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Disconnected rational homotopy theory

We construct two algebraic versions of homotopy theory of rational disconnected topological spaces, one based on differential graded commutative associative algebras and the other one on complete differential graded Lie algebras. As an application of the developed technology we obtain results on the structure of Maurer-Cartan spaces of complete differential graded Lie algebras.


Introduction
The purpose of this paper is to construct an algebraic theory of rational disconnected topological spaces (or simplicial sets), alluded to in [21, p. 67]. The corresponding theory for connected spaces was constructed in the seminal papers [23,24]. The paper [23] related rational connected spaces to differential graded Lie algebras (dglas) whereas [24] took the perspective of commutative differential graded algebras (cdgas). In the present paper we pursue both points of view and construct both the dgla and cdga algebraic models for disconnected spaces. It is interesting that the differences between the two algebraic categories (which were somewhat hidden in the Quillen and Sullivan approaches) become more pronounced in our more general context; in particular we are naturally led to consider dglas endowed with a linearly compact topology, while our cdgas are still discrete.
The definitive reference establishing a correspondence between rational connected spaces and cdgas is [5], and our treatment relies heavily on the results of that paper. Recall that op. cit. constructed a closed model category structure on non-negatively graded cdgas and a Quillen adjunction between this category and the category of simplicial sets. This adjunction restricts to an equivalence between the homotopy categories of connected cdgas of finite Qtype and connected rational nilpotent simplicial sets of finite type.
Our main innovation in establishing a cdga version of disconnected rational homotopy theory is that we remove the restriction that our cdgas be non-negatively graded and use the closed model structure on the category of Z-graded cdgas [12]. This seemingly innocent modification has quite dramatic consequences. For example, any commutative algebra concentrated in degree zero is cofibrant in the Bousfield-Gugenheim category, but not in this extended category, unless it is a retract of a polynomial algebra. The closed model category of all cdgas appears more natural than the Bousfield-Gugenheim category; for instance it allows one to define Harrison-André-Quillen cohomology, cf. [3].
There is still a Quillen adjoint pair of functors between the categories of Z-graded cdgas and simplicial sets giving rise to an adjunction on the level of homotopy categories. This adjunction restricts to an equivalence between the homotopy category of simplicial sets having finite number of connected components, each being rational, nilpotent and of finite type, and a certain subcategory of the homotopy category of Z-graded cdgas. We give an explicit characterization of this subcategory.
To construct the second version of the disconnected rational homotopy theory (based on dglas) we need to relate the homotopy theory of commutative and Lie algebras. This relationship, which is sometimes referred to as Koszul duality was established in the work of Quillen [23]; it was formulated as the duality between differential graded coalgebras and dglas under certain (fairly severe) restrictions on the grading of the objects under consideration. These restrictions were subsequently removed in the seminal paper of Hinich [13]. In our context we need a result that is Koszul dual to Hinich's; it can be regarded as a Quillen equivalence between the categories of cdgas and differential graded Lie coalgebras (which can be dualized and viewed as complete dglas). We prove this result by a suitable adaptation of Hinich's methods. As a consequence we obtain an algebraic model of disconnected rational homotopy theory based on complete dglas. The condition of completeness is essential; it [May 7, 2013] cannot be removed even when restricted to connected spaces (or simplicial sets). In the latter case our theory is close to, but still different from, the one constructed in the papers by Neisendorfer and Baues-Lemaire [2,22]. In particular, our complete dglas admit minimal models even in the nonsimply-connected case, whereas simply-connectedness is an essential requirement for the existence of minimal models constructed by Neisendorfer and Baues-Lemaire.
One important application of the developed theory that we give in this paper, concerns the structure of Maurer-Cartan spaces. Recall (cf. [10]) that associated to any dgla g is a simplicial set MC • (g); in the case when g is non-negatively graded and nilpotent, this is a simplicial set corresponding to g under the Quillen-Sullivan correspondence. When g is not non-negatively graded, the simplicial set MC • (g) is much more mysterious; this is a fundamental object of study for deformation theory [15,19,21] and also comes up in modern approaches to quantum field theory [8]. It turns out that for differential graded Lie algebras satisfying an appropriate completeness condition there exists an operation of disjoint product corresponding to the operation of disjoint union of simplicial sets. The disjoint product of two dglas is never non-negatively graded and we prove that the Maurer-Cartan simplicial set of the disjoint product of complete dglas is weakly equivalent to the disjoint union of the corresponding simplicial sets. Furthermore, for an arbitrary complete dgla g the simplicial set MC • (g) naturally decomposes up to homotopy as a disjoint union of Maurer-Cartan spaces of certain connected dglas naturally associated with g. Along the way we establish a general result of independent interest, expressing the Chevalley-Eilenberg cohomology of free products of dglas through the Chevalley-Eilenberg cohomology of the individual pieces. It appears that this result is new even for ordinary Lie algebras.
Similar results on the Maurer-Cartan spaces of dglas were contained in the recent paper [6] by U. Buijs and A. Murillo, who used completely different methods, see, in particular, Theorem 5.5, Proposition 6.2 and Theorem 6.4 in op. cit. Their statements do not include any completeness assumptions. We have been unable to verify the claims made in that paper.
Notation and conventions. All algebraic objects will be considered over a fixed field k of characteristic zero, at some places we require specifically k to be the field Q of rational numbers. The abbreviation 'dg' stands for 'differential graded'. Further, we will write 'cdga' and 'dgla' for 'commutative differential graded unital associative algebra' and 'differential graded Lie algebra' respectively. We allow also the terminal algebra 0 in which 1 = 0. For a (co)cycle c, the symbol [c] denotes its cohomology class. The degree of a homogeneous element a is denoted |a|.
All our cdgas will have cohomological grading with upper indices while dglas will have homological grading with lower indices. There will, however, be one important exception from this rule. At some places, we will need the tensor product of a homologically graded Lie algebra with the cohomologically graded cdga Ω of the Sullivan-de Rham forms. In this context, we consider Ω as homologically graded by Ω * := Ω − * . The tensor product will then be a homologically graded dgla as expected.
The suspension ΣV of a homologically graded vector space V is defined by the convention ΣV i = V i−1 resp. ΣV i = V i−1 for V cohomologically graded. The functor of taking the [ May 7, 2013] linear dual takes homologically graded vector spaces into cohomologically graded ones so that (V * ) i = (V i ) * ; further we will write ΣV * for Σ(V * ); with this convention there is an isomorphism (ΣV ) * ∼ = ΣV * . Regarding spectral sequences, we will use the terminology of [4]. We will often refer to a duality between the category of discrete vector spaces (colimits of finite-dimensional ones) and linearly compact spaces (limits of finite-dimensional ones), see e.g. [18].
Let A be the category of unital cdgas and A ≥0 its subcategory consisting of non-negatively graded cdgas. By [12] and [5] respectively, both A and A ≥0 are closed model categories whose weak equivalences are morphisms inducing isomorphisms of cohomology, and fibrations are surjective morphisms.
Note that A ≥0 has more cofibrations than A . For instance, every cdga concentrated in degree 0 is cofibrant in A ≥0 . Proposition 3.1 presents a wide class of cdgas that are cofibrant in A ≥0 but not in A .
As usual, a cdga A = (A, d) is connected (resp. homologically connected ) 1 if A 0 = k (resp. H 0 (A, d) = k) and A n = 0 (resp. H n (A) = 0) for n ≤ 0. Each connected algebra A ∈ A ≥0 admits a unique minimal model [5, §7]. Such an algebra A is of finite Q-type if A is defined over Q and its minimal model has finitely many generators in each degree. Equivalently, the cohomology of H(I/I 2 ), where I ⊂ A is the ideal of positive-degree elements of A, is finite-dimensional in each degree, see [5, §9.2] Let S denote the category of simplicial sets and fNQ-S c its subcategory of connected nilpotent rational simplicial sets of finite type. We denote fQ-A c ≥0 ⊂ A ≥0 the subcategory of homologically connected cdgas of finite Q-type. It is well-known [5,Theorem 9.4] that there exists a pair of adjoint functors (0.1) F : A ≥0 i q S : A that, for k = Q, induces an equivalence of the homotopy categories fQ-hoA c ≥0 and fNQ-hoS c . A similar adjunction and induced equivalence hold also for augmented and pointed versions of the above categories, see again Theorem 9.4 in op. cit.
We denote by L the category of dglas and byL the category of complete dglas, i.e. inverse limits of finite-dimensional nilpotent dglas; the morphisms inL are, naturally, continuous dgla maps. We will show in Section 9 thatL is a closed model category whose weak equivalences are maps f : g ′ → g ′′ such that C(f ) : C(g ′′ ) → C(g ′ ) is a weak equivalence; here C(−) is the Chevalley-Eilenberg functor recalled in Definition 7.3. Fibrations inL are surjective morphisms.
The free product of two dglas g and h will be denoted by g * h. If g and h are complete dglas then g * h will stand for the completed free product of g and h; it is thus a categorical coproduct of g and h. Given a dgla g, a Maurer-Cartan element in g is an element ξ ∈ g −1 satisfying the Maurer-Cartan equation: dξ + 1 2 [ξ, ξ] = 0. We will abbreviate the expression 'Maurer-Cartan' as 'MC'. The set of all MC elements in g will be denoted by MC(g). This definition can be extended to give a simplicial set MC • (g) whose vertices are just the MC elements in g; a precise definition is recalled in the main text. Furthermore, given a dgla g and an MC element ξ ∈ MC(g) we can define a twisted differential d ξ in g by the formula d ξ (?) = d(?) + [?, ξ]; we will write g ξ for the dgla g supplied with the twisted differential.
Given a dgla g, we denote by g x the dgla obtained from g by freely adjoining the variable Clearly, x ∈ MC(g x ) and we will write g ⊔ 0 for the twisted dgla (g x ) x . One should view the construction g ⊔ 0 as the Lie analogue of adjoining an isolated base point to a topological space. Furthermore, for two dglas g and h we set g ⊔ h := (g ⊔ 0) * h. The dgla g ⊔ h will be called the disjoint product of g and h; in the case when g and h are complete we will write g ⊔ h for the corresponding completion. The operation of disjoint product equips the category of complete dglas with a non-unital monoidal structure. A non-complete version of the disjoint product was considered in [6].
For convenience of the reader, we include a glossary of notation at the end of Section 1.

Main results
We call a cdga A = (A, d) ∈ A homologically disconnected if H n (A) = 0 for n < 0 and if H 0 (A) is isomorphic to the direct product i∈J k of copies of the ground field indexed by some finite set J. Let us denote by A dc (resp. A dc ≥0 ) the full subcategory of A (resp. A ≥0 ) consisting of homologically disconnected cdgas. We also denote by hoA dc (resp. hoA dc ≥0 ) the full subcategory of hoA (resp. hoA ≥0 ) whose objects are homologically disconnected cdgas. 2 Theorem A. The inclusion A ≥0 ⊂ A induces an equivalence of the homotopy categories hoA dc ≥0 and hoA dc . 1.1. Remark. A related question is whether hoA ≥0 is a full subcategory of hoA . It is not difficult to show, for example, that for cdgas A and B with A concentrated in degree zero, the sets hoA ≥0 (A, B) and hoA (A, B) are in natural bijective correspondence. In general however, we see no compelling reason for the homotopy classes of maps in both categories to be the same.
For a category C denote by coProd(C) the category whose objects are formal finite coproducts A 1 ⊔ · · · ⊔ A s , s ≥ 1, of objects of C and the Hom-sets are with the obvious composition law.

1.2.
Example. Denote by S c the category of connected simplicial sets and by S dc the category of simplicial sets with finitely many components. Then clearly S dc ∼ = coProd(S c ) and the same is obviously true also for the homotopy categories, i.e. hoS dc ∼ = coProd(hoS c ).
Our next main theorem that states a similar result also for homotopy categories of cdgas requires a contravariant version of coProd(C). Namely, denote by Prod(C) the category whose object are formal finite products A 1 × · · · ×A s , s ≥ 1, of objects of C, and morphisms are In the following theorem, A c (resp. A c ≥0 ) denotes the full subcategory of A (resp. A ≥0 ) consisting of homologically connected cdgas.  Theorem B. Each homologically disconnected non-negatively graded cdga A ∈ A dc ≥0 is isomorphic to a finite product i∈J A i of homologically connected non-negatively graded cdgas A i ∈ A c ≥0 . This isomorphism extends to a natural equivalence of categories hoA dc ≥0 ∼ Prod(hoA c ≥0 ). Each homologically disconnected cdga A ∈ A dc is weakly equivalent to a finite product i∈J A i of homologically connected cdgas A i ∈ A c . As above, one has an equivalence hoA dc ∼ Prod(hoA c ).
Our proofs of the above theorems use a proposition describing homotopy classes of maps whose domain is a localization of a cdga. We believe that this statement is of independent interest.
For a cocycle u of a cdga A we denote by A[u −1 ] the localization of A at u, i.e. at the multiplicative subset generated by u, see [1,Section 3]. Since u is a cocycle, A[u −1 ] bears the induced differential. Notice that each χ ∈ [A, D] A := hoA (A, D), for A, D ∈ A , induces a map χ * : H(A) → H(D).

1.3.
Theorem. Let A, D ∈ A be cdga and u ∈ A a cocycle. Denote Observe that each invertible odd-degree element x of a graded commutative algebra, i.e. one for which there exists y such that xy = 1, equals zero since x 2 = 0 by graded commutativity. So the only graded commutative algebra admitting invertible elements in odd degrees is the terminal one. Theorem 1.3 is thus meaningful only when the degree |u| is even.
The next two applications of our theory describe the homotopy category of spaces with finitely many rational nilpotent components of finite type. We need the following definition related to the algebraic side.
1.4. Definition. Let A be a cdga. We say that a dg ideal I in A is an augmentation ideal if A/I ∼ = k, the ground field. Further, A is said to have a finite type if for any augmentation ideal I of A the space H(I/I 2 ) is finite dimensional in every degree.
Assume that the ground field k is the field Q of rational numbers and denote by fQ-A dc ≥0 , resp. fQ-A dc , the subcategory of A dc ≥0 , resp. A dc , consisting of algebras having a cofibrant replacement of finite type. In Section 4 we prove that this definition does not depend on the choice of a cofibrant replacement and relate it to the definition of finite Q-type given in [5].
Theorem C. The following three categories are equivalent.
-The homotopy category fNQ-hoS dc of simplicial sets with finitely many components that are rational and of finite type, -the homotopy category fQ-hoA dc ≥0 of homologically disconnected non-negatively graded cdgas of finite type over Q, and -the homotopy category fQ-hoA dc of homologically disconnected Z-graded cdgas of finite type over Q.
[ May 7, 2013] Let us denote by fQ-hoL dc the full subcategory of the homotopy category ofL consisting of disjoint products of complete non-negatively graded dglas with finite-dimensional homology in each dimension. We call objects of fQ-hoL dc disconnected dglas. We have Theorem D. The following categories are equivalent: -the homotopy category fNQ-hoS dc + of pointed simplicial sets with finitely many components that are rational and of finite type, and -the homotopy category fQ-hoL dc of disconnected complete dglas of finite type.
Neisendorfer in [22,Proposition 7.3] proved that the subcategory fNQ-hoS c + of fNQ-hoS dc + consisting of connected spaces is equivalent to the homotopy category ho(nDGLA) of nonnegatively graded (discrete) dglas L whose homology H(L) is of finite type and nilpotent. As a particular case of Theorem D we get another description of fNQ-hoS c + . Denote by fQ-hoL ≥0 the full subcategory of fQ-hoL dc of complete non-negatively graded dglas with finite dimensional homology in each degree.
A nice feature of the categoryL is that each g ∈L has a minimal model , unique up to isomorphism, see Definition 9.17 and Theorem 9.18. To objects of fQ-hoL ≥0 there correspond non-negatively graded minimal dglas M with homology of finite type. Corollary 1.5 implies a one-to-one correspondence between rational homotopy types of connected nilpotent spaces of finite Q-type and isomorphism classes of minimal dglas M as above.
The description of fNQ-hoS c + given in Corollary 1.5 substantially differs from Neisendorfer's. Notice, for instance, that the category ho(nDGLA) has more objects than fQ-hoL ≥0 . For example, the contractible free Lie algebra L(x, ∂x), |x| = 1, belongs to ho(nDGLA) but not to fQ-hoL ≥0 ; L(x, ∂x) is not complete.
Tracing the functors in [22], one can associate to a dgla g ∈ fQ-L ≥0 the corresponding L ∈ nDGLA as follows. The cdga C(g) is connected and non-negatively graded, so it has the minimal model M A . By assumption, M A is a cdga of finite type, so we may take L := L(M A ), the uncompleted Quillen functor.
On the other hand, starting from L ∈ nDGLA, we take the cobar construction C c (L) on the dgla A, i.e. the obvious coalgebra version of the uncompleted functor C(−), and its linear dual C c (L) * . 3 Then g :=L C c (L) * is the corresponding dgla in fQ-L ≥0 .
Another application concerns the general structure of the MC simplicial sets. The ground field k may again be an arbitrary field of characteristic zero.
1.6. Theorem. Let g i , i ∈ J, be a collection of complete dglas indexed by a finite set J. Then the simplicial set MC • i∈J g i is weakly equivalent to the disjoint union i∈J MC • (g i ).
Theorem 1.6 yields the following elementary corollary on the sets M C (−) := π 0 MC • (−) of connected components of the simplicial MC spaces. We do not know if it has a direct proof. 1.7. Corollary. Let g i , i ∈ J, be a collection of complete dglas indexed by a finite set J. Then there is a bijection M C i∈J g i ∼ = i∈J M C (g i ) of the MC moduli sets.
The following theorem in a certain sense reverses Theorem 1.6. It uses the twisting g ξ of a complete dgla g by an MC element ξ ∈ MC(g) and its connected cover g ξ defined in (7.6).
1.8. Theorem. For a complete dgla g, one has a weak equivalence where the disjoint union in the right hand side runs over chosen representatives of the isomorphism classes in M C (g). If M C (g) is finite, one furthermore has a weak equivalence of simplicial sets.
Theorems A, B and C are proved in Section 5, Theorem D in Section 10. Theorem 1.3 is proved in Section 2 and Theorem 1.8 in Section 7. The proof of Theorem 1.6 occupies Section 8; it is surprisingly involved in that it relies, essentially, on all of the technology developed in the previous sections and the Appendix. The pointed (or augmented) versions of Theorems A, B and C are formulated in Section 6. We finish this part by a Glossary of notation. We use the following notation for various categories: A , the category of (Z-graded) cdgas, A ≥0 , the category of non-negatively graded cdgas, A c , the category of homologically connected cdgas, A c ≥0 , the category of non-negatively graded homologically connected cdgas, A dc , the category of homologically disconnected cdgas, A dc ≥0 , the category of non-negatively graded homologically disconnected cdgas, L , the category of dglaŝ L , the category of complete dglas, S , the category of simplicial sets, S c , the category of connected simplicial sets, S dc , the category of simplicial sets with finitely many components.
The prefix 'fQ-' applied to a category of algebras means 'finite type over Q' while the prefix 'fNQ-' applied to a category of simplicial sets abbreviates 'nilpotent, rational components of finite type.' The subscript ' + ' means 'pointed' for simplicial sets and 'augmented' for algebras.
Part 1. The de Rham-Sullivan approach.
In this part we describe our first version of disconnected rational homotopy theory based on cdgas. Recall [9, Sections 5,6] that the homotopy category hoA of the model category A of Z-graded unital cdgas has the same objects as A , and the morphism sets where QX resp. QY is a cofibrant replacement of X resp. Y 4 and π(−, −) A denotes the set of homotopy classes. By [9,Proposition 5.11], if A is cofibrant and Y fibrant, which in our situation means that Y is arbitrary, one has an isomorphism There is a functor γ : A → hoA which is the identity on objects and, for a morphism f : X → Y in A , γ(f ) is the homotopy class of a liftf : QX → QY of f . The homotopy category hoA ≥0 of A ≥0 has an obvious similar description. Let A be a cdga and S ⊂ A a multiplicative subset of cocycles. Then the localization [1, Section 3] 5 S −1 A of A is a cdga and the canonical map A → S −1 A is a morphism of cdgas. The property crucial for us is the exactness [1, Proposition 3.3] of the functor A → S −1 A. If S is multiplicatively generated by a cocycle u ∈ A we will write A[u −1 ] for S −1 A.

2.1.
Remark. It is easy to see, using the exactness of the localization, that the homotopy type of A[u It is immediate to see that the set [A, D] u A has the same description. The rest of this section is devoted to the proof of Theorem 2.2. We say that We call a cocycle u ∈ X ∈ A cohomologically invertible if its cohomology class [u] ∈ H(X) is invertible.
2.4. Proposition. Assume that A ∈ A is cofibrant and D ∈ A an cdga whose each cohomologically invertible cocycle is invertible. Let u ∈ A be a cocycle.
Proof. Let us look at the left half of the diagram The symbols y, z, s, ds are new free generators with |z| = |s| := −1, |y| := −|u|. The cdga A[u −1 ],d is the localization of A = (A, d) at u with the induced differential. The differential   We are going to prove that f represents φ. Applying the functor γ to the equation f c = f irq gives Since i and r are weak equivalences and ri = 1 1, γ(i) and γ(r) are mutually inverse isomorphisms in hoA . As c is our chosen cofibrant replacement of The proof is finished by recalling thatf q =φ, hence φ = γ(φ) = γ(f ).

2.5.
Lemma. For each cdga X ∈ A there exists X ∈ A and a weak equivalence q : X → X such that (i) each cohomologically invertible cocycle u ∈ X is invertible, and (ii) each morphism f : X → Z whose target is an cdga Z ∈ A in which all cohomologically invertible cocycles are invertible, uniquely factorizes via q : X → X.
Denote by S the multiplicative set of all cohomologically invertible cocycles u ∈ X and by X := S −1 X the localization of X = (X, d) at S with the induced differentiald. Let q : X → X be the localization map. To prove (i) it is, by the above observation, enough to show that each cocycle x ∈ X cohomologous to 1 is invertible. Let x = 1 +db, b ∈ X. Clearly, b = q(s) −1 q(a) for some a ∈ X and s ∈ S, so q(s)x = q(s) + q(da) = q(s + da). Since [s + da] = [s], q(s + da) is invertible in X by the definition of S. We can therefore take Part (ii) follows from the standard universal property of the localization.  of D are invertible, D × D has the same property therefore, by (ii) of Lemma 2.5, (p 1 ,p 2 ) factorizes as We claim that (2.4) with i := qī is a good path object for D.
Firstly, i, being the composition of two weak equivalences, is a weak equivalence. Secondly, since P is a good path object, (p 1 ,p 2 ) is a fibration in A i.e. an epimorphism, hence (p 1 , p 2 ) must be an epimorphism, i.e. a fibration, as well.
commutative. Being a cofibrant replacement, the map c is as weak equivalence. By the exactness of the localization,c is a weak equivalence as well. Consider the induced diagram in which p ♯ (resp.p ♯ , resp. c ♯ , resp.c ♯ ) are the pre-compositions with γ(p) (resp. γ(p), resp. γ(c), resp. γ(c)).
Since c andc are weak equivalences, the induced maps c ♯ andc ♯ are isomorphisms in hoA (c ♯ is in fact the identity). Clearly, c ♯ restricts to an isomorphism We conclude that the theorem will be proved for the localization p : . We may thus assume from the beginning that A is cofibrant. An even simpler argument based on Lemma 2.5 shows that we may also assume, without loss of generality, that each cohomologically invertible cocycle in D is invertible. To Since A is cofibrant this, by (2.1), means that f 1 p and f 2 p are homotopic. By Proposition 2.7, f 1 and f 2 are right homotopic, which implies that

Maps of direct products of dg commutative associative algebras
In this section we study maps, up to homotopy, whose source is a finite direct product of cdgas. Somewhat unexpectedly, it turns out that these maps can be completely understood in terms of (homotopy classes of) maps out of individual components of these direct products.
Here is the first surprise.
Proof. We prove the proposition for J = {1, 2}, the proof for an arbitrary finite indexing set will be similar. Let thus A = A 1 ×A 2 . We need to prove that, for any epimorphism p : E ։ B in A ≥0 which is also a weak equivalence, and for each f : A → B, there exists a lift f : A → E making the diagram Let e 1 + e 2 = 1 (the unit of A), e 2 1 = e 1 , e 2 2 = e 2 and e 1 e 2 = 0. Let u i := f (e i ), i = 1, 2. These elements satisfy an obvious analogue of (3.2), moreover de 1 = de 2 = 0. There are three possibilities: In other words, f factorizes via the projection π 1 : f := f 1 π then clearly solves the lifting problem (3.1). Case 2.: u 1 = 0, u 2 = 1. This 'mirror image' of Case 1 can be treated analogously.
Case 3.: u 1 , u 2 = 0. Since p is a weak equivalence and since there are no 0-dimensional boundaries, p induces an isomorphism of 0-cocycles Z 0 (E) ∼ = Z 0 (B). 6 In particular, one has cocycles u 1 , u 2 ∈ E 0 such that p( u i ) = u i , i = 1, 2, satisfying conditions analogous to (3.2). Put B i := u i B and E i := u i E, i = 1, 2. It is clear that then B ∼ = B 1 ×B 2 , E ∼ = E 1 ×E 2 and that, under the above isomorphisms, also the maps f and p split, The maps p 1 and p 2 are defined in the obvious similar way. Since both p 1 and p 2 must clearly be weak equivalences and epimorphisms, one has, for i = 1, 2, the lifts f i in the diagrams The map f := ( f 1 , f 2 ) : A 1 ×A 2 → E 1 ×E 2 solves the lifting problem (3.1). 6 At this place we need B and E to be non-negatively graded. 7 These inclusions are homomorphisms of non-unital cdgas, but the composition f i preserves units.

[May 7, 2013]
Let us formulate the following simple principle whose proof is straightforward.

3.2.
Principle. Assume A i , i ∈ J, are arbitrary (unital) cdgas indexed by a finite set J and D a cdga such that 1 is the only nontrivial idempotent in Z 0 (D). Then the projections from the cartesian product induce monomorphisms of the homomorphisms sets which in turn induce a decomposition Together with Proposition 3.1, Principle 3.2 gives: be cdgas indexed by a finite set J and D ∈ A ≥0 be such that 1 ∈ H 0 (D) is the only nontrivial idempotent. Then the projections (3.3) induce a decomposition of the set of homotopy classes The same statement holds also with A in place of A ≥0 .
Proof. Let us prove the first part. Since D is non-negatively graded, where the vertical arrows are natural quotient maps, associating to a morphism its homotopy class. It follows that the lower horizontal arrow, making the diagram commutative, exists, is unique and and bijective; this finishes the proof for the category A ≥0 .
The second part with A in place of A ≥0 must be proved differently. The reason is that, firstly, no statement analogous to Proposition 3.1 holds in A and, secondly, even if 1 ∈ H 0 (D) is the only nontrivial idempotent, there may be many nontrivial idempotents in Z 0 (D), so Principle 3.2 does not apply.
To simplify the exposition, we assume again that J = {1, 2}, the proof for a general finite J is similar. Put u 1 := 1×0 ∈ A 1 ×A 2 and u 1 : So all we need to prove is that the projections induce an isomorphism with the projections p 1 , p 2 : D[t, dt] → D given by the evaluation at 0 resp 1, is a good path object in the sense of [9, §4.12] for D in both categories A ≥0 and A . Therefore, if so the pre-composition with c defines a natural map Proof. The cdga A admits a minimal model M by [5,Proposition 7.7]. As M is, by definition, weakly equivalent to A, The cdga M is clearly cofibrant in both categories A ≥0 and A , therefore is a good path object for D in both categories A ≥0 and A , The lemma is an obvious combination of the above isomorphisms.

3.5.
Proposition. The categories hoA c ≥0 and hoA c are equivalent.

[May 7, 2013]
Proof. If C ′ is a full subcategory of C with the property that each object of C ′ is isomorphic to some object of C, then both categories are equivalent. In light of Lemma 3.4, it is enough to prove that each A ∈ A c is weakly equivalent to an cdga in A ≥0 . Proposition 7.7 of [5] states the existence of a minimal model generated by elements of degrees ≥ 1 of each cdga A ∈ A c ≥0 . One can easily verify that the proof of this proposition leads to a minimal model of an arbitrary homologically connected cdga A, i.e. of an arbitrary A ∈ A c . The minimal model M of A ∈ A c is then a connected cdga belonging to A ≥0 , weakly equivalent to A. This finishes the proof. Theorem 3.3 has the following important consequence.
The rest follows from Theorem 3.3.

DG commutative associative algebras of finite type
In this section we investigate properties of cdgas having finite type in the sense of Definition 1. Note that having an augmentation ideal I in A is equivalent to specifying a map ǫ : A → k (an augmentation). Furthermore, the dual dg space (I/I 2 ) * is naturally identified with the dg space Der ǫ (A, k). Thus, A is of finite type if and only if for any augmentation ǫ : A → k the dg space Der ǫ (A, k) has finite-dimensional cohomology in each positive degree.
We now discuss the homotopy invariance of the notion of finite type. To this end, note that for a cofibrant cdga  Proof. Let A be of finite type and I ′ be an augmentation ideal of A ′ . The given quasi- Comparing the corresponding spectral sequences in [3, Corollary 2.5] we obtain that Der(A ′ , k) is quasi-isomorphic to Der(A, k) and thus, Der(A ′ , k) has finite-dimensional cohomology, as desired. Since each quasi-isomorphism of cofibrant algebras is homotopy invertible, the rôles of A and A ′ can be exchanged. This finishes the proof.
In the following statementLA is the completed Harrison complex of an augmented cdga A recalled in Definition 7.3.

Lemma.
Suppose that A is a cofibrant cdga and I is an augmentation ideal in A. Then (I/I 2 ) * is quasi-isomorphic to ΣLA.
Proof. Without loss of generality one can assume that the cdga A is free as a graded associative commutative algebra. It is easy to show that it is then isomorphic, as a non-differential algebra, to the graded polynomial ring generated by I. Lemma 4.3 therefore appears as a version of [22, Proposition 4.2(a)], so we omit its proof.
Observe that a non-negatively graded homologically connected cdga A is of finite type if and only if it is of finite type in the sense of [5, §9.2]. We can now formulate a criterion for a homologically disconnected cdga to be of finite type.
Proof. We assume, without loss of generality, that A is cofibrant and each A i is also cofibrant. Suppose that every A i is of finite type and let I be an augmentation ideal in A. By Theorem B, the map A → A/I ∼ = k must factor in the homotopy category through a map A i → k for some i = 1, . . . , n. Since by Lemma 4.1 the homology of the space Der(A, k) does not depend on the homotopy class of the map A → k we might as well assume that the map A → k factors through A i on the nose. Denote by I i the kernel of the corresponding map A i → k.
Consider the dga B :=T ΣI * , the reduced cobar-construction of A with the augmentation given by A → A/I ∼ = k. Note that B computes Ext A (k, k), the differential Ext of the cdga A. Similarly, the reduced cobar-construction k). Looking at the primitive elements in the spaces Ext A (k, k) resp. Ext A i (k, k) that are described as the homology of B resp. B i we conclude that the complete dglasL(A) andL(A i ) are quasi-isomorphic. Therefore, by Lemma 4.3, I/I 2 is quasi-isomorphic to I i /I 2 i . This proves that if each A i is of finite type, then so is A.
Conversely, suppose that A is of finite type and let I i be an augmentation ideal in A i ; then k becomes an A i -module via the augmentation which π i realizes the projection p i : A 1 × · · · × A n → A i in the homotopy category hoA . The same argument as above shows that the dg spaces I i /I 2 i and I/I 2 are quasi-isomorphic and thus A i is of finite type. [May 7, 2013] 4.5. Remark. The quasi-isomorphismL(A) ∼L(A i ) was crucial for the proof of Proposition 4.4. It can be established differently. Assume, for simplicity, that n = i = 2. Since, by a standard spectral sequence argument,L(−) preserves quasi-isomorphisms, we only need to prove thatL( . This gives the requisite statement. 4.6. Remark. It is not true that for a cofibrant cdga A of finite type and any maximal ideal I of A the quotient I/I 2 has finite dimensional cohomology. This is not even true for a homologically connected A. Indeed, let A be cofibrant cdga supplied with an augmentation A → k. Let K ⊃ k be an infinite dimensional field extension of k. The factorization axiom in the closed model category A expresses the composite map A → K as On the other hand, Der(A, K) is quasi-isomorphic to Der(Ã, K) ∼ = (Ĩ/Ĩ 2 ) * . We see that H(Ĩ/Ĩ 2 ) may be infinite-dimensional even whenÃ is of finite type.

Proofs of Theorems A, B and C
Proof of Theorem A. It is an obvious combination of Proposition 3.5 and Theorem B which we prove below.
Proof of Theorem B. We start by proving the more difficult second part of the theorem. Assume that A ∈ A dc and let ι i : k → i∈J k ∼ = H 0 (A) be the canonical inclusion into the ith factor. Denote by e i ∈ H 0 (A), i ∈ J, the idempotent ι i (1) and by H(A)[e −1 i ] the localization of H(A) at the multiplicative subset generated by e i . By elementary algebra, induces the isomorphism (5.1) of cohomology.
is homologically connected; its cohomology in negative degrees clearly vanishes. Therefore (5.2) shows that each A ∈ A dc is weakly equivalent to a finite product of cdgas from A c . Denote, for the purposes of this proof, [May 7, 2013] the full subcategory of A dc whose objects are these products byĀ dc . It follows from the above that the corresponding homotopy categories hoA dc and hoĀ dc are equivalent. Theorem 3.3 then implies that hoĀ dc is equivalent to Prod(hoA c ). This finishes the proof of the second part.
The first part can be proved in exactly the same way, but the situation admits a simplification. Assume the same notation as above. The cdga A is now non-negatively graded, so there are no 0-boundaries in A 0 , thus u i is unique and it is an idempotent. The multiplication with u i defines the projection π i : A ։ u i A which represents the localization map Since {u i } i∈J are orthogonal idempotents whose sum is 1, the system {π i } i∈J defines, instead of just a weak equivalence (5.2), a strict isomorphism is equivalent to its full subcategoryĀ dc ≥0 whose objects are finite products of cdgas from A c ≥0 , and the same is true also for the corresponding homotopy categories, that is, hoA dc ≥0 ∼ hoĀ dc ≥0 . The proof is finished with the aid of Theorem 3.3. Proof of Theorem C. The equivalence of Example 1.2 clearly restrict to the equivalence fNQ-hoS dc ∼ coProd(fNQ-hoS c ).
By Proposition 4.4, the equivalences of Theorem B restrict to the equivalences ). The equivalence fNQ-hoS dc ∼ fQ-hoA dc ≥0 then follows from the classical equivalence between fQ-hoA c ≥0 and fNQ-hoS c , cf. [5,Theorem 9.4]. It is easy to show that it is in fact induced by adjunction (0.1). The equivalence fQ-hoA dc ≥0 ∼ fQ-hoA dc is a combination of (5.3) with Proposition 3.5.
In this section we briefly outline the relationship between the homotopy theory of augmented cdgas and pointed spaces. The results formulated here will be used in Section 10. They are more or less obvious analogues of the non-augmented theory developed in the previous sections.
Recall (e.g. [14, Proposition 1.1.8]) that for a given closed model category C and an object O ∈ C, the overcategory of O and the undercategory of O are themselves closed model categories with fibrations, cofibrations and weak equivalences created in the category C. Now consider the overcategory A + of the initial object k in A ; its objects are augmented unital cdgas; i.e. cdgas A supplied with an augmentation A → k. The morphisms in A + are cdga maps respecting the augmentation. The category A + has a closed model structure inherited from A .
Inside the category A + is the category A c + consisting of homologically connected cdgas and the category A dc + consisting of augmented homologically disconnected cdgas. We also have the corresponding subcategories of non-negatively graded augmented cdgas, indicated by ≥ 0 in the subscript. Similarly we have the category S + of pointed simplicial sets, the undercategory of the terminal simplicial set; it has a closed model structure inherited from S . Let us formulate an augmented version of Theorem A: Theorem A + . The inclusion A ≥0 + ⊂ A + induces an equivalence of the homotopy categories hoA dc ≥0 + and hoA dc + .
For a category C with the terminal object * denote by coProd(C) + the category whose objects are the formal finite coproducts A 1 ⊔ · · · ⊔ A s , s ≥ 1, such that A 1 , . . . , A s−1 are objects of C, A s is an object of the undercategory * /C of C, and the Hom-sets are with the obvious composition law.
Assume that C is a closed model category and consider * /C with the induced closed model category structure. Since the categories ho( * /C) and * /hoC are not equivalent in general, unlike the un-pointed case, we cannot apply the above construction to the homotopy category ho( * /C) directly. We need to modify the above definition by taking coProd ho( * /C) + the category with the same objects as coProd( * /C), but with the morphism sets coProd ho( * /C) where B t denotes the object B t ∈ * /C considered as an object of C by forgetting the coaugmentation * → B t . The above definitions are designed to model the category of disconnected pointed simplicial sets: 6.1. Example. Denote by S c + the category of pointed connected simplicial sets and by S dc + the category of pointed simplicial sets with finitely many components. One then has S dc + ∼ = coProd + (S c ) and the same is obviously true for the homotopy categories, i.e. hoS dc + ∼ = coProd + (hoS c + ). Note that the weak equivalences in S dc + are pointed maps inducing bijections on the sets of connected components as well as a weak equivalence on each connected component.
Dually, let C be a category with an initial object * . Denote by Prod(C) + the category whose object are formal finite products A 1 × · · · ×A s , s ≥ 1, where A 1 , . . . , A s−1 are objects of C and A s is an object of the overcategory C/ * of C. The morphism sets are The category Prod ho(C/ * ) + is defined by an obvious dualization of the definition of coProd ho( * /C) + given above. The following theorems are proved analogously to the nonaugmented case.
Theorem B + . Each homologically disconnected non-negatively graded augmented cdga A ∈ A dc ≥0 + is isomorphic to a finite product A 1 × · · · × A s of homologically connected cdgas A 1 , . . . , A s−1 ∈ A c ≥0 and a homologically connected augmented cdga A s ∈ A c ≥0 + . This isomorphism extends to a natural equivalence of categories Each homologically disconnected augmented cdga A ∈ A dc + is weakly equivalent to a finite product A 1 ×· · ·×A s of homologically connected cdgas A 1 , . . . , A s−1 ∈ A c and a homologically connected augmented cdga A s ∈ A c + . As above, one has an equivalence hoA dc + ∼ Prod(hoA c + ) + . Theorem C + . The following three categories are equivalent.
-The homotopy category fNQ-hoS dc + of pointed simplicial sets with finitely many components that are rational and of finite type, -the homotopy category fQ-hoA dc ≥0 + of homologically disconnected non-negatively graded augmented cdgas of finite type over Q, and -the homotopy category fQ-hoA dc + of homologically disconnected Z-graded augmented cdgas of finite type over Q.

Part 2. The Lie-Quillen approach
In this part we give an application of the developed theory to the structure of MC spaces and describe the second version of disconnected rational homotopy theory based on dglas.

The simplicial Maurer-Cartan space
We write A + for the category of augmented cdgas; it is, thus, an overcategory of k ∈ A . As such, it inherits from A the structure of a closed model category. The weak equivalences and fibrations are still quasi-isomorphisms and surjective maps respectively. Note that the product of two augmented cdgas A and B is their fiber product A × k B. We denote the augmentation ideal of A ∈ A + by A + ; it is a possibly non-unital cdga. Conversely, given a non-unital cdga B one can form a unital algebra B e obtained by adjoining the unit; B e ∼ = k ⊕ B. Thus, the category A + is equivalent to the category of non-unital cdgas. 7.1. Definition. A complete dgla is an inverse limit of finite-dimensional nilpotent dglas. The category of complete dglas and their continuous homomorphisms will be denoted byL .

7.2.
Remark. The functor of linear duality establishes an anti-equivalence between the categoryL and that of conilpotent Lie coalgebras, cf. [3] where complete Lie algebras were called pronilpotent Lie algebras; we feel that this terminology might not be ideal since, e.g. an abelian Lie algebra on a countably dimensional vector space is not pronilpotent under this convention.
Let us show, however, that complete dglas g ∈L are pronilpotent in the classical sense, where g = g 1 ⊃ g 2 ⊃ · · · is the lower central series. Assume g = lim n g n where g n , n ≥ 0, are finite-dimensional nilpotent. Since the filtered limit of finite-dimensional vector spaces is exact, we easily verify that (g n ) k ∼ = lim n (g n ) k and that g/g k ∼ = lim n g n /(g n ) k for each k ≥ 1.
The nilpotence of g n implies lim k g n /(g n ) k ∼ = g n , therefore lim k g/g k ∼ = lim k lim n g n /(g n ) k ∼ = lim n lim k g n /(g n ) k ∼ = lim n g n = g as desired.
[May 7, 2013] 7.3. Definition. LetL : A + →L be the functor associating to an augmented cdga A the complete dglaL (A) whose underlying graded Lie algebra isLΣ −1 A * + , the completed free Lie algebra on Σ −1 A * + . The differential d inL (A) is defined as d = d I + d II where d I is induced by the internal differential in A + and d II is determined by its restriction onto Σ −1 A * , which is, in turn, induced by the product map A ⊗ A → A.
Dually, let C :L → A + be the functor associating to a complete dgla g the (discrete) augmented cdga C(A) whose underlying graded Lie algebra is SΣg * , the symmetric algebra on Σg * . The differential d in C(g) is defined as d = d I + d II where d I is induced by the internal differential in g and d II is determined by its restriction onto Σg * , which is, in turn, induced by the bracket map g ⊗ g → g.
7.4. Remark. The dglaL(A) is also known as the Harrison complex of an augmented cdga A whereas C(g) is an analogue of the Chevalley-Eilenberg complex of g, except that usually g is not assumed to be complete (and in that case C(g) has to be completed).
The following result follows directly from the definition. 7.5. Proposition. The functors C andL are adjoint, so there is a natural isomorphism for A ∈ A + and g ∈L :L L (A), g ∼ = A + C(g), A .
7.6. Remark. In Section 9 we endowL with the structure of a closed model category, in such a way that the functorsL and C will become inverse equivalences of the corresponding homotopy categories. We will also see that each cdga of the form C(g) is cofibrant.
7.1. Simplicial mapping space for cdgas and dglas. To construct a simplicial Hom in the category of complete dglas we are forced, as an intermediate step, to deal with dglas endowed with a linear topology, but which are not complete. An example of such a dgla is a tensor product A ⊗ g where g is a complete dgla and A a discrete cdga that is infinite dimensional. Our convention is that the tensor product of a discrete vector space W and a complete vector space V = lim α V α is the topological vector space W ⊗ V := lim α (W ⊗ V α ). Whenever two dglas g and h are endowed with a linear topology we will write Hom dgla (g, h) for the set of continuous Lie homomorphisms from g into h.
Let g and h be two complete dglas. We are going to construct a simplicial set of mapŝ L (g, h) • . As expected, the simplicial enrichment will be obtained by tensoring the dgla h with the Sullivan-de Rham algebra Ω(∆ • ) of polynomial forms on the standard topological cosimplicial simplex, cf. [5, §2]. According to our conventions, Ω(∆ • ) will be considered as a homologically graded cdga, so Ω(∆ • ) ⊗ h will be a homologically graded dgla. In particular, Ω p (∆ • ) ⊗ h q will be placed in homological degree q − p.
[ May 7, 2013] Recall that the category A + also has a simplicial Hom: for two augmented cdgas A and B let A + (A, B) • be the simplicial set for which The following result is an enriched version of Proposition 7.5; its proof is a straightforward inspection.
7.9. Proposition. For any augmented cdga A and a complete dgla g there is a natural isomorphism of simplicial sets: 7.10. Remark. We defined the simplicial mapping space between two augmented cdgas in this way in order for the adjunction (7.2) to hold. The simplicial mapping space in the category of (unital) cdgas is different; for two (unital) cdgas A and B we have It is easy to see that if A is an augmented cdga and B is a (unital) cdga then there is an isomorphism between simplicial sets Denote by s the dgla spanned by two vectors x and [x, x] with |x| = −1 and the dif- The dgla s is designed to model the topological space S 0 , the zero-dimensional sphere (which is a disjoint union of two points). Note that s is isomorphic toL(k × k), where k × k is the augmented cdga (with vanishing differential) obtained from the ground field k by adjoining a unit. 7.11. Definition. Let g be a complete cdga. Its MC simplicial set MC • (g) is the simplicial mapping space: MC • (g) :=L (s, g) • . In other words, MC • (g) is the simplicial space of MC elements in g ⊗ Ω(∆ • ). The set π 0 MC • (g) is called the MC moduli set of g and will be denoted by M C (g). 7.12. Example. Fix a finite set S and assign, to each s ∈ S, a degree −1 generator x s . Consider the complete dgla g S :=L(x s ; s ∈ S) freely generated by the set {x s } s∈S , with the differential given by dx s := − 1 2 [x s , x s ]. Let us describe the MC elements in g S ⊗Ω(∆ n ). The vector space g S ⊗Ω(∆ n ) −1 consists of expressions The MC condition for x reads By the freeness of g S , the above equation is satisfied if and only if dα s = 0 for each s ∈ S, (7.4a) α s (1 − α s ) = 0 for each s ∈ S and (7.4b) α s ′ α s ′′ = 0 for each s ′ , s ′′ ∈ S, s ′ = s ′′ . We conclude that MC • (g S ) is the constant simplicial set representing the discrete space S ∪ { * }. If S is arbitrary, we define g S := lim F g F , the obvious limit over finite subsets F ⊂ S. It is easy to check that the above calculation remains valid, so we see that each non-empty discrete space can be realized as the MC space of some complete dg-Lie algebra. An interpretation of this example is given in Remark 7.19 below. 7.13. Remark. In light of Proposition 7.9 we see that the MC simplicial set of a complete dgla g is isomorphic to It is easy to see that if two complete g and h are related by a filtered quasi-isomorphism, then MC • (g) and MC • (h) are weakly equivalent simplicial sets. We will not need this result.

Disjoint products of dglas and their MC spaces.
Recall that an MC element ξ ∈ h −1 in a dgla h allows one to twist the differential d in h according to the formula d ξ (?) = d(?) + [?, ξ]. The graded Lie algebra h supplied with the twisted differential d ξ will be denoted by h ξ . The same construction applies when h is a complete dgla, in that case h ξ will likewise be complete.
Given an arbitrary complete dgla g consider the complete dgla g * s where * stands for the coproduct in the categoryL . Abusing the notation, we will write x for the image of x ∈ s inside g * s under the inclusion of dglas s ֒→ g * s. It is clear that x is an MC element in g * s. The twisted dgla (g * s) x is analogous to adjoining a base point to a topological space. Generalizing this construction, we define a disjoint product of two complete dglas; note that it also makes sense for ordinary (non-complete) dglas. 7.14. Definition. Let g and h be two complete dglas. Their disjoint product is the complete dgla (g * s) x * h; it will be denoted by g ⊔ h.
In particular, for a dgla g the twisted free product (g * s) x is now denoted as g ⊔ 0. Even if g is non-negatively graded, g ⊔ 0 is not; this obviously applies also to more general disjoint products of dglas. 7.15. Remark. The operation of a disjoint product of two dglas was considered (without this name) in [6] and, in a more general, operadic context in [7]. It is not hard to prove that for any dgla g (complete or not) the dgla g ⊔ 0 is acyclic, cf. [7, Theorem 5.7] or [6, Lemma 6.1]. It follows that for any two dglas g and h the dgla g ⊔ h is quasi-isomorphic to h.
In particular, the operation of a disjoint product of two (complete) dglas is very far from being commutative. This phenomenon has the following topological explanation, or analogue. Given two pointed topological spaces X and Y define their 'disjoint product' as the disjoint union X ⊔ Y having its base point in Y . Clearly, the operation of disjoint product of two topological spaces is likewise noncommutative, owing to the asymmetric placement of the base point.
The following statement 'measures' the non-commutativity of the disjoint product.
[May 7, 2013] 7.16. Proposition. The disjoint products g ⊔ h and h ⊔ g are related by a twist; namely the element −x is an MC element in g ⊔ h ∼ = (g * s) x * h and there is an isomorphism of dglas Proof. The isomorphism in the proposition is the identity map on g and h and takes x ∈ g⊔h to −x ∈ h ⊔ g. It is straightforward to check that it has the desired property.
However, the following result shows that the operation of the disjoint product of dglas is associative, just as its topological counterpart. 7.17. Proposition. For any three dglas g, h, a there is a natural isomorphism of dglas: Proof. We have an isomorphism of graded Lie algebras, disregarding the differential: Denote the generator of the first copy of s by x and the generator of the second copy of s by y in (7.5). Then we have by definition an isomorphism of dglas: Similarly there is an isomorphism of dglas where x ′ and y ′ have an obvious meaning.
Now consider the map f : g ⊔ (h ⊔ a) ∼ = (g ⊔ h) ⊔ a which is the identity on g, h and a, f (x) := x ′ + y ′ and f (y) := y ′ . A straightforward calculation shows that f is compatible with the differentials and is, therefore, an isomorphism as claimed. 7.18. Remark. The operation of disjoint product does not make the category of dglas into a monoidal category, since there is no unit object. However, one can check that the pentagon condition for the operation ⊔ is satisfied and thus, in any expression involving multiple disjoint products the arrangement of brackets does not matter, up to a natural isomorphism. 7.19. Remark. As an exercise we recommend to verify that for finite S the algebra g S from Example 7.19 is isomorphic to the disjoint product of 0 with itself, iterated card(S)-times, that is For an MC element ξ ∈ h −1 denote by φ ξ : MC(h ξ ) → MC(h) the isomorphism η → η + ξ, η ∈ MC(h ξ ), and by φ ξ • : MC • (h ξ ) → MC • (h) the obvious induced map of simplicial sets. Our proof of Theorem 1.8 will be based on 7.20. Proposition. Let g be a complete dgla and ξ ∈ g −1 an MC element. Then the composition induces a weak equivalence between MC • (g ξ ) and the connected component of MC • (g) containing ξ ∈ MC(g).
Proof. Let h be a complete dgla. By Remark 7.13 we have an isomorphism of simplicial As shown in Section 9, the cdga C(h) is cofibrant, so the homotopy groups of the simplicial space A C(h), k • can be, for n ≥ 1, calculated using [5, Proposition 8.12] with X := C(h). One gets Since all isomorphisms above are functorial, they combine into a functorial isomorphism valid for an arbitrary complete dgla. This isomorphism is for n ≥ 2 an isomorphism of abelian groups, but we will not need this result.
Let us return to the proof of our theorem. It is clear that the the simplicial isomorphism φ ξ • : MC • (g ξ ) ∼ = MC • (g) induces an isomorphism between the connected component containing the trivial MC element 0 ∈ MC(g ξ ) and the connected component containing ξ ∈ g −1 . Furthermore, it is clear that the simplicial set MC • (g ξ ) is connected, with the trivial MC element 0 ∈ g ξ −1 its only 0-simplex. It is therefore enough to show that the inclusion ι : g ξ ֒→ g ξ induces and isomorphism for arbitrary n ≥ 1. By (7.7), this is the same as showing that ι induces an isomorphism for each n ≥ 1. The last isomorphism easily follows from the definition (7.6) of the connected cover g ξ of g ξ .
Proof of Theorem 1.8. The isomorphism (1.1) follows from Proposition 7.20, the weak equivalence from (1.1) and Theorem 1.6. We start this section by the following auxiliary statement. 8.1. Lemma. Let A, B and C be unital cdgas with A and B cofibrant and such that H 0 (C) has no idempotents different form 1. Denote by A×B a cofibrant approximation of A×B. Then there is a weak equivalence of simplicial sets Proof. Let S be a finite, connected simplicial set and let Ω(S) be the cdga of Sullivande Rham forms on S (denoted by A(S) in [5]). Then we have the following standard adjunction isomorphism, cf. [5, Lemma 5.2]: Observe that this adjunction automatically extends to simplicially enriched Homs, that is Indeed, by (8.1) one has for each n ≥ 0, , Ω(S) ⊗C n for each n ≥ 0. By naturality, the above individual isomorphisms assemble into an isomorphism (8.2) of simplicial sets. Since A × B is cofibrant, the simplicial set A (A × B, C) • is Kan. Adjunction (8.2) induces an isomorphism between the sets of connected components of the corresponding simplicial sets, therefore The cdga Ω(S) ⊗ C is bigraded and the first grading is non-negative. It is easy to show that in this situation any idempotent in its cohomology must be contained in the (0, 0)graded part which is isomorphic to H 0 (S) ⊗ H 0 (C) ∼ = H 0 (C). But H 0 (C) has no non-trivial idempotents other than 1 by our assumption. According to Theorem 2.3 Note that, because S is connected, we have a natural bijection Thus, we have a natural bijection in the homotopy category of simplicial sets, for any connected finite simplicial set S: The desired conclusion follows.
[May 7, 2013] 8.2. Remark. Note that the above lemma is an enriched analogue of Theorem 3.3. Since A×B is not cofibrant in general, the result will not be true without taking a cofibrant replacement. Also note that the lemma can be easily generalized to a finite direct product of cofibrant cdgas.
We need one preliminary result on the MC twisting of dglas. Let g be a complete dgla and consider the cdga C(g). It is clear that the set of MC elements in g is in 1-1 correspondence with cdga maps (augmentations) C(g) → k, in particular the canonical augmentation corresponds to the zero MC element.
Let us explain this correspondence explicitly. The evaluation at ξ ∈ g −1 determines a degree +1 linear map α ξ : g * → k which in turn defines a degree 0 linear map (denoted by the same symbol) α ξ : Σg * → k from the space of generators of C(g) to the ground field. The latter map extends to a unique morphisms of graded algebras ǫ ξ : C(g) → k which commutes with the differentials, i.e. makes C(g) an augmented cdga, if and only if ξ is Maurer-Cartan.
Likewise, define a degree 0 linear map β ξ : Σg * → k ⊕ Σg * ⊂ C(g ξ ) as β ξ := α ξ + 1 1 Σg * . One can easily verify that β ξ extends to a cdga's isomorphism φ ξ : C(g) → C(g ξ ) such that the diagram The maps ǫ ξ and φ ξ above have a nice geometric meaning if we interpret elements f ∈ C(g) or f ∈ C(g ξ ) as polynomial functions on Σg. Then ǫ ξ is the evaluation at ϑ := Σξ while φ ξ is given by the shift by ϑ, that is Checking the commutativity of (8.3) in this language boils down to Given ξ ∈ MC(g) there is a bijective correspondence MC(g) → MC(g ξ ) defined by η → η − ξ. In particular, the element ξ ∈ MC(g) corresponds to the zero element in MC(g ξ ).
8.3. Lemma. Let g be a complete dgla. Then the augmented cdga C(g ⊔ 0) is quasi-isomorphic to C(g) × k, with the augmentation given by the projection to the second factor.
Proof. Consider the cdga C(g * s). According to Corollary A.7 the augmented cdga C(g * s) is quasi-isomorphic to C(g) × k (k × k) ∼ = C(g) × k; moreover, the augmentation of C(g) × k factors through the projection onto the first factor. On the other hand, there is, as in (8.3), an isomorphism of cdgas (not respecting the augmentation) φ x : C(g * s) → C(g * s) x such that the induced augmentation C(g * s) → C(g * s) x → k corresponds to the MC element x ∈ MC(g * s).
Note that the augmentation of C(g * s) corresponding to the MC element x ∈ MC(g * s) translates via the above quasi-isomorphism into the augmentation of C(g) × k given by the projection onto the second factor. This proves the desired statement.
[ May 7, 2013] We now return to the proof of Theorem 1.6. It suffices to consider the case when J consists of two elements. Let g and h be two complete dglas. Then by Theorem A.7 and Lemma 8.3 the augmented cdga C(g ⊔ h) = C (g ⊔ 0) * h is quasi-isomorphic to We conclude that the simplicial set MC • (g ⊔ h) = A + (C(g ⊔ h), k× k) • is weakly equivalent to A + (C(g) × C(h), k× k) • . The latter simplicial Hom is taken in the category of augmented cdgas; it is isomorphic to the simplicial mapping space A (C(g) × C(h), k) • taken in the category of (unital) non-augmented cdgas. Now we have the following weak equivalence of simplicial sets, which follows from Lemma 8.1: This finishes the proof of Theorem 1.6.
We close this section by two versions of an example which is a dgla analogue of a disjoint union of a circle S 1 and an isolated point. This example is simple enough to be worked out by hand, although the calculations are still nontrivial. We will see that, in this particular case, an analogue of Theorem 1.6 actually holds without a completion. The general claim, in the non-completed context, was made in [6, Theorem 6.4], but we have not been able to parse the proof in op. cit. for some α i ∈ Ω 0 (∆ n ), i ≥ 0, and ω ∈ Ω 1 (∆ n ), with the assumption that only finitely many α i 's are = 0. The MC equation for t reads a]⊗ω 2 clearly vanishes). Singling out the parts in f 0 ⊗Ω 2 (∆ n ), f −1 ⊗Ω 1 (∆ n ) and f −2 ⊗Ω 0 (∆ n ), respectively, one gets the equations Since, by definition, e i+1 = [a, e i ], (8.5b) implies that dα 0 = 0, i.e. α 0 ∈ k is a constant, while [May 7, 2013] Equation (8.5c) means that i≥0 e i ⊗α i is an MC element in f⊗Ω 0 (∆ n ), with Ω 0 (∆ n ) taken with the trivial differential. It expands to the system . . .
The Lie algebra f 0 is one-dimensional, spanned by a. Consider, only for the purposes of this example, the completionf of f with respect to the grading by the number of a's. Let now t be an MC element in f ⊗ Ω(∆ n ) with α 0 = 1 and α 1 = 0. Then exp(−a⊗α 1 )t is clearly an MC element inf ⊗ Ω(∆ n ) with α 1 = 0, so exp(−a⊗α 1 )t = x⊗1 or, equivalently, In particular, t has infinitely many non-trivial elements, so it is not an MC element in the uncompleted dgla f⊗Ω(∆ n ). So α 1 must be 0 therefore the only MC element of f⊗Ω(∆ n ) with α 0 = 1 is x⊗1. We arrive at the following conclusion.
Claim. The only MC elements of f⊗Ω(∆ n ) are either x or closed 1-forms in Ω 1 (∆ n ). In other words, one has an isomorphism of simplicial sets where 0 is the trivial Lie algebra and a the one-dimensional abelian Lie algebra generated by a. So f serves as an algebraic model for the disjoint union S 1 ∪ { * }.

8.5.
Example. Let us analyze the completed version of the previous example, i.e. the MC elements inf ⊗Ω(∆ n ), wheref is the completion of the Lie algebra f = L(x, a). Of course, we know the answer from Theorem 1.6 but the explicit calculation that could be performed in this simple case is still instructive. Most of the work done in the previous example applies in the complete situation with only small modifications. For instance, elements of (f ⊗Ω(∆ n )) −1 are of the form (8.4), but the sum may now have infinitely many nontrivial terms.
Equations (8.5a)-(8.7d) still take place, as does the separation into Case α 0 = 0 and Case α 0 = 1. Precisely as in Example 8.4 we show that the only MC element with α 0 = 0 is the trivial one. In contrast with the uncomplete case we, however, do not require the sum (8.9) to be finite, so α 1 there may be an arbitrary function from Ω 0 (∆ • ).

Dual Hinich correspondence
The purpose of this section is to establish an analogue to the main result of Hinich's paper [13], giving a closed model category structure on cocomplete cocommutative dg coalgebras. A formal dualization of Hinich's result states that the category of complete cdgas has a closed model structure, cf. [16] where this formulation was explicitly spelled out. The result proved in this section should be viewed as the Koszul dual to Hinich's. We prove this Koszul dual version by suitably adapting Hinich's methods. The use of the associative version of this result contained in [17] allows us to shorten the proof in several places.
Recall that in Definition 7.3 we introduced an adjoint pair of functors C andL between the categories A + of augmented cdgas andL of complete dglas. We will write i g :LC(g) → g and i A : CL(A) → A for the counits of this adjunction. Our goal is to establish a closed model category structure onL in such a way that the functors C andL would induce an equivalence on the level of homotopy categories.
The adjoint functorsL and C can be 'embedded' into an adjunction between bigger categories of associative algebras. We will denote the category of augmented dg associative (discrete) algebras (dgas) by Ass + ; algebras in Ass + will be cohomologically graded . The augmentation ideal in an augmented dga A will be denoted by A + , as in the commutative case. Let us remind the details of the corresponding adjunction following [17]. First, we need a relevant definition. 9.1. Definition. A complete augmented dga is, by definition, an inverse limit of finitedimensional nilpotent augmented dgas. 8 The category of complete dgas and their continuous homomorphisms will be denoted by Ass + . Algebras in Ass + are assumed to be homologically graded.
Algebras of Definition 9.1 are complete in the sense of [23, Appendix A.1]. For instance, repeating the arguments used in Remark 7.1 to prove that complete dglas are pronilpotent, one may easily show that lim k A/A k + ∼ = A, which is Condition (c) on page 265 of that paper. 9.2. Remark. The functor of linear duality establishes an anti-equivalence between the category Ass + and that of conilpotent dg coalgebras, cf. [11] where complete dgas were called formal dgas; again, we opted to change this terminology since formality often has a different meaning in homological algebra. 9.3. Definition. LetB : Ass + → Ass + be the functor associating to a dga A the complete dgaB(A) whose underlying graded algebra isT Σ −1 A * + , the completed tensor algebra on Σ −1 A * + . The differential d inT Σ −1 A * + is defined as d = d I + d II where d I is induced by the internal differential in A and d II is determined by its restriction onto Σ −1 A * + , which is, in turn, induced by the product map A + ⊗ A + → A + .
Likewise, let B : Ass + → Ass + be the functor associating to a complete dga C the (discrete) dga B(C) whose underlying graded algebra is T ΣC * + , the tensor algebra on ΣC * + . The differential d in B(C) is defined as d = d I + d II where d I is induced by the internal differential in C and d II is determined by its restriction onto ΣC * + , which is, in turn, induced by the product C + ⊗ C + → C + . 9.4. Remark. We will refer to either functor B orB as the cobar-construction. Note that for a dga A its cobar-constructionB(A) is a complete Hopf algebra, whose space Σ −1 A * + of algebra generators consists of primitive elements. If A is commutative, then the differential ofB(A) takes Σ −1 A * + to primitives and thus makesB(A) a dg Hopf algebra.
(1) Let A be a cdga. Then there is a natural isomorphism of complete dglas: (2) Let g be a complete dgla andÛ g be its completed universal enveloping algebra. Then there is a quasi-isomorphism of dgas Proof. The first statement is a consequence of the completed version of the well-known fact that the primitive elements in the tensor algebra on a graded vector space form the free Lie algebra on the same vector space which easily follows from the Appendix to [23].
To prove the second statement, note that the inclusion of g intoÛ g as the space of primitive elements induces a map of dgas B(Û g) → C(g). To see that the latter map is a quasiisomorphism it suffices to assume that g is finite-dimensional nilpotent; the general result will be obtained by passing to the limit. Denote byg the graded Lie algebra with the same underlying space as g and the vanishing differential. Then we have spectral sequences with E 1 terms H B(Ûg) and H C(g) , which converge to B(Û g) and C(g) respectively. The map B(Û g) → C(g) gives a map between these spectral sequences. It is therefore enough to prove that the map B(Ûg) → C(g) is a quasi-isomorphism.
As in the proof of Lemma A.5 we use the filtration ofg * induced by the shifted lower central series ofg. This filtration induces, in the usual way, increasing exhaustive and complete filtrations of B(Ûg) and C(g) compatible with the map B(Ûg) → C(g). This brings us to the case wheng is abelian, the desired result then follows from the calculation of Tor C(g) (k, k) via the Koszul complex.
The category Ass + has a closed model category structure, by a general result of Hinich, cf. [12]. Namely, weak equivalences in Ass + are quasi-isomorphisms of augmented dgas and fibrations are surjective homomorphisms. The category Ass + of complete dgas also admits the structure of a closed model category, as follows. (3) a cofibration if f has the left lifting property with respect to all acyclic fibrations. That means that in any commutative square / / D where g is an acyclic fibration there exists a dotted arrow making the whole diagram commutative. 9.7. Theorem. The category Ass + is a closed model category with fibrations, cofibrations and weak equivalences defined as above. Moreover, it is Quillen equivalent to the closed model category Ass + via the adjunctions B andB.
Proof. This is just a reformulation of Théorème 1.3.1.2 of [17], taking into account the antiequivalence between complete dgas and conilpotent dg coalgebras. 9.8. Remark. The above result is an associative analogue of Hinich's theorem [13] on the existence of a closed model category on dg conilpotent coalgebras or, equivalently, on complete cdgas. All objects in Ass + are fibrant and cofibrant objects correspond precisely to A ∞ algebras, cf., for example, [11] for a treatment of A ∞ algebras relevant to the present context.
We will now construct a closed model category structure onL using Theorem 9.7 as a shortcut; a more direct approach, essentially repeating the original Hinich's argument in the dual context is also possible. We will prove that the above structures makeL a closed model category. Our proof will be based on the following results. 9.10. Proposition.
(1) Let A be an augmented cdga; then i A : CL(A) → A is a quasi-isomorphism.
(2) Let g be a complete dgla; then i g :LC(g) → g is a weak equivalence, i.e. C(i g ) : Proof. There is a quasi-isomorphism CL(A) ≃ B ÛL (A) in Ass + by Proposition 9.5(2). Furthermore, the dga B ÛL (A) = BB(A) is quasi-isomorphic to A by Theorem 9.7. This proves (1).
Let us prove that the induced map C(i g ) : C L C(g) → C(g) is a quasi-isomorphism of augmented cdgas. By Proposition 9.5(2) we have a quasi-isomorphism of dgas C L C(g) ≃ BÛ L C(g) = BBC(g) and, by Theorem 9.7, BBC(g) is quasi-isomorphic to C(g) as required.
[ May 7, 2013] The following statement is an analogue of the 'Key Lemma' of [13, p. 223]. 9.11. Lemma. Let A be a cdga, g be a complete dgla and f : A → C(g) be a surjective map. Consider the pushout diagramL Then C(j) : C(a) → CL(A) is a quasi-isomorphism of cdgas.
In the proof of Lemma 9.11 we will use the following technical statement.
be a chain map. Assume that A ′ = p,q F p,q A ′ is a finite double filtration which is descending in the sense that Suppose that A ′′ has a double filtration with the similar properties and that φ is compatible with these filtrations. If the induced map φ p,q : of the quotients, with the differentials induced by the 'untwisted' parts d ′ 1 resp. d ′′ 1 , is a quasiisomorphism for each p and q, then φ is a quasi-isomorphism, too.
Proof. Consider the spectral sequences in the q-direction. The E 1 -sheets of these spectral sequences are clearly the same as if the twisted parts d ′ 2 resp. d ′′ 2 of the full differentials vanish. We may therefore assume d ′ 2 = d ′′ 2 = 0 from the beginning, in which case is the claim obvious.
We will also need to know that the functor C(−) preserves filtered quasi-isomorphisms. The proof of the following statement is a harmless modification of [13,Proposition 4.4.4]. 9.13. Lemma. Assume that u and v are filtered complete dglas, with complete filtrations Let φ : u → v be a morphism, compatible with the filtrations, such that the induced map φ n : u/F n u → v/F n v is a quasi-isomorphism for each n ≥ 1. Then C(φ) : C(v) → C(u) is a quasi-isomorphism, too.
9.14. Remark. The map φ n : u/F n u → v/F n v is a chain map in the category of linearly compact spaces. By saying it is a quasi-isomorphism we mean that it is a quasi-isomorphism in the underlying category of dg vector spaces. It is easy to show that φ n is a quasiisomorphism if and only if its dual φ * n is a quasi-isomorphism in the usual sense, compare Remark 10.3.

[May 7, 2013]
Proof of Lemma 9.11. Our proof will be similar to that of [13, pp. 224-225] or [17, pp. 42-43]. The kernel B of the map f : A → C(g) is a non-unital cdga. Since C(g) is free as a non-differential algebra, choosing an algebra splitting of f , we obtain an isomorphism of graded algebras, in which B is closed under the differential. ThereforeL(A) is isomorphic, as a complete graded Lie algebra, toL(B e ) * LC(g), the (completed) free product ofL(B e ) andLC(g). The differential inL(A) consists of three parts: the differential inL(B e ), the differential inLC(g) and the 'twisted' part mapping the generators ofL(B e ) intoLC(g).
We clearly have the following isomorphisms of complete graded Lie algebras (again, disregarding the differentials): The differential inL(B e ) * g is, under this isomorphism, also the sum of three parts: the differential inL(B e ), the differential in g and the twisted part, which maps the generators of L(B e ) into g. It is clear that the map j :L(A) → a, under the above identifications, equals (9.2) 1 1 * i g :L(B e ) * LC(g) →L(B e ) * g, but the differentials differ by the twistings from those of the free products.
Before we proceed, we need some notation. If u and v are filtered complete dglas as in (9.1), then their completed free product u * v possesses an induced double filtration with F p,q (u * v) the closure of the subspace generated by all iterated brackets of u i ∈ F p i u and v j ∈ F q j v, 1 ≤ i ≤ a, 1 ≤ j ≤ b, such that p 1 + · · · + p a = p and q 1 + · · · + p b = q.
One also has the associated total filtration with F n (u * v) := p+q=n F p,q (u * v).
For graded dglas one defines the bigrading on their completed free product in the obvious analogous manner. We then have the following formula for the associated bigraded dgla: Let us return to our proof. The lower central series g 1 ⊃ g 2 ⊃ g 3 ⊃ · · · of g induces, in the standard manner, a filtration ofLC(g) while the complete free Lie algebraL(B e ) is filtered by its lower central series. As explained above, one has the induced double filtrations of the completed free productsL(B e ) * LC(g) andL(B e ) * g. The associated total filtrations are stable under the differentials and the map (9.2) factorizes, for each n, into a map (9.4) φ = φ n :L (B e ) * LC(g) F n (L(B e ) * LC(g)) −→L (B e ) * g F n (L(B e ) * g) of dglas. By Lemma 9.13, it suffices to prove that φ is a quasi-isomorphism for each n.
Fix n and consider the double filtrations of the quotients in (9.4) induced by the double filtrations F p,q L (B e ) * LC(g) resp. F p,q L (B e ) * g . It is straightforward though technically involved to prove that these filtrations and the map φ satisfy the assumptions of Lemma 9.12. Taking the quotients was needed to make these double filtrations finite.
Since the gradings of both Gr(L(B e )),LC(Gr(g)) and Gr(g) are positive, the (p, q)components of the free products in (9.5) are spanned by iterated brackets of length ≤ p + q. We may thus in (9.5) disregard the topologies and consider uncompleted free products.
In the proof of Theorem A.4 we established that the uncompleted free product is an exact functor. To finish the proof, it is therefore enough to establish that the canonical mapLC Gr(g)) p → Gr(g) p is a quasi-isomorphism for each p. It is immediate to see that LC(Gr(g)) p coincides with the degree p-part L C (Gr(g)) p of the composition of the adjoint functors C and L (see [13, 2.2] for their definitions) applied to Gr(g) considered as a discrete dgla. The proof is finished by of [13,Proposition 3.3.2] by which the canonical map L C (Gr(g)) → Gr(g) is a quasi-isomorphism.
Let us prove another auxiliary statement. in which f is a fibration in A + by assumption. Since C(u) is a quasi-isomorphism by the definition of weak equivalences inL , all we need to show is that C(u) is a cofibration in A + .
Denote by g = g 1 ⊂ g 2 ⊂ g 2 ⊂ · · · the lower central series of g. In Remark 7.2 we verified that g = lim g/g n . Since K := Ker(f ) ⊂ g is a closed subspace, by standard properties of filtered limits we verify that the limit of the tower (9.6) h ∼ = g/(g 1 ∩ K) և · · · equals g, therefore the colimit of the diagram C(h) ֒→ C(g/g 2 ∩ K) ֒→ C(g/g 3 ∩ K) ֒→ C(g/g 4 ∩ K) ֒→ · · · equals C(g). It would therefore suffice to prove that the maps C(π n ), where π n are as in (9.6), are cofibrations in A + for each n ≥ 1.
To this end, observe that the kernel of π n is an abelian dgla (g n ∩ K)/(g n+1 ∩ K). With this knowledge, it is easy to see as in the proof of the lemma in [13, §5.2.2] that C(π n ) is a standard cofibration obtained by adding free generators to C(g/g n ∩ K).
To prove the last part of the lemma, note the standard fact that cofibrations in A + are monomorphisms, whileL(−) clearly converts monomorphisms to epimorphisms, i.e. fibrations inL .
In the proof of Lemma 9.15 we established that the functor C(−) converts fibrations to cofibrations. As each g ∈L is fibrant, this in particular implies that the dglas C(g) are cofibrant in A + . 9.16. Theorem. The categoryL is a closed model category with fibrations, cofibrations and weak equivalences as in Definition 9.9. Moreover, it is Quillen equivalent to the closed model category A + via the adjunctionsL and C.
Proof. The arguments are precisely dual to those of Hinich [13, pp. 223-224]. The categorŷ L admits arbitrary limits and colimits: the limits and coequalizers of pairs of maps are created in the category of dg vector spaces while a coproduct of a family of complete dglas is constructed by taking their free product, and then completing. This proves axiom CM1; the axioms CM2 and CM3 are obvious, and one half of CM4 holds by definition.
Let us prove the factorization axiom CM5. Given a map f : g → h inL , consider the corresponding map of cdgas C(f ) : C(h) → C(g). Suppose we factorize C(f ) as where p is a surjective map, i.e. a fibration, and i a cofibration in A + . Then one has an induced factorization of f : wherep is easily seen to be an epimorphism, i.e. a fibration inL . By construction,ĩ is obtained by a cobase-change fromL(p) which is a cofibration by Lemma 9.15. Thusĩ is a cofibration as well.
Since A + is a closed model category, factorization (9.7) exists and can be chosen such that p is a quasi-isomorphism. Then the correspondingĩ in (9.8) is acyclic; this follows (just [ May 7, 2013] as in [13]) from Lemma 9.11 and Proposition 9.10(1). This proves the first factorization of CM5.
The proof of the second factorization property of CM5 is similar, we only choose this time i in (9.7) to be a quasi-isomorphism. The mapp is then acyclic by Lemma 9.11. It remains to prove the second half of CM4, i.e. that any acyclic cofibration inL has the LLP (left lifting property) with respect to all fibrations. Let f : g → h be such a cofibration. Factorize it as in (9.8): f =p •ĩ wherep is an acyclic fibration andĩ is an acyclic cofibration obtained by a cobase-change from a mapL(p), where p is an acyclic fibration in A + .
By Lemma 9.15,L(p) is an acyclic cofibration inL , so it has the LLP with respect to all fibrations inL , thusĩ has the same property. Sincep has the LLP with respect to f it follows that f is a retract ofĩ and so, it has the LLP with respect to all fibrations inL as required. This finishes our proof thatL is a closed model category.
The statement that the adjoint functorsL and C form a Quillen pair, i.e. satisfy assumptions of [9,Theorem 9.7], follows at once from Proposition 9.10 and Lemma 9.15.
An interesting feature of the closed model structure of Definition 9.9 is the existence and uniqueness of minimal models. Proof. For g ∈L denote by H the cohomology of C(g) and choose a homotopy equivalence f : C(g) → H in the category of dg vector spaces; here H is considered with the trivial differential. By [20, Move M1, p. 133], there exists a C ∞ -algebra C with the underlying dgvector space H, and a C ∞ -morphism F : C(g) → C extending f . Let M = L (Σ −1 H * ), ∂ be the complete dgla corresponding to the C ∞ -algebra C. It is minimal and the C ∞ -morphism F translates into a weak equivalence ϕ : M →LC(g). The canonical map i g :LC(g) → g is, by Proposition 9.10(2), a weak equivalence, too. The composition is then the desired minimal model of g.
Let us prove uniqueness. Suppose that ρ ′ : M ′ → g is another minimal model. Since C(g) is fibrant as every object of A + ,LC(g) is cofibrant by Proposition 9.15. Although ρ ′ need not be a fibration, it is still a map between cofibrant objects. By e.g. [25,Lemma 3.5], there exists the dotted arrow α in the diagram g making the right triangle homotopy commutative. The composition αϕ is a weak equivalence of minimal dglas, so it induces an isomorphism of the generators; it is therefore itself an isomorphism.
[May 7, 2013] 9.1. Non-unital monoidal structures onL and A + . In Subsection 7.2 we constructed a non-unital monoidal structure on the categoryL of complete dglas. An analogous, but simpler, structure on the category A + is given by the product A × B of augmented cdgas A, B ∈ A + equipped with the augmentation through the projection onto the second factor. The operation A, B → A × B is coherently associative, but not commutative and not unital. The two structures correspond to each other under the adjoint functorsL and C: 9.19. Proposition.
(1) For two augmented cdgas A and B there is a natural isomorphism inL : (2) For two complete dglas g and h there is a natural quasi-isomorphism in A + : Proof. Starting from part (1), take B to be the trivial augmented cdga: B = k. Disregarding the differentials, we have the following isomorphisms of complete graded algebras: Direct inspection shows that, under the above isomorphism, the differential inL(A × k) corresponds to the twisted differential d Here we used the fact thatL transforms categorical products in A + into categorical coproducts inL , which follows formally from its adjointness property.
Part (2) was established in the proof of Theorem 1.6 on page 30.

Disconnected spaces and dg Lie algebras
In this section we prove Theorem D. The ground field will always be the field Q of rational numbers. Proof. Note that, by definition, there is an isomorphism of simplicial sets MC(g) ∼ = F C(g) , where F is the Bousfield-Kan functor associating to a cdga B the simplicial set A (B, Q) • recalled in (7.3).
Note also that if B is an augmented cdga then F (B) is naturally a pointed simplicial set. The proposition now follows from the fact that the functors MC and Q are the compositions of the upper resp. lower functors in the diagram in which the functorsL and C are adjoint by Proposition 7.5, while the functors F and Ω form an adjoint pair by [5, §8.9] (the functor Ω of Sullivan-De Rham forms was denoted by A in [5]).

10.2.
Definition. The subcategory in the homotopy category ofL formed by the disjoint products of finitely many non-negatively graded complete dglas whose homology are finitedimensional in each degree will be denoted by fQ-hoL dc .
10.3. Remark. Note that g = (g, ∂) ∈L is a chain complex in the category of linearly compact spaces. Since ∂ is continuous, Ker(∂), and also Im(∂) as a continuous image of a linearly compact space, are closed subspaces of g. So H(g) = Ker(∂)/ Im(∂) is a linearly compact graded space. Clearly H(g) * ∼ = H(g * ), where H(g * ) is the ordinary cohomology of the discrete dual (g * , ∂ * ). 10.4. Remark. The functor g, h → g ⊔ h lifts to the homotopy category ofL . Indeed, there is a quasi-isomorphism C(g) × C(h) ≃ C(g ⊔ h), by Proposition 9.19 (2). If g and g ′ are weakly equivalent complete dglas then, by definition, the cdgas C(g) and C(g ′ ) are quasi-isomorphic, thus the cdgas C(g) × C(h) and C(g ′ ) × C(h) are likewise quasi-isomorphic. Therefore the complete dglas g ⊔ h and g ′ ⊔ h are weakly equivalent.
It follows from Theorem 9.16 resp. [5,Lemma 8.5] that the composite functors MC • and Q induce an adjoint pair of functors between the homotopy categories ofL and S + . The following result implies Theorem D. 10.5. Theorem. The functors MC • and Q determine mutually inverse equivalences between the categories fQ-hoL dc and fNQ-hoS dc + .
Proof. We use the pointed (resp. augmented) versions of the results of Part 1 formulated in Section 6. As the first step, observe that the category fQ-hoL dc is equivalent to the auxiliary 'extended' category extfQ-hoL dc whose objects are dglas weakly equivalent to objects of fQ-hoL dc . We claim that the left adjunction of (10.1) restricts to the adjunction where fQ-hoA dc + is the full subcategory of hoA + consisting of augmented homologically disconnected algebras of finite type. To this end, we must check that whenever g ∈ fQ-hoL dc and A ∈ fQ-hoA dc + . Let us look at the first condition. Assume that g = g 1 ⊔ · · · ⊔ g k , where g i ∈ fQ-hoL dc are non-negatively graded. Since, by Proposition 9.19(2), C(g 1 ⊔ · · · ⊔ g k ) is quasi-isomorphic to C(g 1 )×· · ·×C(g k ), it is enough to show that C(g) ∈ fQ-hoA dc + whenever g ∈ fQ-hoL dc is nonnegatively graded. By Remark 7.6, C(g) is cofibrant non-negatively graded connected cdga whose only augmentation ideal I is the ideal generated by Σg * , so I/I 2 ∼ = Σg * . Therefore C(g) is connected of finite type since H(g * ) is such by assumption. This proves the first condition of (10.3).
To prove the second condition, observe that, by definition, the algebra A is weakly equivalent (i.e. related by a ziz-zag of quasi-isomorphisms) to a finite product of non-negatively graded connected algebras of finite type, A ≃ A 1 × · · · × A k , with the augmentation given by the projection to the last factor. Since, by Proposition 9.19(1), it is enough to verify thatL(A) ∈ extfQ-hoL dc whenever A is connected, cofibrant nonnegatively graded cdga of finite type. Since in this caseL(A) is non-negatively graded, it suffices to prove that the homology ofL(A) is finite-dimensional in each degree. This, however, immediately follows from Lemma 4.3.
As the functorsL and C form a Quillen pair by Theorem 9.16, their restrictions in (10.2) are mutually inverse equivalences of categories. The rest of the proof immediately follows from Theorem C + of Section 6 by which the category fQ-hoA dc + is equivalent to the category fNQ-hoS dc + .
10.6. Remark. The reader may wonder why Theorem 10.5 relates augmented algebras to dg-Lie algebras with no additional structure. The answer is that the dg Lie-analogue of an augmentation is a choice of an MC-element. From this perspective, each dgla is canonically augmented by the trivial MC-element 0. Our definition of the disjoint product g ⊔ h is such that its augmentation is induced by the augmentation of the last factor. 10.7. Remark. Note that Theorem 10.5 allows one to say something new even for connected spaces (or simplicial sets). Indeed, the category of unpointed connected spaces is a subcategory of disconnected pointed spaces. Namely, this subcategory consists of spaces, consisting of two connected components, one of which is the basepoint. We see, therefore, that the homotopy category of rational connected unpointed spaces is equivalent to a certain subcategory of fQ-hoL dc , whose objects are of the form g ⊔ 0 for a non-negatively graded complete dgla g. This subcategory is not full; in fact it is easy to see that there is a natural bijection where * denotes an isolated basepoint corresponding to the zero map g ⊔ 0 → h ⊔ 0.
[May 7, 2013] Appendix A. Cohomology of free products of dg Lie algebras The purpose of this appendix is to express the Chevalley-Eilenberg cohomology of free products of dglas in terms of the Chevalley-Eilenberg cohomology of the factors. The main results are Theorem A.4 for the non-complete case and Theorem A.7 for the compete one. These results seem completely standard but we have not found them in the literature. There is an analogous result for group cohomology, but it relies on the construction of the classifying space of a group; such a construction has no analogue for an arbitrary (i.e. not necessarily nilpotent) dgla, so we needed to develop a certain algebraic machinery instead. First, introduce some notation.
Recall that L is the category of (discrete) dglas. We denote byÂ the category of complete cdgas; its objects are inverse limits of finite dimensional nilpotent non-unital cdgas.
A.1. Definition. LetĈ : L →Â be the functor associating to a discrete dgla g ∈ L the complete cdgaĈ(g) whose underlying non-unital cdga isŜ + Σg * , the completed non-unital symmetric algebra on Σg * . The differential d inĈ(g) is defined as d = d I + d II , where d I is induced by the internal differential in g and d II is determined by its restriction onto Σg * , which is, in turn, induced by the bracket map g ⊗ g → g.
A.2. Remark. The constructionĈ(g) is a double complex with the horizontal differential d II and the vertical differential d I . As such, it has two spectral sequences associated with it; one converging to the direct product totalization, the other to the direct sum totalization. SinceĈ(g) is constructed using direct products, only the first spectral sequence is relevant. We denote this spectral sequence by E ′ (g); we have E ′ 2 (g) = H Ĉ (H(g) . Using this spectral sequence, we easily prove that the functorĈ : L →Â preserves quasi-isomorphisms.
The proof of Theorem A.4 below will use the following lemma.
A.3. Lemma. Let f : u → v be a morphism of dglas such that the induced morphism U(f ) : U(u) → U(v) of their universal enveloping algebras is a quasi-isomorphism. Then f is a quasi-isomorphism too.
Proof. By [23,Theorem 4.5], the functor U from dglas to cocommutative connected dg Hopfalgebras admits a quasi-inverse, denoted by P, which associates to a dg Hopf algebra its dgla of primitive elements. Consider the following commutative diagram of canonical maps: A.4. Theorem. Let g and h be two discrete dglas. Then there is a quasi-isomorphism C(g * h) ∼ =Ĉ(g) ×Ĉ(h) where g * h is the free (non-completed) product of g and h.
[May 7, 2013] Proof. As the first step we prove that the functor g → g * h preserves quasi-isomorphisms. Our proof of this fact will use the isomorphism U(g * h) ∼ = U(g) * U(h) for universal enveloping algebras, where the symbol * in the right hand side stands for the free product of dgas.
Let g ′ be a dgla quasi-isomorphic to g. Since U preserves quasi-isomorphisms by [23,Theorem 2.1], it suffices to prove that the functor of taking the free product on the category of dgas preserves quasi-isomorphisms. Indeed, if it is so, then we have a chain of isomorphisms and quasi-isomorphisms: U(g * h) ∼ = U(g) * U(h) ≃ U(g ′ ) * U(h) ∼ = U(g ′ * h).
Lemma A.3 then implies that g * h is quasi-isomorphic to g ′ * h.
To show the exactness of the free product functor for associative algebras, observe that the product A * B of dgas A and B decomposes as a direct sum of tensor products of A and B: so the statement follows from the exactness of the tensor product of dgas over a field of characteristic zero.
Recall that the functor g →Ĉ(g) preserves quasi-isomorphisms by Remark A.2. Since L is a closed model category, we conclude that it suffices to prove the statement of our theorem in the case when both g and h are standard cofibrant dglas, i.e. when they are obtained from the trivial dgla by a sequence of cell attachments. In that case g * h is likewise a standard cofibrant dgla.
Further, we claim that if a is a cofibrant dgla thenĈ(a) is quasi-isomorphic to the space Der(a, k) of derivations of a with coefficients in the trivial one-dimensional a-module knote that Der(a, k) is isomorphic to (a/[a, a]) * , the dual space of indecomposables of a. This is a standard fact which can be proved e.g. by k-linear dualization of [13, Proposition 9.1.1]. Finally, it is clear that Der(g * h, k) ∼ = Der(g, k) × Der(h, k). The desired result is proved.
Recall that we are really interested in complete dglas whereas the last result concerns noncomplete ones. Somewhat surprisingly, it also holds for complete dglas, as a consequence of Theorem A.4. In the formulation of the complete case we have to take the fiber product of the corresponding CE complexes C(g) and C(h) since the latter are augmented cdgas. Let us prove some preliminary statements. From now on we will write C + (g) for the augmentation ideal of the cdga C(g).
A.5. Lemma. Let g and h be finite dimensional nilpotent graded Lie algebras, viewed as complete dglas. Then there is a quasi-isomorphism C + (g * h) ≃ C + (g) × C + (h).
Proof. The statement of the lemma is similar to the quasi-isomorphism of Theorem A.4, however the essential difference is that g and h are regarded as complete dglas (with vanishing differentials) and so, their free product g * h is likewise completed. However, it turns out not to influence the result. The natural maps g → g * h and h → g * h induce a map (A.2) C + (g * h) → C + (g) × C + (h); we want to show that this map is a quasi-isomorphism.
[ May 7, 2013] We start the proof by reducing the statement to the case when g and h are abelian. First observe that g, being finite dimensional nilpotent, has finite filtration, whose associated graded is abelian (graded) Lie algebra which will be denoted by g ♮ .
Indeed, consider the lower central series g = g 1 ⊃ g 2 ⊃ · · · ⊃ g s+1 = 0 of g. Although the associated graded of this descending filtration need not be abelian, the shifted filtration (A.3) g = F 0 g ⊃ F 1 g ⊃ · · · ⊃ F s g = 0 with F n g := g n+1 , 0 ≤ n ≤ s, does have this property. Let G n g * := (F n+1 g) ⊥ = g ⊥ n+2 be the annihilator of F n+1 g in g * . Then (A.4) 0 ⊂ G 0 g * ⊂ G 1 g * ⊂ · · · ⊂ G s g * = g * is a finite ascending filtration of g * . The subspaces G n C + (g) := k≥1 n=n 1 +···+n k ΣG n 1 g * · · · ΣG n k g * ⊂ C + (g) form an ascending filtration 0 ⊂ G 0 C + (g) ⊂ G 1 C + (g) ⊂ G 2 C + (g) ⊂ · · · ⊂ C + (g) which is exhaustive, Hausdorff and complete. The E 1 -term of the induced spectral sequence clearly equals C + (g ♮ ) and this spectral sequence strongly converges to C + (g). Let us turn our attention to the completed free product g * h. One has a natural epimorphism π : L(g, h) ։ g * h, where L(g, h) is the free graded Lie algebra generated by g ⊕ h, and g * h denotes, only in this proof, the uncompleted free product. Let L ≥m (g, h) be the ideal spanned by products of at least m elements, and (g * h) ≥m := π L ≥m (g, h) . Since π is an epimorphism, (g * h) ≥m is an ideal, and g * h = lim m (g * h)/(g * h) ≥m .
Recall that (g * h) * , by definition, consists of continuous linear functionals. By continuity, every such functional α factors through the canonical epimorphism (g * h) ։ (g * h)/(g * h) ≥m for m >> 0. By the finiteness of (A.3), the induced filtration of (g * h)/(g * h) ≥m is finite, so α annihilates F n (g * h) for n >> 0. This implies that the filtration (A.5) is exhaustive. Now we proceed as in the case of C + (g). The filtration (A.5) induces an exhaustive, Hausdorff and complete filtration of C + (g * h). The induced spectral sequence strongly converges to C + (g * h) and its E 1 -term is C + (g ♮ * h).
[ May 7, 2013] Since the canonical map (A.2) is compatible with the filtrations, by the comparison theorem for spectral sequences, it is enough to prove that (A.2) is a quasi-isomorphism with g replaced by g ♮ . Repeating the same steps with h in place of g, we prove that the desired statement would follow if we can prove that the map C + (g ♮ * h ♮ ) → C + (g ♮ ) × C + (h ♮ ) is a quasi-isomorphism. In other words, we reduced the statement of the lemma to the case when both g and h are abelian.
The dg space C(g * h) = SΣ(g * h) * consists of symmetric tensors in Σ(g * h) * ; we can introduce a grading on C(g * h) as follows. Let x ∈ g * h. Then x has weight n + 1 if x is a sum of Lie monomials of bracket length n; this grading lifts to C(g * h).
It is clear that the differential in C(g * h) preserves the weight grading and so the dg space C(g * h) decomposes as an infinite direct sum of subcomplexes consisting of elements of fixed weight. It follows that the map (A.2) is a quasi-isomorphism if and only if it is a quasi-isomorphism for each weight component. This, in turn, holds if and only if and only if a similar statement holds when the infinite direct sum over all positive weights is replaced by the corresponding infinite direct product. Observe that the obtained completed dg space has the formĈ(g * h) * =ŜΣ(g * h) * where in the last formula g * h stands for the uncompleted free product. It is, therefore, nothing but the standard complex computing the Chevalley-Eilenberg cohomology of the uncompleted Lie algebra g * h. Thus, the statement is reduced to computing the usual Chevalley-Eilenberg cohomology of g * h and, therefore, follows from Theorem A.4.
A.6. Lemma. Let a be a complete dgla and letã be the complete dgla with vanishing differential and the same graded Lie bracket as a. Then there exists a spectral sequence E ′′ (a) converging strongly to C + (a) such that E ′′ 1 (a) = H C + (ã) .
Proof. As graded vector spaces, C + (a) = p≥1 S p (Σ −1 a * ), where S p (−) denotes the subspace of the symmetric algebra consisting of elements of homogeneity p. It is obvious that (A.6)  Clearly F n C + (a) m = 0 if n > m, the filtration (A.6) is thus Hausdorff and complete. The induced spectral sequence therefore converges strongly and obviously has the properties stated in the lemma.
A.7. Theorem. Let g and h be two complete dglas. Then there is a quasi-isomorphism C + (g * h) ≃ C + (g) × C + (h).

[May 7, 2013]
Proof. Assume first that g and h are finite-dimensional nilpotent. Arguing as in the proof of Lemma A.5 we construct a map C + (g * h) −→ C + (g) × C + (h); the desired statement is equivalent to proving that this is a quasi-isomorphism. Consider a map of spectral sequences E ′′ 1 (g * h) = H(C + ( g * h)) → E ′′ 1 ( g) × E ′′ 1 ( h) = H C + (g) × H C + (h) of Lemma A.6. The map in the middle is an isomorphism by Lemma A.5, so the desired result for finite dimensional nilpotent g and h follows.
Since the canonical map (A.2) is compatible with these colimits and the colimits are exact functors, the general case follows from the finite-dimensional nilpotent one.
A.8. Remark. Note that the quasi-isomorphism of Theorem A.7 can also be formulated as C(g * h) ≃ C(g) × k C(h).