in the definition of the Chevalley–Eilenberg chain complex, is identified with
a symmetric algebra as a unitary commutative algebra. We therefore obtain that
our cochain complex C∗CE (g) forms a quasi-free object in the category of unitary
commutative algebras in the sense of §6.2.7.
The twisting differential ∂ in our expression (1) can be determined from the
formula of the Chevalley–Eilenberg differential in §13.2.2(2), after making explicit
the duality relation between the symmetric algebra and the complete symmetric
algebra.
For this aim, we consider a basis of the Lie algebra g, whose elements, let ξα ,
form representatives of homogeneous basis elements ξ¯α ∈ E0 g of the weight graded
Lie algebra E0 g. To be explicitly, we assume that each ξ¯α has a homogeneous
weight sα and a homogeneous degree dα = deg(ξ¯α ). Note that the basis elements
ξ¯α with a prescribed weight sα = s and degree dα = d form a finite sets by our
assumption on g. Let (ξ α )α denote the dual of the basis (ξα )α in the module g∨ .
The Lie bracket on g is defined by an expression of the form
X γ
(2)
[ξα , ξβ ] =
cαβ ξγ
γ
for structure constants cγαβ ∈ k such that cγαβ = (−1)dα dβ +1 cγβα . The inclusion
relation [Fp g, Fq g] ⊂ Fp+q g implies
(3)
cγαβ = 0
when sγ < sα + sβ and dγ 6= dα + dβ .
Then we get the expression
(4)
∂(Σ−1 ξ γ ) =
X
(−1)dα dβ +1 cγαβ · (Σ−1 ξ α ) · (Σ−1 ξ β )
αβ
for the differential of a generating element Σ−1 ξ γ ∈ Σ−1 g∨ in C∗CE (g). Note that
this formula makes sense because we only have a finite number of pairs (ξ α , ξ β )
such that cγαβ 6= 0.
The result of Theorem 13.1.3 has the following analogue for the Chevalley–
Eilenberg cochain complex of a complete Lie algebra in chain graded modules:
Theorem 13.2.5. Let g ∈ gr ∗ fˆ Lie be a complete Lie algebra in chain graded
modules. We assume that this complete Lie algebra satisfies our usual connectedness
assumption g = F1 g, and that the associated weight graded object E0 g satisfies the
local finiteness condition of §13.2.1 (explicitly, the homogeneous components of this
weight graded object E0s g are finitely generated modules degreewise).
(a) The Chevalley–Eilenberg cochain complex C∗CE (g) = (S(Σ−1 g∨ ), ∂) associated to g defines a cofibrant object of the category of unitary commutative cochain
dg-algebras.
ˆ Ω∗ (∆• ) ∈ s dg Lie
(b) We consider the simplicial Lie algebra in dg-modules g ⊗
which we form by taking the tensor product of the object g ∈ gr ∗ fˆ Lie with the
Sullivan cochain dg-algebra Ω∗ (∆• ) ∈ s dg ∗ Com + (we give more details on the
definition of this object in the proof of our statement). We have an identity of
13.2. THE CHEVALLEY–EILENBERG COMPLEX OF COMPLETE LIE ALGEBRAS
413
simplicial sets
G• C∗CE (g) = MC• (g),
when we take the image of the cochain dg-algebra C∗CE (g) ∈ dg ∗ Com + under the
ˆ Ω∗ (∆• )) defunctor G• : dg ∗ Com + → sSet op of §7.4, and where MC• (g) = MC(g ⊗
∗
•
ˆ Ω (∆ ) which are homogeneous of (upnotes the simplicial set of elements γ ∈ g ⊗
per) degree 1 and satisfy the Maurer–Cartan equation
1
δ(γ) + [γ, γ] = 0
2
ˆ Ω∗ (∆• ) ∈ dg Lie.
in g ⊗
These assertions are classical (we notably refer to [58] for the second assertions).
The equation δ(γ) + 12 [γ, γ] = 0 in this theorem is obviously a generalization of the
classical Maurer–Cartan equation which occurs in Lie theory (see for instance [83,
§I.4], [114, §2.4]). We therefore call this equation the Maurer–Cartan equation.
ˆ Ω∗ (∆• )) is the Maurer–Cartan
We also say that the simplicial set MC• (g) = MC(g ⊗
space associated to the complete chain graded Lie algebra g ∈ gr ∗ fˆ Lie.
Explanations and proof. We address the proof of each assertion of this
theorem separately.
Proof of assertion (a). We may adapt the arguments of Theorem 13.1.3 in
order to establish the first assertion of our theorem (a), but we will rather use
this construction in a subsequent statement, where we examine the correspondence
between the Chevalley–Eilenberg complex of an (ungraded) complete Lie algebra
and the cobar complex of Theorem 13.1.3. We prefer to use the characterization
of the cofibrant cell complexes of unitary commutative dg-algebras of §6.2.7 in
order to give a simple a direct verification of our assertion for the moment. In
short, we already observed that the Chevalley–Eilenberg cochain complex C∗CE (g) =
(S(Σ−1 g∨ ), ∂) forms a quasi-free unitary commutative dg-algebra. We just have to
check that the dg-module Σ−1 g∨ , which defines the generating object of this quasifree algebra, is equipped with a filtration of the form considered in §6.2.7(1) and
that the Chevalley–Eilenberg differential fulfills our filtration condition §6.2.7(2).
For this purpose, we use our basis (Σ−1 ξ α )α of the dg-module Σ−1 g∨ and the
expression of the Chevalley–Eilenberg differential on this basis §13.2.4(4). We take
the module spanned by elements Σ−1 ξ α ∈ Σ−1 g∨ associated to homogeneous basis
elements ξ¯α ∈ E0 g of weight
L sα ≤ s to define the sth layer of our filtration. We
explicitly set Fs (Σ−1 g∨ ) = sα ≤s k(Σ−1 ξ α ), for each s > 0.
The connectedness assumption E00 g = 0 implies that the weight sα associated to
any such Σ−1 ξ α ∈ Σ−1 g∨ satisfies sα > 0. The homogeneity relation in §13.2.4(3)
also implies that the weight of the factors Σ−1 ξ α , Σ−1 ξ β ∈ Σ−1 g∨ in the expression
of the Chevalley–Eilenberg differential §13.2.4(4) of any element Σ−1 ξ γ ∈ Σ−1 g∨
satisfies sα , sβ < sγ . We accordingly have Σ−1 ξ γ ∈ Fs (Σ−1 g∨ ) ⇒ ∂(Σ−1 ξ γ ) ∈
S(Fs−1 (Σ−1 g∨ )), and this inclusion relation is exactly the requirement of our filtration condition in §6.2.7. We therefore have, according to §6.2.7, all structures required to define a cofibrant cell complex decomposition on C∗CE (g) = (S(Σ−1 g∨ ), ∂)
from our filtration.
Explanations on the definition of the Maurer–Cartan space and proof of assertion (b). We identify the Lie algebra g with a graded module concentrated in
414
13. COMPLETE LIE ALGEBRAS AND CLASSIFYING SPACES
non-negative degree and the cochain dg-algebra Ω∗ (∆• ) with a (lower graded) dgmodule concentrated in non-positive degree when we form the tensor product of our
statement. To be more precise, we use these identities, where we forget about the
filtration of our Lie algebra g, to form the tensor product g ⊗ Ω∗ (∆• ) in the category
of dg-modules dg Mod in a first step. Then we provide this tensor product with the
filtration such that Fs (g ⊗ Ω∗ (∆• )) = Fs (g) ⊗ Ω∗ (∆• ) for every s ≥ 1. The notation
ˆ Ω∗ (∆• ), which we use in our statement, refers to the consideration of a complete
g⊗
ˆ Ω∗ (∆• ) = lims g ⊗ Ω∗ (∆• )/ Fs (g ⊗ Ω∗ (∆• )) in our constructions. Let
dg-module g ⊗
us simply observe that we have the relation:
(1)
lim(g ⊗ Ω∗ (∆• )/ Fs (g) ⊗ Ω∗ (∆• )) = (lim g / Fs g) ⊗ Ω∗ (∆• )
s
s
ˆ Ω∗ (∆• ) = g ⊗ Ω∗ (∆• )
⇒ g⊗
since the Sullivan dg-algebra Ω∗ (∆• ) forms a finitely generated module degreewise.
If we use the correspondence between upper grading and lower grading, then
we get the formula
Y
ˆ Ω∗ (∆• ))k =
(2)
(g ⊗
gl ⊗ Ωk+l (∆• )
l≥0
ˆ Ω∗ (∆• ). This dgfor the component of upper degree k ∈ Z of our object g ⊗
∗
∗
•
ˆ Ω (∆ ) is equipped with the differential δ : g ⊗
ˆ Ω (∆• ) → g ⊗
ˆ Ω∗ (∆• )
module g ⊗
∗
•
inherited from the Sullivan dg-algebra Ω (∆ ) (since we assume that g is equipped
with a trivial differential). Then we use the obvious extension of the Lie bracket
[−, −] : g ⊗ g → g to the tensor product with the cochain dg-algebra Ω∗ (∆• ) in
ˆ Ω∗ (∆• ) with a Lie algebra structure. We explicitly
order to provide this object g ⊗
set [ξα ⊗ ωα , ξβ ⊗ ωβ ] = ±[ξα , ξβ ] ⊗ ωα ωβ , for any pair of tensors ξα ⊗ ωα , ξβ ⊗ ωβ ∈
ˆ Ω∗ (∆• ), where we consider the Lie bracket of the elements ξα , ξβ ∈ g on the one
g⊗
hand, and the product of the forms ωα , ωβ ∈ Ω∗ (∆• ) on the other hand.
Recall that the functor G• : dg ∗ Com + → sSet op is defined by the formula
G• (A) = Mordg ∗ Com + (A, Ω∗ (∆• )), for any A ∈ dg ∗ Com + . To get the identity of
our theorem, we use that any morphism φ : C∗CE (g) → Ω∗ (∆• ) is determined by
its restriction to the generators of the symmetric algebra C∗CE (g)[ = S(Σ−1 g∨ )
when we forget about differentials, and the homomorphism φ|Σ−1 g∨ : Σ−1 g∨ →
Ω∗ (∆• ) which we consider in this construction can also be determined by an element
ˆ Ω∗ (∆• ) of upper degree deg∗ (γ) = 1. To be explicit, if we go back to
γ ∈ g⊗
ˆ Ω∗ (∆• ) has an
the notation ofP§13.2.4, and we assume that our element γ ∈ g ⊗
expansion γ = α ξα ⊗ ωα , where we still set (ξα )α for the basis of the Lie algebra
g and ωα ∈ Ω∗ (∆• ), then we get the formula
φ(Σ−1 ξ α ) = ωα
(3)
for the image of the basis elements Σ−1 ξ α ∈ Σ−1 g∨ ⊂ S(Σ−1 g∨ ) under the morˆ Ω∗ (∆• ).
phism φ : S(Σ−1 g∨ ) → Ω∗ (∆• )[ associated to γ ∈ g ⊗
−1 α
Then we readily check that the relation φ∂(Σ ξ ) = δφ(Σ−1 ξ α ), where we consider the differential of a generating element Σ−1 ξ α ∈ Σ−1 g∨ in C∗CE (g) is equivalent
to the equation of the theorem:
X
1X
±[ξα , ξβ ] ⊗ ωα ωβ .
(4)
±ξα ⊗ δωα =
2
α
αβ
{z
}
|
|
{z
}
=δγ
=[γ,γ]
13.2. THE CHEVALLEY–EILENBERG COMPLEX OF COMPLETE LIE ALGEBRAS
415
To complete this verification, we just use that the validity of the relation φ∂(Σ−1 ξ α ) =
δφ(Σ−1 ξ α ), for generating elements Σ−1 ξ α ∈ Σ−1 g∨ implies that φ preserves the differential on the whole quasi-free dg-algebra C∗CE (g) = (S(Σ−1 g∨ ), ∂). We finally get
ˆ Ω∗ (∆• )) =: MC• (g) asserted in our statement.
the identity G• C∗CE (g) = MC(g ⊗
This assertion of Theorem 13.2.5(a) has the following refinement, which parallels the observations of Proposition 13.1.4 about the decomposition of the cobar
construction of the dual of the complete enveloping algebra of a Lie algebra:
Proposition 13.2.6. The Chevalley–Eilenberg complex C∗CE (g) on our complete chain graded Lie algebra g ∈ dg ∗ fˆ Lie in Theorem 13.2.5 is identified with
the colimit of a sequence of cofibrations
k = C∗CE (g / F1 g) · · ·
· · · C∗CE (g / Fs g) C∗CE (g / Fs+1 g) · · ·
· · · colim C∗CE (g / Fs+1 g) = C∗CE (g)
s
which fit in pushouts of the form:
S(E0s (g)∨ ⊗ B2 )
S(E0s (g)∨ ⊗ E2 )
/ C∗ (g / Fs g) ,
CE
/ C∗ (g / Fs+1 g)
CE
for all s > 0.
Explanations. We may again use a generalization of the arguments of Theorem 13.1.3 to get such a refinement of our result, but we do not need this construction yet. We still prefer to rely on the observations of §6.2.7, where we explain the
general correspondence between a cofibrant cell complex structure in the category
of unitary commutative cochain dg-algebras and a filtration of the form considered
in the proof of Theorem 13.2.5.
We immediately obtain, from the general correspondence of §6.2.7, that the
algebras C∗CE (g / Fs g) = (S(Σ−1 (g / Fs g)∨ ), ∂) represent the layers of a cell complex structure on C∗CE (g) = (S(Σ−1 (g)∨ ), ∂) and the cells, which attach to get
C∗CE (g / Fs+1 g) from C∗CE (g / Fs g), are symmetric algebra morphisms S(Bdα +2 ) →
S(Edα +2 ) associated to the basis elements Σ−1 ξ α ∈ Σ−1 g∨ with weight sα = s.
α
Recall that dα denotes the degree of the
Edα +2 =
Nelement ξdα.+2We have 0an identity
2
2
α
∨
k ξ ⊗ E , and we immediately get that sα =s S(E
) = S(Es (g) ⊗ E ). We simN
dα +2
2
0
∨
ilarly have sα =s S(B
) = S(Es (g) ⊗ E ), and we therefore readily see that we
deal with cell attachments equivalent to the pushouts of our proposition when we
apply the correspondence of §6.2.7 to get the cell decomposition of our dg-algebra
C∗CE (g).
13.2.7. The decomposition of the Maurer–Cartan space associated to a complete
Lie algebra. We can use the result of the previous proposition to retrieve a decomposition of the space MC• (g) = G• C∗CE (g) which we associate to our complete Lie
algebra g ∈ gr ∗ fˆ Lie.
416
13. COMPLETE LIE ALGEBRAS AND CLASSIFYING SPACES
First, we can apply the result of Theorem 13.2.5(b) to the quotients g / Fs g of
our complete Lie algebra g = lims g / Fs g. We then get the identity
(1)
G• C∗CE (g / Fs g) = MC(g / Fs g ⊗ Ω∗ (∆• )) = MC• (g / Fs g),
for every s ≥ 1. We moreover obtain that the functor G• : dg ∗ Com + → sSet op
carries the sequence of cofibrations of Proposition 13.2.6 to the tower of fibrations
(2)
MC• (g) = lim MC• (g / Fs g) · · ·
s
· · · MC• (g / Fs+1 g) MC• (g / Fs g) · · ·
· · · MC• (g / F1 g) = pt,
where we consider the morphisms of Maurer–Cartan spaces induced by the canonical morphisms of Lie algebras g / Fs+1 g → g / Fs g for s ≥ 1.
We can, as in the context of cosimplicial algebras (see §13.1.5), identify the
image of a symmetric algebra S(M ) under our functor G• : dg ∗ Com + → sSet op
with the simplicial module Γ• (D(M )) ∈ s Mod , where we consider the dual chain
graded dg-module D(M ) ∈ dg ∗ Mod of the object M ∈ dg ∗ Mod . We more precisely
have an identity G• S(M ) = Mordg ∗ Mod (M, Ω∗ (∆• )) = ΓΩ• (D(M )), where we consider
the variant of the Dold–Kan functor ΓΩ• : dg ∗ Mod → s Mod which we associate to
the simplicial object Ω∗ (∆• ) in §7.3.12. Recall simply that we have a natural weak∼
equivalence ΓΩ• (N ) −
→ ΓN• (N ), which connect this functor ΓΩ• : N 7→ ΓΩ• (N ) to the
standard Dold–Kan functor Γ• = ΓN• : N 7→ ΓN• (N ), for any N ∈ dg ∗ Mod .
We then have the relations G• (S(E0s (g)∨ ⊗B2 )) = ΓΩ• (E0s g ⊗ B2 ) and G• (S(E0s (g)∨ ⊗
2
E )) = ΓΩ• (E0s g ⊗ E2 ) for the symmetric algebras occurring in the pushout square
of Proposition 13.2.6. We can identify the tensor product E0s g ⊗ B2 with the double suspension of the object E0s g in the category of chain graded dg-modules. We
equivalently have E0s g ⊗ B2 = Σ2 E0s g. We similarly have E0s g ⊗ E2 = Cyl Σ E0s g =
Σ Cyl E0s g, where Cyl M denotes the standard cylinder object functor, defined by
the tensor product Cyl M = M ⊗ E1 , on the category of chain graded dg-modules
M ∈ dg ∗ Mod . (The dg-module E1 is obviously given by the same construction as
E2 , with a generator e0 in lower degree 0, a generator b1 in lower degree 1, and the
differential such that δ(b1 ) = e0 .)
We finally get that the functor G• : c Com + → sSet op carries the pushout of
Proposition 13.2.6 to the pullback:
(3)
MC• (g / Fs+1 g)
/ ΓΩ• (Cyl Σ(E0s g))
MC• (g / Fs g)
/ ΓΩ• (Σ2 (E0s g))
in the category of simplicial sets. In each case, we consider the morphism of simplicial modules induced by the projection M ⊗ E1 → M ⊗ k b1 = Σ M on the cylinder
object Cyl M = M ⊗ E1 .
We can identify any dg-module M with the summand M ⊗ k e0 of the cylinder Cyl M = M ⊗ E1 . We accordingly have an inclusion ΓΩ• (M ) ⊂ ΓΩ• (Cyl M )
when we apply our Dold–Kan functor ΓΩ• , and we can also identify the simplicial
module ΓΩ• (Σ E0s g) with the fiber of the map MC• (g / Fs+1 g) MC• (g / Fs g) in our
diagram (3). Then, from our pullback (3), we immediately conclude that each map
13.2. THE CHEVALLEY–EILENBERG COMPLEX OF COMPLETE LIE ALGEBRAS
417
MC• (g / Fs+1 g) MC• (g / Fs g) in our tower (2) forms a principal fibration with this
simplicial module ΓΩ• (Σ E0s g) as fiber.
We can give simple conditions on the weight graded Lie algebra E0 g in order to apply the equivalence of rational homotopy theory to the object C∗CE (g)
and to retrieve the model of the Maurer–Cartan space MC• (g) from the result of
Theorem 13.2.5. To be explicit, we have the following statement:
Proposition 13.2.8. If the components of the weight graded module E0 (g) associated to the Lie algebra g of Theorem 13.1.3 satisfy the connectedness relation
n < ns ⇒ E0s (g)n = 0 for a sequence of integers such that ns → ∞, then we have a
weak-equivalence of unitary commutative cochain dg-algebras
∼
C∗CE (g) −
→ Ω∗ (MC• (g)),
where we consider the cochain dg-algebra Ω∗ (−) associated to the Maurer–Cartan
space X = MC• (g) on the right-hand side.
Proof. We adapt the argument lines of the proof of Proposition 7.5.9. We
easily check that we have a weak-equivalence
(1)
∼
C∗CE (g / Fs g) −
→ Ω∗ (G• C∗CE (g / Fs g)) ' Ω∗ (MC• (g / Fs g))
at each level s ≥ 0, by using the correspondence of §7.5.4(a-b) and of §7.5.8(a-b).
We then use the connectedness assumption of the proposition to pass to the colimit
s → ∞. We fix m ∈ N, and s0 such that s ≥ s0 ⇒ ns ≥ m. We readily check, by
using the cartesian squares of §13.2.7, that the map MC• (g) → MC• (g / Fs g) induces
an isomorphism in degree n < m on homotopy groups when s ≥ s0 . We use
the Hurewicz isomorphism theorem to deduce from this statement that we have
an identity H∗ (MC• (g), Q) = colims H∗ (MC• (g / Fs g), Q) at the (rational) cohomology
level, and this assertion implies that we can perform the colimit s → ∞ in the
level-wise weak-equivalence (1) to get the weak-equivalence of the proposition. We go back to the case of a complete Lie algebra in plain (ungraded) modules
g ∈ fˆ Lie, which we identify with a complete chain graded Lie algebra concentrated
in degree 0. We then have the following relationship between the cobar construction
of Theorem 13.1.3 and the Chevalley–Eilenberg cochain complex of Theorem 13.2.5:
Theorem 13.2.9. In the situation of Theorem 13.1.3, where g ∈ fˆ Lie is a
complete Lie algebra in plain (ungraded) modules such that E0s g forms a finitely
generated module in each weight s ≥ 1, we have a canonical isomorphism of unitary
commutative cochain dg-algebras:
'
N∗] Bc (Û(g)∨ ) −
→ C∗CE (g),
where we consider the image of the cobar complex of Theorem 13.1.3 under the
unitary commutative algebra upgrade of the conormalization functor of §6.4 on the
left-hand side, the Chevalley–Eilenberg cochain complex of the complete Lie algebra
g on the right-hand side.
Recall that we have a weak-equivalence G• N∗] Bc (Û(g)∨ ) ∼ G• Bc (Û(g)∨ ), where
we consider the image of the cochain dg-algebra N∗] Bc (Û(g)∨ ) ∈ dg ∗ Com + under
the functor G• = GΩ• : dg ∗ Com + → sSet op on the left-hand side, and the image of
the cosimplicial algebra Bc (Û(g)∨ ) ∈ c Com + under the functor G• = GA• : c Com + →
418
13. COMPLETE LIE ALGEBRAS AND CLASSIFYING SPACES
sSet op on the right-hand side (see Theorem 7.4.4). Theorem 13.2.9 therefore implies that, for a plain complete Lie algebra g ∈ fˆ Lie, the Maurer–Cartan space
MC(g ⊗ Ω∗ (∆• )) = G• C∗CE (g) in Theorem 13.2.5 is weakly-equivalent to the classifying space B(G) of the Malcev complete group G = G Û(g) associated to g (see
Theorem 13.1.3 and §13.1.5). We refer to [37? ] and to [58] for another approach
and for further generalizations of this statement.
Proof. We divide this proof in several steps. We first deal with chain graded
dg-modules dual to the cochain graded dg-modules considered in the theorem. We
define a comparison map at this level. We check that this comparison map gives rise
to a morphism of cochain dg-algebras after dualizing. Then we revisit the definition
of the cell decomposition of the Chevalley–Eilenberg cochain complex C∗CE (g), by
using constructions which parallels the definition of our cell decomposition of the
cobar complex in the proof of Theorem 13.1.3, and we prove that our comparison
morphism actually defines an isomorphism of cell complexes.
Step 1. The construction of a comparison morphism on chain complexes. We
consider the normalized chain complex of the complete bar construction of the
enveloping algebra Û(g) on the one hand, and the complete Chevalley–Eilenberg
complex ĈCE
∗ (g) on the other hand.
Recall that we have an identity Nn B̂(Û(g)) = I Û(g)⊗n , where we write I Û(g)
for the augmentation ideal of the complete enveloping algebra Û(g) (see the proof
of Theorem 13.1.3). For the Chevalley–Eilenberg complex of a complete Lie algebra
g, concentrated in degree 0, we also have an identity ĈCE
n (g) = Ŝn (Σ g), where we
write Ŝn (Σ g) for the completion of the module of symmetric tensors Sn (Σ g) =
(Σ g⊗n )Σn with respect to the filtration inherited from g. For each n ∈ N, we
consider the map κ : ĈCE
n (g) → Nn B̂(Û(g)) such that:
X
(1)
κ(Σ ξ1 · · · Σ ξn ) =
sgn(σ) · ξσ(1) ⊗ · · · ⊗ ξσ(n)
σ∈Σn
for any monomial Σ ξ1 · · · Σ ξn ∈ Sn (Σ g). We easily check that these maps intertwine
differentials, and hence, define a morphism of chain graded dg-modules
κ : ĈCE
∗ (g) → N∗ B̂(Û(g)).
(2)
Step 2. The definition of the comparison morphism of the theorem. We readily
see that our morphism (2) fits in a commutative diagram
(3)
ĈCE
∗ (g)
/ N∗ B̂(Û(g))
κ
∆∗
∆
κ⊗κ
/ N∗ B̂(Û(g))⊗
ˆ CE
ˆ N∗ B̂(Û(g))
ĈCE
∗ (g)⊗Ĉ∗ (g)
,
ˆ
∇
/ N∗ (B̂(Û(g))⊗
ˆ B̂(Û(g)))
where the morphism on the left-hand side, denoted by ∆, is the coproduct of the
complete Chevalley–Eilenberg complex ĈCE
∗ (g), the morphism on the right-hand
side, denoted by ∆∗ , is the image of the coproduct of bar complex B̂(Û(g)) under
ˆ for the extension of the
the normalized complex functor N∗ (−), and we write ∇
Eilenberg–MacLane morphism of Theorem 5.3.3 to our completed tensor products
(see §I.7.3.3). We also have an obvious commutation relation κ = when we
consider the augmentation morphisms associated with our coalgebra structures.
13.2. THE CHEVALLEY–EILENBERG COMPLEX OF COMPLETE LIE ALGEBRAS
419
We take the dual map κ : N∗ (Bc (Û(g)∨ )) → C∗CE (g) of our morphism (2). We deduce from the commutativity of diagram (3), and from our adjunction constructions
(see Proposition 6.4.2), that this map factors through the upgraded conormalized
complex N∗] Bc (Û(g)∨ ), so that we actually get a morphism of unitary commutative
cochain dg-algebras
κ : N∗] Bc (Û(g)∨ ) → C∗CE (g)
(4)
from the morphism constructed in Step 1.
Step 3. The pushout decomposition. We revisit the definition of the cellular
decomposition of the Chevalley–Eilenberg complex in order to establish that our
morphism (4) is an isomorphism. We aim to retrieve the pushouts of Proposition 13.2.6
S(E0s (g)∨ ⊗ B2 )
(5)
S(E0s (g)∨ ⊗ E2 )
/ C∗ (g / Fs g) ,
CE
/ C∗ (g / Fs+1 g)
CE
by a construction which parallels our cellular decomposition of the cobar construction in the proof of Theorem 13.1.3.
We work in the category of complete chain graded modules in the first step of
CE
this construction. We write ĈCE
∗ (−) = Ĉ∗ (g / Fs+1 g) for short, and we consider
0
0
the dg-modules Es g ⊗ B2 and Es g ⊗ E2 , dual to the generating cochain graded dgmodules of the symmetric algebras in our pushout diagram (5), and where we set
B2 = D(B2 ) and E2 = D(B2 ) as in the proof of Theorem 13.1.3. We again form a
diagram
(6)
···
δ
/ ĈCE
3 (−)
ψ]
···
···
δ
/ ĈCE
2 (−)
δ
ψ]
/0
/ E0s g ⊗ b2
/0
/ E0s g ⊗ b2
/ ĈCE
1 (−)
δ
/ ĈCE
0 (−) ,
ψ]
'
/ E0s g ⊗ e1
/0
ψ]
/0
/0
where we consider the chain complexes equivalent to these dg-modules E0s g ⊗ B2 ,
CE
E0s g ⊗ E2 , and ĈCE
∗ (−) = Ĉ∗ (g / Fs+1 g).
We use the splitting g / Fs+1 (g) = g / Fs (g) ⊕ E0s g fixed in the proof of Theorem 13.1.3 and we consider the morphism of chain complexes ψ] : ĈCE
∗ (−) →
E0s g ⊗ E2 defined by the map
(7)
0
0
ĈCE
1 (g / Fs+1 g) = Σ g / Fs+1 g → Es g ' Es g ⊗ e1 ,
420
13. COMPLETE LIE ALGEBRAS AND CLASSIFYING SPACES
which we deduce from the choice of this splitting, in degree 1. We still use the
preservation of the differential to determine the degree 2 component of our morphism. If we unravel definitions, then we just get the map
(8)
ĈCE
2 (g / Fs+1 g)
=
/ E0s g ⊗ b2
[−,−]
δ
ĈCE
(g
/
Fs+1 g)
1
/ Ŝ2 (Σ g / Fs+1 g)
=
/ Σ g / Fs+1 g
'
/ E0s g
given by the bracket of the Lie algebra g on the module Ŝ2 (Σ g / Fs+1 g), followed
by the map g / Fs+1 g = g / Fs (g) ⊕ E0s g → E0s g associated to our splitting.
In the proof of Theorem 13.1.3, we use that our Lie algebra g satisfies [g, Fs g] ⊂
Fs+1 g, for each s > 0, to form the pushout of cosimplicial algebras in our decomposition of the cobar construction. From this inclusion relation, we now deduce that
the map (8) factors through the module ĈCE
2 (g / Fs g) = Ŝ2 (Σ g / Fs g), where we
CE
use the morphism ĈCE
∗ (g / Fs+1 g) → Ĉ∗ (g / Fs g) induced by the natural quotient
map g / Fs+1 g → g / Fs g to identify the dg-module ĈCE
∗ (g / Fs g) with a quotient
object of the chain complex ĈCE
∗ (g / Fs+1 g). This observation implies that the com0
0
posite morphism of chain complexes ĈCE
∗ (−) → Es g ⊗ E2 → Es g ⊗ B2 in (6) factors
CE
through Ĉ∗ (g / Fs g), and our construction therefore yields a commutative square:
(9)
ĈCE
∗ (g / Fs+1 g)
(g
/ Fs g)
ĈCE
∗
ψ]
φ]
/ E0s g ⊗ E2
/ E0s g ⊗ B2
in the category of chain graded dg-modules. We dualize this square to get a commutative square of morphisms of cochain graded dg-modules
(10)
E0s (g)∨ ⊗ B2
E0s (g)∨ ⊗ E2
/ C∗ (g / Fs g) ,
CE
/ C∗ (g / Fs+1 g)
CE
from which we deduce our diagram (5) in the category of unitary commutative
cochain dg-algebras.
We have the identity:
(11)
C∗CE (g / Fs+1 g)[ = S(Σ−1 (g / Fs+1 g)∨ )
' S(Σ−1 (g / Fs g)∨ ⊕ E0s (g)∨ ⊗ e1 )
' C∗CE (g / Fs g)[ ⊗S(E0s (g)∨ ⊗B2[ ) S(E0s (g)∨ ⊗ E2[ )
when we forget about differentials. We immediately conclude from this observation
that our diagram (5) does form a pushout.
Step 4. The comparison of pushout diagrams. The comparison morphism of (4)
is functorial, and hence, commutes with the morphisms induced by the canonical
quotient map g / Fs+1 g → g / Fs g on our cochain complexes. We moreover readily
see that the morphism (2), which we use to define this comparison morphism in
Step 1, extends to a morphism of diagrams between our commutative square (9)
13.2. THE CHEVALLEY–EILENBERG COMPLEX OF COMPLETE LIE ALGEBRAS
421
and its counterpart §13.1.3(8) in the proof of Theorem 13.1.3. We deduce from this
observation that our comparison morphism (4) extends to a morphism of diagrams
relating the image of the pushout of cosimplicial algebras §13.1.3(2) under the
functor N∗] to our new pushout (5).
Let K ∈ c Mod be any cosimplicial module. The definition of the functor N∗]
as a left adjoint implies that we have an identity N∗] (S(K)) = S(N∗ (K)), where we
consider the conormalized cochain complex associated to K (see the proof of Proposition 6.4.1). For the symmetric algebras S(K) = S(Γ• (E0s (g)∨ ⊗ B2 )), and S(K) =
S(Γ• (E0s (g)∨ ⊗ B2 )), which occur on the left-hand side of our pushout of cosimplicial
algebras §13.1.3(2), we get an identity N∗] (S(K)) = S(N∗ (K)) = S(E0s (g)∨ ⊗ B2 ) and
similarly N∗] (S(K)) = S(E0s (g)∨ ⊗B2 ). We readily see that our comparison morphism
'
reduces, at this level, to the canonical isomorphism N∗] S(K) −
→ S(N∗ (K)) given by
∗
the definition of the functor N] on symmetric algebras S(K).
We then obtain by a straightforward induction that our comparison morphism
of cochain dg-algebras (4) defines an isomorphism
(12)
'
N∗] Bc (Û(g / Fs g)∨ ) −
→ C∗CE (g / Fs g),
at each stage s > 0 of our cellular decompositions, and we pass to the limit g =
lims g / Fs g to get the result of our theorem.
13.2.10. Outlook: The correspondence between the Sullivan model, the Quillen
model, and the Maurer–Cartan equation. The constructions studied in this chapter
have an extension to differential graded Lie algebras, which can be used to relate
the Sullivan model of a space, of which we reviewed the definition in §7, and the
Lie algebra model introduced by Quillen in [119] (the Quillen model ).
Let us give an overview of this correspondence. The Quillen model of a (simply connected) space X is a Lie algebra in chain graded dg-modules g(X) such
that H∗ (g(X)) = π∗+1 (X) ⊗ Q, where we consider the homology of this Lie algebra g(X) on the left-hand side, the rational homotopy groups of the space X
on the right-hand side. If X is of finite rational type (see §7.5.3), then the Lie
algebra g(X) is locally finitely generated (as a dg-module), and we may see that
the associated Chevalley–Eilenberg cochain complex C∗CE (g(X)) defines a cofibrant
resolution of the Sullivan model Ω∗ (X) of the space X. The other way round, we
already mentioned that the Harrison complex CHarr
(A), which computes the homol∗
ogy of the derived indecomposables of a unitary commutative cochain dg-algebra
A ∈ dg ∗ Com + , inherits a Lie coalgebra structure (see §7.5.15). The cochain complex C∗Harr (A), dual to CHarr
(A), forms a Lie dg-algebra. We may see that the
∗
Harrison cochain complex C∗Harr (Ω∗ (X)) associated to the Sullivan model of a (simply connected) space of finite rational type X represents a resolution of the Quillen
model of this space g(X).
In these constructions, we implicitly use duality operations, and when we pass
to Chevalley–Eilenberg cochains we basically change a counitary cocommutative
coalgebra into a unitary commutative algebra, but we have an analogous equivalence
of homotopy categories between the category of connected Lie dg-algebras and
the category of simply connected counitary cocommutative dg-coalgebras. This
equivalence of homotopy categories is used in this form by Quillen, in [119], in
order to associate a unitary cocommutative coalgebra model to a Lie algebra model
of the rational homotopy of spaces. This correspondence between Lie algebras
422
13. COMPLETE LIE ALGEBRAS AND CLASSIFYING SPACES
and cocommutative coalgebras also reflects the fact that the operads governing
these categories of algebras are dual in the sense of the Ginzburg–Kapranov Koszul
duality theory for operads [62].
Let us observe that the correspondence given by the isomorphism of Theorem 13.2.5(b)
ˆ Ω∗ (∆• ))
G• C∗CE (g) = Mordg ∗ Com + (C∗CE (g), Ω∗ (∆• ) ' MC(g ⊗
has a straightforward extension to morphisms φ : C∗CE (g) → Ω∗ (X), where g is
any complete Lie algebra (possibly differential graded) and we consider the Sullivan dg-algebra of piecewise linear forms on a simplicial set X ∈ sSet. To be
explicit, we get that any such morphism φ = φγ : C∗CE (g) → Ω∗ (X) is equivalent
ˆ Ω∗ (X) which satisfies the Maurer–Cartan equation in the
to an element γ ∈ g ⊗
∗
ˆ Ω (X). We may also interpret such a tensor γ ∈ g ⊗
ˆ Ω∗ (X) as a
tensor product g ⊗
piecewise linear form on X with coefficients in the Lie algebra g.
If we take the cochain dg-algebra of de Rham’s forms instead of the Sullivan
cochain dg-algebra piecewise linear forms, then we retrieve the classical notion of
a connection form with value in a Lie algebra and the classical theory of Maurer–
Cartan forms (we also refer to [83, Note 1] for a historical survey of the theory of
connections). This interpretation is used by K.T. Chen in [26] in order to define
a model of loop spaces. He considers a quasi-free Lie algebra g = (L(M ), ∂), such
that M = Σ−1 H̄∗ (X) is the desuspension of the reduced homology of our space X,
ˆ Ω∗ (X) whose
together with a solution of the Maurer–Cartan equation γ ∈ L(M )⊗
∗
∗
ˆ Ω (X) = M ⊗
ˆ Ω (X) reflects the duality pairing between
component γ1 ∈ L1 (M )⊗
the homology module H∗ (X) and the cohomology of the dg-module of piecewise
∗
linear forms H∗ (X) = H∗ Ω∗ (X). The morphism φγ : CCE
∗ (L(M ), ∂) → Ω (X),
∗
ˆ Ω (X), is a weak-equivalence
associated to our Maurer–Cartan form γ ∈ L(M )⊗
under this assumption and the Lie dg-algebra g = (L(M ), ∂) defines a Quillen
model of the space X. The result obtained by Chen asserts that the enveloping
algebra of this Lie dg-algebra U(L(M ), ∂) is weakly-equivalent to the bar complex
of the dg-algebra Ω∗ (X) and to the dg-algebra of piecewise linear forms Ω∗ (ΩX),
where ΩX denote the loop space associated to X. Chen moreover gives an explicit
geometrical construction of this weak-equivalence by using a notion of iterated
integration and a generalization, for Maurer–Cartan forms with values in the Lie
dg-algebra g = (L(M ), ∂), of the concept of parallel transport (we refer to [27] for a
comprehensive survey of this construction). The Chen model actually predates the
development of the Sullivan rational homotopy theory. We also refer to [145] for
a comprehensive study of the relationship between the Sullivan model, the Quillen
model, the Chen model and for the interpretation of the equivalence between the
Sullivan and Quillen models involving the Chen model of loop spaces.
Maurer–Cartan spaces are also used to set up a homotopical version of the theory of Lie groups (we refer to [58] and [71] for this subject) and to study deformation
problems from a homotopy theory viewpoint (we refer to [72] for the general setup,
to [88, 87] for striking applications of this idea to the deformation-quantization
problem of Poisson manifolds).