Realization of Lie algebras and classifying spaces of crossed modules

The category of complete differential graded Lie algebras provides nice algebraic models for the rational homotopy types of non-simply connected spaces. In particular, there is a realization functor, $\langle -\rangle$, of any complete differential graded Lie algebra as a simplicial set. In a previous article, we considered the particular case of a complete graded Lie algebra, $L_{0}$, concentrated in degree 0 and proved that $\langle L_{0}\rangle$ is isomorphic to the usual bar construction on the Malcev group associated to $L_{0}$. Here we consider the case of a complete differential graded Lie algebra, $L=L_{0}\oplus L_{1}$, concentrated in degrees 0 and 1. We establish that the category of such two-stage Lie algebras is equivalent to explicit subcategories of crossed modules and Lie algebra crossed modules, extending the equivalence between pronilpotent Lie algebras and Malcev groups. In particular, there is a crossed module $\mathcal{C}(L)$ associated to $L$. We prove that $\mathcal{C}(L)$ is isomorphic to the Whitehead crossed module associated to the simplicial pair $(\langle L\rangle, \langle L_{0}\rangle)$. Our main result is the identification of $\langle L\rangle$ with the classifying space of $\mathcal{C}(L)$.


Introduction
In this text, we pursue the study of the rational homotopy type of spaces with models in the category cdgl of complete differential graded Lie algebras, as developed in [4].We emphasize that in this approach, there are no requirements concerning simply connectivity or nilpotency.In particular, to any finite simplicial complex is associated a cdgl M X whose homology in degree 0 is the Malcev completion of π 1 (X) ([4, Theorem 10.5]).
One of the main tools in this theory is a cosimplicial cdgl L • = {L n } n≥0 , where L 0 is the free Lie algebra on a Maurer Cartan element in degree -1, and L 1 is the Lawrence-Sullivan interval (see below for more details).This cosimplicial cdgl plays a role similar to the simplicial algebra of PL-forms on ∆ • .It enables us to construct a realization functor from the category of complete differential graded Lie algebras to the category of simplicial sets, − : cdgl → Sset, defined by If a Lie algebra L is concentrated in degree 0, we proved in [6, Theorem 0.1] that its realization L is isomorphic to the usual bar construction on the group exp L, constructed on the set L with the Baker-Campbell-Hausdorff product.
Here we consider the next step: L is a connected cdgl with non-trivial homology only in degrees 0 and 1. Geometrically, this corresponds to the notion of homotopy 2-types and, by analogy, a connected cdgl L such that H * L = H 0 L ⊕ H 1 L is called a 2-type cdgl.First of all, if L = L ≥0 and H ≥2 L = 0, then the Lie subalgebra I = L ≥2 ⊕ dL 2 is an ideal because if a ∈ L 0 and b ∈ L 2 , then da = 0 and [a, We have thus reduced the problem to considering only cdgl's L of the form L = L 0 ⊕L 1 and denote by cdgl ≤1 the corresponding subcategory of cdgl.We associate to such L a natural crossed module C(L) and denote by CrMod the category of crossed modules.Our main result, which extends [6, Theorem 0.1], can be formulated as follows.
Theorem 1.If L is a complete differential graded Lie algebra such that L = L 0 ⊕ L 1 , then its geometric realization L is naturally isomorphic to the classifying simplicial set BC(L); i.e., the diagram commutes up to natural isomorphisms.
This theorem shows that the functor − generalizes many classical constructions.
Geometrically, crossed modules appear in the work of Whitehead ([14]).If (X, A) is a pair of topological spaces, based in A, Whitehead proved that the boundary map d : π 2 (X, A) → π 1 (A), together with the action of π 1 (A) on π 2 (X, A), defines a crossed module.Then, in [11], MacLane and Whitehead showed that the spaces X with π q (X) = 0, q ≥ 2, are determined by the crossed module of the pair (X, X 1 ), where X 1 is the 1-dimensional skeleton of X.For any cdgl L = L 0 ⊕ L 1 , the geometric realization L is determined by the crossed module associated to the pair ( L , L 0 ).Our second main result identifies this crossed module with C(L).
Theorem 2. The Whitehead crossed module associated to the simplicial pair ( L , L 0 ) is isomorphic to the crossed module C(L) introduced above.
In short, these two theorems unify the geometric realizations of complete differential graded Lie algebras of the form L = L 0 ⊕ L 1 and of crossed modules.In the last section, we extend the correspondence between Malcev groups and pronilpotent Lie algebras to crossed modules.We introduce the categories of Malcev crossed modules and of pronilpotent Lie algebra crossed modules and prove an isomorphism of categories.Theorem 3. The three following categories are isomorphic: (1) the category of pronilpotent differential graded Lie algebras of the form L = L 0 ⊕ L 1 , (2) the category of pronilpotent Lie algebra crossed modules, (3) the category of Malcev crossed modules.Moreover, the equivalence between (1) and ( 3) is given by the functor C.
As a next step for the future, we can consider a connected cdgl L such that H ≥n+1 L = 0 for some n ≥ 1.Using the ideal J = L ≥n+1 ⊕ dL n+1 , the same argument used above gives a weak homotopy equivalence

Conventions and notation
In a graded Lie algebra L, the group of elements of degree i is denoted by L i .A Lie algebra differential decreases the degree by 1, i.e., dL i ⊂ L i−1 .If x ∈ L, we denote by ad x the Lie derivation of L defined by ad If there is no ambiguity, the product of two elements m, m ′ of a group M is denoted mm ′ .Sometimes, if several laws are involved, we can use some specific notation, such as m ⊥ m ′ or m * m ′ , to avoid confusion.An action of a group N on a group M is always a left action and is denoted by (n, m) → n m.We denote then by M ⋊ N the semi-direct product whose multiplication law is defined by

Background on Lie models
A complete differential graded Lie algebra (henceforth cdgl) is a differential graded Lie algebra L equipped with a decreasing filtration of differential Lie ideals, such that If no filtration is specified, it is understood that we consider the lower central series.
Let V = ⊕ i∈Z V i be a rational graded vector space.We denote by L(V ) the free graded Lie algebra on V , and by L ≥n (V ) the ideal of L(V ) generated by the brackets of length greater than or equal to n.The completion of L(V ) is the inverse limit This is a cdgl for the filtration given by the ideals The correspondence V → L(V ) gives a left adjoint to the forgetful functor to graded rational vector spaces ([4, Proposition 3.10]).We call L(V ) the free complete graded Lie algebra on V .
If θ is a derivation of degree 0 on a cgl L, the exponential map e θ is a cgl automorphism of L defined by In particular, for any x ∈ L 0 , e adx is a cgl automorphism of L. Therefore, in any cgl L, the sub Lie algebra L 0 admits a group structure whose multiplication law * is given by the Baker-Campbell-Hausdorff product ([1, Ch.II.§6.Proposition 4], [13, §3.4]) and characterized by e adx * y = e adx • e ady .Now we recall the first properties of the cosimplicial cdgl L • ([4, Chapter 6]).Denote as usual by ∆ n the simplicial set in which ∆ n p is the set of p + 1-uples of integers (j 0 , . . ., j p ) such that 0 ≤ j 0 ≤ • • • ≤ j p ≤ n.We also denote by ∆ n the simplicial complex formed by the non-empty subsets of {0, . . ., n}.The subcomplex ∆n of ∆ n is the simplicial complex containing the proper non-empty subsets of {0, . . ., n}.
Finally s −1 C * ∆ n denotes the desuspension of the simplicial chain complex on ∆ n and s −1 C * ∆ n the desuspension of the complex of simplicial chains on ∆ n , which is isomorphic to s −1 N * ∆ n , the complex of non-degenerate chains on ∆ n .Then, as a graded Lie algebra (without differential), we set In other words, L n is the free complete graded Lie algebra on elements a i 0 ...i k of degree The family ∆ • = {∆ n } n≥0 is a cosimplicial object in the category of simplicial sets.It follows that the family s −1 N * ∆ • is a cosimplicial object in the category of chain complexes.The identification s object in the category of chain complexes.The extension of the cofaces and codegeneracies as morphisms of Lie algebras gives morphisms of complete graded Lie algebras δ i : L n → L n+1 and σ i : L n → L n−1 .More precisely, we have Proposition 1.1.[4, Theorem 6.1]Each L n can be endowed with a differential d satisfying the following properties.
(i) The linear part d 1 of d is given by (ii) The generators a i are Maurer-Cartan elements; i.e., (iii) The cofaces δ i and the codegeneracies σ i are cdgl morphisms.
Let us specify the cdgl L n in low dimensions.
• L 0 = (L(a 0 ), d) is the free Lie algebra on a Maurer-Cartan element a 0 .
• L 1 = ( L(a 0 , a 1 , a 01 ), d) is the Lawrence-Sullivan interval (see [9]) with The cosimplicial cdgl L • leads naturally to the definition of cdgl models for any simplicial set and to a geometric realization for any given cdgl, see [4,Chapter 7].For our purpose, we only need the realization of a cdgl L, defined as the simplicial set which satisfies properties of the classical Quillen realization.For instance, for any n ≥ 1, we have π n L = H n−1 L, where the group law of H 0 L is the BCH product (see [4,Section 4.2] or [1, §II.6.4]).

Crossed modules and cdgl's
For general background on crossed modules, we refer the reader to the historical papers of Whitehead ([14], [11]) or to more modern presentations, such as [2], [3] or [10].We recall only the basics we need.(1) for all m ∈ M and If the group N acts on itself by conjugation, the first property means that d is compatible with the N -action.It also implies that the group d(M ) is a normal subgroup of N and that ker d is a sub N -module of M .
On the other hand, we remark that if d(m) = 1, the second property implies mm ′ = m ′ m which means that ker d is included in the center of M .The same property shows that Im d acts trivially on ker d and induces thus an action of coker d on kerd.
In what follows L 0 is always considered as a group equipped with the BCH product denoted by * .We will prove that d : L 1 → L 0 is a crossed module.The first step consists in defining a group structure on L 1 .This construction was originally carried out in [4, Definition 6.14].Proof.The different possibilities for a definition of this law are described in [4, Section 6.5].We recall here the construction for the convenience of the reader, beginning with the "universal" example, the cdgl L ′ = L(u 1 , u 2 , du 1 , du 2 ), with u i in degree 1.Since HL ′ = 0 there is an element Of course such an element is not unique.If ω ′ is another element satisfying (2.1), the difference ω − ω ′ is a boundary since H ≥1 L ′ = 0.This shows that the class of ω is well defined in the cdgl quotient (L ′ /(L ′ ≥2 ⊕ dL ′ 2 ), d).We denote this class by Among all the different possible choices for ω, one starts with the Baker-Campbell-Hausdorff series for du 1 * du 2 .Replacing in each term one and only one du i by u i we get an element ω with dω = du 1 * du 2 .This gives, Now, let L be a cdgl with L = L 0 ⊕ L 1 , e 1 , e 2 ∈ L 1 , and f : L ′ → L the unique cdgl map sending u i to e i .Therefore the element e 1 ⊥ e 2 := f (u 1 ⊥ u 2 ) is a well defined element in L 1 .By construction, if e 1 and e 2 are cycles, using the image of the formula (2.2) in L, we have e 1 ⊥ e 2 = e 1 + e 2 .
For the associativity of ⊥, we consider L ′′ = L(u 1 , u 2 , u 3 , du 1 , du 2 , du 3 ) and observe that in ) because both have the same boundary.The same is thus true in L 1 .
With this group structure on L 1 we can now prove that Proof.Recall from [4,Definition 12.40] that the group L 0 acts on L 1 by From [4, Corollary 4.12]) it follows that, for any x ∈ L 0 , y ∈ L 0 , z ∈ L 1 , we have (x * y) z = e adx * y (z) = e adx (e ady z) = x ( y z).
To prove that the function y → x y is a group homomorphism, as in Proposition 2.2, we consider a universal example.Let E = L(x, z, t, dz, dt) with x in degree 0, z and t in degree 1, and dx = 0. Since the injection L(x) → E is a quasi-isomorphism, we have Thus in E 1 /dE 2 , we get The same is therefore also true in L 1 .
As x is a cycle, by [4, Propositions 4.10 and 4.13] we have and Property (1) of Definition 2.1 is satisfied.For Property (2), we use once again the universal example , and thus the same is true in L 1 .
Remark 2.4.By Proposition 2.2, under the hypotheses of Proposition 2.3, we deduce that the group structures ⊥ and + coincide on H 1 L = ker d.

The crossed module of a realization and Theorem 2
In this section, in the case L = L 0 ⊕ L 1 , we establish the isomorphism between C(L) and the Whitehead crossed module of ( L , L 0 ).Proposition 7.13]).We first compute π 1 ( L 0 ) and π 2 ( L , L 0 ), and for that, we use the homotopy relation introduced in [12, §3].

Proof of Theorem 2. The realization
Since L 1 = ( L(a 0 , a 1 , a 01 ), d), the map f → f (a 01 ) induces an isomorphism of sets Since ∂ i f = 0, for i = 0, 1, each element of L 0 defines an element of π 1 ( L 0 ).Now, two such 1-simplices, g and f , are homotopic in L 0 if there exists a map h : L 2 → L 0 such that ∂ 1 h = g, ∂ 2 h = f and ∂ 0 h = 0.The simplex h is called a homotopy from f to g.

Consider now the action of π
and a b the element of L 1 corresponding to this action.Recall ([4, Lemma 4.23]) that y = e ad a b is also an element of L 1 such that dy = a * db * a −1 .Both constructions, a b and e ada b, are natural, so that to prove a b = e ada b, we have only to prove it for the cdgl The required identification follows from d( a u) = d(e ada u) and the injectivity of d : L ′′ 1 → L ′′ 0 .We have thus recovered the crossed module C(L).

The classifying space of a crossed module
By definition, the classifying space of a crossed module C is the classifying space of the nerve of the categorical group associated to C. Let us specify this association.
Recall that a categorical group is a group object in the category of groups (see [10 where N is a subgroup of G, s and t are homomorphisms such that s| N = t| N = id N and [ker s, ker t] = 1. In [10], J. L. Loday defines a categorical group associated to a crossed module C = (d : (M, ⊥) → (N, * )) as follows: • G = M ⋊ N is the product M × N with the semi-direct product given by the action of N on M .Thus, the product of (m ′ , n ′ ) and • An element (m, n) of G has for source and target, respectively, Thus, the group N is interpreted as the group of objects viewed in G as {1} × N .The group G = M ⋊ N is the group of arrows with the morphisms s and t giving the source and the target.Two elements (m ′ , n ′ ) and (m, n) are composable if In this case the composition is defined by We deduce easily from Property (1) of Definition 2.1 that s and t are group homomorphisms.We also verify that the source of a composite is the source of the first factor and the target is the target of the second factor: Finally, composition is a group homomorphism, see [10,Lemma 2.2].
The usual nerve of a category is a simplicial set.When the category is a categorical group, we obtain naturally a simplicial group.Let us describe the nerve of the categorical group associated to a crossed module C = (d : (M, ⊥) → (N, * )).We have As the n i , for i ≥ 2, are determined by n 1 and the family (m i ) 1≤i≤k , the sequence (m i , n i ) i≤k can be identified with the sequence In particular, Ner k = M k × N. (4.1)Each Ner k is a group, the multiplication being given component wise.With the identification (4.1), this product is given by ).The boundary and degeneracy maps of Ner * are morphisms of groups defined as usual by: Recall from [5,Definition 3.20] or [7, Page 255] the classifying functor W which goes from the category of simplicial groups to the category of reduced simplicial sets.The classifying space BC of the crossed module C is the space obtained by composing Ner * with W : BC = W (Ner * ).By definition of W , we have The boundaries and degeneracies are given by In particular, in low dimensions, we have

The classifying space functor W and twisting functions
Let A * be a simplicial set.By [12,Corollary 27.2], there is a bijective correspondence between morphisms of simplicial sets ϕ : A * → W • Ner * = BC and twisting functions Recall that ([12, Definition 18.3]) a twisting function τ is a family of maps The simplicial map ϕ k : A k → (BC) k associated to the twisting function τ is given by

Proof of Theorem 1
First we compute the simplicial set L • = Hom cdgl (L • , L) in the case L = L 0 ⊕ L 1 .By L ≥2 = 0 and [4, Corollary 6.5], we have isomorphisms ).Since any morphism of codomain L vanishes on elements of negative degree, we can quotient by the differential ideal generated by the generators of degree -1.This gives as free cgl Finally, in view of the differential in L 2 , recalled in (1.1), the differential of L k satisfies da ij = 0 and da 0st = a 0s * a st * a −1 0t .In the rest of this text, we will use that for all k, there exists an isomorphism given by Ψ(f ) = ((f (a r r+1 )) 0≤r<k , (f (a r,r+1,s )) r+1<s≤k ) is an isomorphism.
Proof.For the sake of simplicity write for i < j, a ji = a −1 ij , and for 0 ≤ i < j < r ≤ k, With this notation, when the integers i, j, r are all different from each other and between 0 and k, we have da ijr = a ij * a jr * a ri .
This shows that Ψ is injective.The same construction process shows that Ψ is also surjective.
The isomorphism of our main theorem is based on a family τ of maps In low dimensions, this gives: Proposition 6.2.The family τ is a twisting function.
Proof.Observe that m i+1 ⊥ m i = f (a 01(i+2) ) −1 ⊥ f (a 01i ).Thus, the index i + 1 disappears in the expression of d i τ k f and we get A similar argument gives also the result for d k−1 .We have reduced the problem to proving the more subtle equality involving d 0 .We use an induction, supposing the result is true for τ j , j < k, and considering τ k .Due to the inductive step, we can concentrate the computations on the left hand factor.From the definitions, we have We determine the product of the two last terms, To obtain the equality with τ k−1 d 1 f , we consider the following computation in L k : ).Similar computations give the corresponding equalities for degeneracy maps.
Denote by ϕ the morphism of simplicial sets induced by the previous twisting function τ , ϕ : Hom cdgl (L, L) → BC(L).
The following result finishes the proof of the Theorem.
Proposition 6.3.The morphism ϕ is an isomorphism of simplicial sets.
Proof.Recall from (5.1) that . By iteration from (d 0 ) ℓ f = f (δ 0 ) ℓ , we deduce that the image of ϕ k is the linear subspace generated by the elements f (a r,r+1 ), for 0 ≤ r < k, and f (a r,r+1,s ), for r + 1 < s ≤ k.The result follows thus from Proposition 6.1.

Malcev crossed modules and Theorem 3
In this section, we establish an isomorphism of categories between cdgl ≤1 and a subcategory of crossed modules.We use the Lie algebra crossed modules introduced by Kassel and Loday in [8].We begin with a reminder of [8].By definition, since L is pronilpotent the associated Lie algebra crossed module is also pronilpotent.
(2) ⇒ (1).We start with a pronilpotent Lie algebra crossed module u : m → n with action v : n → Der(m) and we construct a pronilpotent cdgl L = L 0 ⊕ L 1 .We define L 0 = n as Lie algebra and L 1 = m as vector space.For a ∈ L 1 and x ∈ L 0 , we set [x, a] = v(x).aand d = u.We check easily that d is a derivation and L = L 0 ⊕ L 1 is pronilpotent.
The associations (1) ⇒ (2) and ( 2) ⇒ (1) give the desired isomorphism of categories for the two first points of the statement.
(2) ⇒ (3).We start with a pronilpotent Lie algebra crossed module u : m → n with action v : n → Der(m) and we construct a Malcev crossed module d : M → N .We define M and N to be the vector spaces m and n respectively, with the group structure given by the Baker-Campbell-Hausdorff product, and set d = u.The action v extends in an action by e v : for n ∈ N = n, m ∈ M = m, we set As v is a morphism of Lie algebras, we have v[n, n ′ ] = [v(n), v(n ′ )], for all n, n ′ ∈ N , and so, the Baker-Campbell-Hausdorff formula implies v(n * n ′ ) = v(n) * v(n ′ ) and (n * n ′ ) m = e v(n * n ′ ) (m) = e v(n) (e v(n ′ ) (m)).
Thus, we have a group action.The two additional properties of Malcev crossed modules are easily deduced from the corresponding properties of Lie algebra crossed modules as well as the pronilpotency conditions.
(3) ⇒ (2).As we do for the cases (1) and ( 2), the previous process can be reversed.We associate a pronilpotent Lie algebra to a Malcev group, replacing the exponential by the functor L → log(1 + L).The only significative point is the construction of the Lie algebra action v : n → Der(m) from the group action ν : N → Aut(M ); this is done by v(n).m = log(1 + ν(n))(m).

4 . 5 .
crossed module of a realization and Theorem 2 8 The classifying space of a crossed module 9 The classifying space functor W and twisting functions 11

Definition 2 . 1 .
A crossed module C = (d : M → N ) is a morphism of groups d together with an action of N on M , given by group automorphisms n → (m → n m) satisfying two conditions:

Proposition 2 . 2 .
For any cdgl (L, d) such that L = L 0 ⊕ L 1 , L 1 admits a natural product ⊥ for which the differential d : (L 1 , ⊥) → (L 0 , * ) is a group morphism.Moreover, a ⊥ b = a + b if a and b are cycles.

In Definition 2 . 1 ,
the group action of N on M corresponds to a homomorphism from N in the group of automorphisms of M .For Lie algebras, n and m, an action of n on m corresponds to a Lie morphism v : n → Der(m) in the Lie algebra of derivations of m.The action of n ∈ n on m ∈ m is denoted v(n).m.We can now state [8, Définition A.1].The action v : n → Der(m) is given by the adjoint action, v(x) = ad x .The formulae (1) and (2) of Definition 7.1 also follow immediately: let a, b ∈ m = L 1 and x ∈ n = L 0 , we have u(v(x).a)= d(ad x (a)) = d[x, a] = [x, da] = [x, u(a)], v(u(a)).b= ad da (b) = [da, b] = [a, b] ′ .