Complete filtered L ∞ -algebras and their homotopy theory
aa r X i v : . [ m a t h . A T ] A ug Complete filtered L ∞ -algebras and their homotopy theory Christopher L. Rogers
Abstract
We analyze a model for the homotopy theory of complete filtered Z -graded L ∞ -algebras, which lendsitself well to computations in deformation theory and homotopical algebra. We first give an explicit proofof an unpublished result of E. Getzler which states that the category c Lie [1] ∞ of such L ∞ -algebras andfiltration-preserving ∞ -morphisms admits the structure of a category of fibrant objects (CFO) for a ho-motopy theory. As a novel application, we use this result to show that, under some mild conditions, every L ∞ -quasi-isomorphism between L ∞ -algebras in c Lie [1] ∞ has a filtration preserving homotopy inverse.Finally, building on previous joint work with V. Dolgushev, we prove that the simplicial Maurer–Cartanfunctor, which assigns a Kan simplicial set to each complete filtered L ∞ -algebra, is an exact functorbetween the respective categories of fibrant objects. We interpret this as an optimal homotopy-theoreticgeneralization of the classical Goldman–Millson theorem from deformation theory. One immediate appli-cation is the “ ∞ -categorical” uniqueness theorem for homotopy transferred structures previously sketchedby the author in [27]. Contents d Ch ∗ as a category of fibrant objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.4 Homotopy inverses for weak equivalences in d Ch ∗ . . . . . . . . . . . . . . . . . . . . . . . 19 L ∞ -algebras 21 L ∞ -algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.3 Filtered L ∞ -algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235.4 Maurer–Cartan elements and twisting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Department of Mathematics & Statistics, University of Nevada, Reno. 1664 N. Virginia Street Reno, NV 89557-0084 [email protected], [email protected] c Lie [1] ∞ c Lie [1] ∞ as a category of fibrant objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326.2 Homotopy inverses of weak equivalences in c Lie [1] ∞ . . . . . . . . . . . . . . . . . . . . . . 36 MC : c Lie [1] ∞ → KanCplx MC ( L ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377.2 MC as an exact functor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 References Introduction
A fundamental guiding principal of deformation theory is that every deformation problem over a field ofcharacteristic zero is governed by the homotopy type of a dg Lie algebra. This dictum, attributed to P. Deligneand V. Drinfeld, now appears as a precise theorem in the work of J. Lurie [20], and J. Pridham [22]. Inapplications, complete L ∞ -algebras, i.e. Z -graded L ∞ -algebras (cid:0) L, d, {· · · } k ≥ (cid:1) equipped with a compatiblecomplete descending filtration L = F L ⊇ F L ⊇ · · · provide computationally useful models for suchhomotopy types.Two different categories of complete L ∞ -algebras frequently appear in the literature. The first, which wedenote as c Lie [1] ∞ , is the category whose morphisms are filtration-compatible weak L ∞ -morphisms (a.k.a.“ ∞ -morphisms”). The second, c Lie [1] str ∞ ⊆ c Lie [1] ∞ , is the wide subcategory of the former whose morphismsare filtration-compatible strict L ∞ -morphisms. The weak morphisms in the larger first category are arguablymore useful in applications, e.g. formal deformation quantization, and they are the ones that we focus on inthis paper.The simplicial Maurer-Cartan functor MC : c Lie [1] ∞ → KanCplx is a construction [13, 17] that producesfrom any complete L ∞ -algebra a Kan simplicial set, or ∞ -groupoid. Its properties, which depend cruciallyon the topology induced by the filtration, generalize those of the Deligne groupoid associated to a nilpotentdg Lie algebra in formal deformation theory. In the non-positively graded case, it can be thought of as a“fattened” non-abelian analog of the classical Dold–Kan functor from chain complexes to simplicial vectorspaces. Indeed, in this case, from the point of view of rational homotopy theory, MC is the Lie theoreticanalog of D. Sullivan’s spatial realization functor.We recall in Def. 6.1 that a morphism between complete L ∞ -algebra L → L ′ in c Lie [1] ∞ is a weak equiv-alence if the restriction of its linear term to each piece of the filtration is a quasi–isomorphism of subcochaincomplexes F n L ∼ −→ F n L ′ . In particular, every weak equivalence is an L ∞ quasi-isomorphism. In previousjoint work [9] with V. Dolgushev, we generalized a result of E. Getzler [13] and established a connectionbetween the homotopy theory of complete L ∞ -algebras and Kan complexes: Theorem (Thm. 1.1 [9]) . If Φ : L → L ′ is a weak equivalence in c Lie [1] ∞ , then MC (Φ) : MC ( L ) →MC ( L ′ ) is a homotopy equivalence between simplicial sets. This is a generalization of the classical theorem of Goldman and Millson [15] from deformation theory,which was first brought into the L ∞ context by Getzler for the special case of nilpotent L ∞ -algebras andstrict L ∞ quasi-isomorphisms [13, Thm. 4.8]. The above theorem has turned out to be useful in a variety ofapplications beyond deformation theory, including the rational homotopy theory of mapping spaces [3, 12],and operadic homotopical algebra [8, 25].The main goals of the present work are twofold. First, we give a more complete and explicit descriptionof the homotopy theory in the category c Lie [1] ∞ . We then extend the aforementioned “ L ∞ Goldman-MillsonTheorem” by establishing the compatibility of this additional homotopical structure in c Lie [1] ∞ with the Kan-Quillen model structure for simplicial sets via the simplicial Maurer-Cartan functor. Overview and main results
In Section 6, we verify that the category c Lie [1] ∞ of complete L ∞ -algebras over a field k with char k = 0 ,and filtration-compatible weak L ∞ -morphisms form “one-half of a model category”, i.e. a category of fibrantobjects for a homotopy theory (Def. 3.1) in the sense of K. Brown [4]. What we call L ∞ -algebras in this paper are technically “shifted” L ∞ -algebras. The conceptual distinction is minimal in the Z -graded context, since the only difference is a suspension of the underlying vector space. See Sec. 5.2.
3e attribute the following statement to unpublished work of E. Getzler. Our proof, however, is novel, tothe best of our knowledge: Theorem 1 (Thm. 6.2) . The category c Lie [1] ∞ admits the structure of a category of fibrant objects in which– a morphism Φ : (cid:0)
L, d, {· · · } (cid:1) → (cid:0) L ′ , d ′ , {· · · } ′ (cid:1) is a weak equivalence if and only if its linear term Φ : ( L, d ) → ( L ′ , d ′ ) induces for each n ≥ a quasi–isomorphism of cochain complexes Φ | F n L : ( F n L, d ) ∼ −→ ( F n L ′ , d ′ ) . – a morphism Φ : (cid:0)
L, d, {· · · } (cid:1) → (cid:0) L ′ , d ′ , {· · · } ′ (cid:1) is a fibration if and only if its linear term Φ : ( L, d ) → ( L ′ , d ′ ) induces for each n ≥ an epimorphism of cochain complexes Φ | F n L : ( F n L, d ) ։ ( F n L ′ , d ′ ) . The proof of the theorem is given in Sec. 6.1. It builds on the following fact which we establish beforehandin Sec. 4.3: the CFO structure on the category Ch ∗ of Z -graded cochain complexes over k induced by thestandard projective model structure [18, 30] lifts to the category of complete filtered Z -graded complexes d Ch ∗ . Our proof also provides explicit descriptions of pullbacks, and a “strictification” result for fibrations.The tractability of this approach relies on technical lemmas from B. Vallette’s work [31] on the homotopytheory of homotopy algebras. Furthermore, we observe that the category c Lie [1] ∞ has a functorial path object,given by the completed tensor product − b ⊗ Ω with the cdga Ω of polynomial de Rham forms on ∆ .As an application, we use Theorem 1 to prove Theorem 2 (Thm. 6.9) . Every weak equivalence
Φ : (cid:0)
L, d, {· · · } (cid:1) ∼ −→ (cid:0) L ′ , d ′ , {· · · } ′ (cid:1) in c Lie [1] ∞ has ahomotopy inverse in c Lie [1] ∞ which can be exhibited by any choice of path objects for L and L ′ .In particular, there exists a weak equivalence Ψ : (cid:0) L ′ , d ′ , {· · · } ′ (cid:1) ∼ −→ (cid:0) L, d, {· · · } (cid:1) and homotopies H L : (cid:0) L, d, {· · · } (cid:1) → (cid:0) L b ⊗ Ω , d Ω , {· · · } Ω (cid:1) , H L ′ : (cid:0) L ′ , d ′ , {· · · } ′ (cid:1) → (cid:0) L ′ b ⊗ Ω , d ′ Ω , {· · · } ′ Ω (cid:1) , which induce equivalences ΨΦ ≃ id L and ΦΨ ≃ id L ′ . The proof of Theorem 2 follows from:(i) abstract properties of CFOs satisfying two additional axioms which recall in Section 3.2, and(ii) our observation that every object in c Lie [1] ∞ is “cofibrant”. That is, every acyclic fibration in c Lie [1] ∞ is a retract (Cor. 6.8).Finally, in section 7.2 we give an optimal homotopy-theoretic generalization of the classical Goldman–Millson theorem, which extends our previous work [9] with V. Dolgushev. Theorem 3 (Thm. 7.4) . The simplicial Maurer-Cartan functor MC : c Lie [1] ∞ → KanCplx is an exact functorbetween categories of fibrant objects (Def. 3.7). In particular:1. MC sends weak equivalences in c Lie [1] ∞ to weak homotopy equivalences between Kan simplicial sets.2. MC sends (acyclic) fibrations in c Lie [1] ∞ to (acyclic) Kan fibrations.3. MC sends pullbacks of (acyclic) fibrations in c Lie [1] ∞ to pullbacks in KanCplx . We note that our proof of Theorem 3 completes the sketch we first provided in the announcement [27]. We learned of this statement during a conversation with E. Getzler at the “Algebraic Analysis & Geometry Workshop” hostedby the University of Padua in the fall of 2013. emark on applications We suggest that these results, while not surprising, give further support for thepoint of view that complete L ∞ -algebras can provide “chain models” for studying obstruction, transfer, andenrichment phenomena in homotopical algebra, in analogy with the Dold-Kan correspondence in the classicalabelian setting. Along those lines, we make some rather simple observations concerning obstruction theoryin Sec. 5.5 which formalize ideas that are surely well known to experts.A more novel application which follows from Theorem 1 is the “ ∞ -categorical” uniqueness theoremfor homotopy transferred structures sketched in [27, Corollary 1]. The statement is roughly the following:Suppose we are given a cochain complex A , a homotopy algebra B of some particular type (e.g, an A ∞ , L ∞ , or C ∞ -algebra) and a quasi-isomorphism of complexes φ : A → B . Then, using the simplicial Maurer–Cartan functor, we can naturally produce an ∞ -groupoid F whose objects correspond to solutions to the“homotopy transfer problem”. By a solution, we mean a pair consisting of a homotopy algebra structure on A , and a lift of φ to a ∞ -quasi-isomorphism of homotopy algebras A ∼ −→ B . The fact that MC preserves bothweak equivalences and fibrations allows us to conclude that: (1) The ∞ -groupoid F is non–empty, and (2) itis contractible. In other words, a homotopy equivalent transferred structure always exists, and this structureis unique in the strongest possible sense. Related work
There is a substantial amount of literature on the homotopy theory of L ∞ -algebras and theproperties of the simplicial Maurer-Cartan functor. What follows is our attempt to carefully give attributionto these previous works, and clearly identify their relationship to the results presented in this paper. • The results of A. Lazarev [19] and B. Vallette [31] imply that the full subcategory of fibrant objectsin V. Hinich’s model category [16] of Z -graded conilpotent dg cocommutative coalgebras over k isequivalent to the category Lie [1] ∞ of (shifted) Z -graded L ∞ -algebras and ∞ -morphisms. This gives Lie [1] ∞ the structure of a CFO (with functorial path objects) in which the weak equivalences are L ∞ quasi-isomorphisms and the fibrations are L ∞ -epimorphisms (see Def. 5.2). When combined with ourTheorem 1, these results imply that the forgetful functor c Lie [1] ∞ → Lie [1] ∞ is an exact functor betweencategories of fibrant objects. • In [17], V. Hinich established statements (1) and (2) of Theorem 3 for the special case of strict morphismsbetween nilpotent dg Lie algebras. • As previously mentioned, in [13], E. Getzler proved statements (1) and (2) of Theorem 3 for the specialcase of strict morphisms between nilpotent L ∞ -algebras. • In [32], S. Yalin established statement (1) of Theorem 3 for the functor MC : c Lie [1] str ∞ → KanCplx ,i.e., for the special case of strict filtration-compatible morphisms between complete L ∞ -algebra. Yalinalso proved in [32, Thm. 4.2] statement (2) of Theorem 3 for the functor MC : c Lie [1] str ∞ → KanCplx ,assuming certain finiteness conditions, as well as strictness of morphisms. • In [2], R. Bandiera proved an analog of Theorem 3 for Getzler’s Deligne ∞ -groupoid functor γ • : c Lie [1] str ∞ → KanCplx , which can be thought of as a “smaller” homotopy equivalent model for the simplicial Maurer-Cartan construction, when restricted to the category of complete L ∞ -algebras and strict filtration-compatiblemorphisms. Similar results were also established by D. Robert-Nicoud in [25].We note that the functor γ • is not well defined on ∞ -morphisms between L ∞ -algebra, and therefore doesnot extend to the category c Lie [1] ∞ . • A detailed analysis of the homotopy type of the simplicial Maurer-Cartan set MC ( L ) and Deligne ∞ -groupoid γ • ( L ) for a complete L ∞ -algebra L ∈ c Lie [1] str ∞ was given by A. Berglund in [3]. Statement(1) of Theorem 3 for the functor MC : c Lie [1] str ∞ → KanCplx , as well as the analogous statement for the5unctor γ • : c Lie [1] str ∞ → KanCplx was proved in [3, Prop. 5.4.]. Statement (2) of Theorem 3 was alsoasserted [3, Prop. 5.4.] for both functors MC and γ • with domain category c Lie [1] str ∞ , but the proof giventhere is unfortunately incomplete.Finally, we note that the CFO structure that we describe in Sec. 4.3 for the category d Ch ∗ of completefiltered Z -graded cochain complexes over k is equivalent to the one obtained by other methods by C. DiNatale in [6] for complexes over more general rings. Acknowledgments
We thank Ezra Getzler for invaluable conversations on the homotopy theory of L ∞ -algebras and relatedtopics. The axioms (EA1) and (EA2) presented in Section 3.2 stemmed from discussions with Aydin Ozbek.We thank him for sharing his insights on homotopy equivalence in categories of fibrant objects.This paper is based upon work supported by the National Science Foundation under Grant No. DMS-1440140, while the author was in residence at the Mathematical Sciences Research Institute in Berkeley,California, during the Spring 2020 semester. Additional support provided by a grant from the Simons Foun-dation/SFARI (585631,CR). Throughout, we work over a field k of characteristic zero. All graded objects are assumed to be Z -gradedand, in general, unbounded. We use cohomological conventions for all differential graded (dg) objects. Wefollow the conventions and notation of [9, Sec. 1] and [10, Sec. 1.1.] for graded linear algebra, Koszul signs,etc. In particular, for a Z -graded vector space V , we denote by s V (resp. by s − V ) the suspension (resp. thedesuspension) of V . In other words, s V i := V [ − i := V i − s − V i := V [1] i := V i +1 . Throughout,
Vect denotes the category of Z -graded vector spaces over k , and Ch ∗ denotes the categoryof unbounded cochain complexes of k -vector spaces. We denote by Ch ∗ proj the category Ch ∗ equipped withthe projective model structure [18, 30]. The weak equivalences in Ch ∗ proj are the quasi-isomorphisms, andthe fibrations are those maps which are surjective in all degrees. Note that every object in Ch ∗ proj is fibrant.Finally, s Set denotes the category of simplicial sets, which we tacitly assume is equipped with the Kan-Quillen model structure, and
KanCplx ⊆ s Set denotes the full subcategory of Kan complexes as a categoryof fibrant objects [14, Sec. I.9].
Definition 3.1 (Sec. 1 [4]) . Let C be a category with finite products, with terminal object ∗ ∈ C , and equippedwith two classes of morphisms called weak equivalences and fibrations . A morphism which is both a weakequivalence and a fibration is called an acyclic fibration . Then C is a category of fibrant objects (CFO) fora homotopy theory iff:1. Every isomorphism in C is an acyclic fibration.2. The class of weak equivalences satisfies “2 out of 3”. That is, if f and g are composable morphisms in C and any two of f, g, g ◦ f are weak equivalences, then so is the third.3. The composition of two fibrations is a fibration.6. The pullback of a fibration exists, and is a fibration. That is, if Y g −→ Z f ←− X is a diagram in C with f a fibration, then the pullback X × Z Y exists, and the induced projection X × Z Y → Y is a fibration.5. The pullback of an acyclic fibration exists, and is an acyclic fibration.6. For any object X ∈ C there exists a (not necessarily functorial) path object , that is, an object X I equipped with morphisms X s −→ X I ( d ,d ) −−−−→ X × X, such that s is a weak equivalence, ( d , d ) is a fibration, and their composite is the diagonal map.7. All objects of C are fibrant . That is, for any X ∈ C the unique map X → ∗ is a fibration. The axioms of a CFO imply Brown’s Factorization Lemma [4, Sec. 1]: Every morphism f : X → Y in C canbe factored as f = p f ◦ f , where p f is a fibration and f is a right inverse of an acyclic fibration (and hencea weak equivalence). Let us briefly recall the construction. Let Y s −→ Y I ( d ,d ) −−−−→ Y × Y be a path object for Y , and let X × Y Y I be the pullback in the diagram X × Y Y I Y I X Y pr f pr ∼ d ∼ (3.1)Then the commutative diagram X id X (cid:15) (cid:15) sf / / Y Id (cid:15) (cid:15) X f / / Y provides a unique weak equivalence f : X ∼ −→ X × Y Y I such that pr f = id X . Finally, define p f : X × Y Y I → Y to be the composition p f := pr ◦ d . See [4], or the proof of [28, Lemma 2.3] for the verificationthat p f is indeed a fibration. Remark . The above construction combined with axiom (2) in Def. 3.1 implies that if C is a CFO, thenany two objects X, Y ∈ C connected by a weak equivalence f : X ∼ −→ Y are connected by a span of acyclicfibrations X × Y Y I X Y pr ∼ p f ∼ We recall from [4, Sec. 2] that two morphisms f, g : X → Y in a CFO are (right) homotopic if and onlyif there exists a path object Y s −→ Y I ( d ,d ) −−−−→ Y × Y and a morphism h : X → Y I such that f = d h and g = d h . In this case we write “ f ≃ g ”. In analogy with the situation in a model category one shows thathomotopy in a CFO is an equivalence relation by taking iterated pullbacks of path objects. See for examplethe dual of [23, Lemma 4].The next proposition, which we recall from [4], describes how homotopy behaves under pre and post-composition. It will be quite useful to us in the next section.7 roposition 3.3 (Prop. 1 [4]) . Let C be a category of fibrant objects. Assume α ≃ β : A → B are homotopicmorphisms in C . Then1. If µ : Z → A is an arbitrary morphism, then αµ ≃ βµ : Z → B .2. If ν : B → C is an arbitrary morphism, and C I is any path object for C , then there exists an acyclicfibration φ : Z ∼ ։ A such that ναφ ≃ νβφ : Z → C via a homotopy ˜ h : Z → C I . We now consider a CFO C which satisfies two additional axioms:(EA1) Every acyclic fibration in C is a retraction , i.e. if φ : X ∼ ։ Y is an acyclic fibration, then there existsa morphism σ : Y → X such that φσ = id Y .(EA2) C is equipped with a functorial path object , i.e., an assignment of a path object X X s X −−→ Path( X ) ( d X ,d X ) −−−−−→ Path( X ) × Path( X ) to each object X ∈ C , and to each f : X → Y , a morphism f I : Path( X ) → Path( Y ) such that thefollowing diagram commutes: X Path( X ) X × XY Path( Y ) Y × Y s X ∼ s Y ∼ f f I ( f, f )( d X , d X )( d Y , d Y ) A weaker version of axiom (A2) above would also be sufficient for our purposes here, but it seems less natural:(EA ′ ) Each object in C is equipped with a distinguished path object , i.e., a choice of a path object X X s X −−→ Path( X ) ( d X ,d X ) −−−−−→ Path( X ) × Path( X ) for each object X ∈ C . Remark .
1. Axiom (EA1) can be understood as saying that all objects in C are “cofibrant”, as well as fibrant. Forthis reason, perhaps a CFO satisfying these additional axioms should be called a “category of bifibrantobjects”.2. Noteworthy examples of CFOs satisfying the above additional axioms include: the category of Z -gradedcochain complexes over a field Ch ∗ proj equipped with the projective model structure; the subcategory offibrant objects in V. Hinich’s model category on Z -graded conilpotent dg cocommutative coalgebras [16](as well as B. Vallette’s generalization to the model category of conilpotent dg P ! -coalgebras [31]); thecategory of Kan simplicial sets equipped with the CFO structure induced by D. Quillen’s model structureon s Set [24]. 8. More generally, the subcategory of fibrant objects of any model category equipped with functorial factor-izations in which all objects are cofibrant is an example of a CFO satisfying Axioms (EA1) and (EA2).The first observation is that the additional axioms imply that every homotopy equivalence can be realizedvia the distinguished path object . Proposition 3.5.
Let C be a CFO satisfying axioms (EA1) and (EA2) . Morphisms α, β : A → B are homo-topic in C if and only if there exists a homotopy h : A → Path( B ) such that α = d B h and β = d B h .Proof. Suppose α ≃ β : A → B . In statement (2) of Prop. 3.3 take C = B , ν = id B and C I = Path( B ) .Then there exists an acyclic fibration φ : Z ∼ ։ A and a homotopy ˜ h : Z → Path( B ) which gives a homotopyequivalence αφ ≃ βφ . Axiom (EA1) implies that there exists a morphism σ : A → Z such that φσ = id Z .Define h : A → Path( B ) to be the composition h := ˜ hσ : A → Path( B ) . Then, as asserted in statement (1)of Prop. 3.3, h gives the desired homotopy equivalence α ≃ β via the path object Path( B ) .Next, we show that axioms (EA1) and (EA2) imply that every weak equivalence has a homotopy inverse. Proposition 3.6.
Let C be a CFO satisfying axioms (EA1) and (EA2) . Suppose f : X ∼ −→ Y is a weakequivalence in C . Then there exist a weak equivalence g : Y ∼ −→ X and homotopies h X : X → Path( X ) , h Y : Y → Path( Y ) which induce equivalences gf ≃ id X , f g ≃ id Y , respectively.Proof. Let f : X ∼ −→ Y be a weak equivalence. Note that Prop. 3.5 implies that it is sufficient to exhibit atwo-sided homotopy inverse to f using any path objects. Choose a factorization f = p f f as constructed inSec. 3.1. As discussed in Remark 3.2, we have a span of acyclic fibrations of the form X X × Y Y I Y pr ∼ d pr ∼ where X × Y Y I , pr and pr are the pullback and projections from diagram (3.1). Axiom (EA1) impliesthat there exists a section σ : Y → X × Y Y I such that d pr σ = id Y . Define g : Y ∼ −→ X and h : Y → Y I to be the morphisms g := pr σ, h := pr σ. (3.2)Then the commutativity of diagram (3.1) implies that d h = f pr σ = f g, and by construction, we have d h = d pr σ = id Y . f g ≃ id Y (3.3)Now, we repeat the above steps mutatis mutandis for the weak equivalence g : Y ∼ −→ X . This producesanother weak equivalence ˜ f : X ∼ −→ Y such that g ˜ f ≃ id X . We thank A. Ozbek for showing us this result.
9e proceed using the standard tricks for this type of situation. Choose a path object Y I for Y , and in statement(2) of Prop. 3.3, take ν = f : X ∼ −→ Y , α = g ˜ f , and β = id X . Then there exists an acyclic fibration φ : Z ∼ ։ X and a homotopy Z → B I which gives an equivalence f g ˜ f φ ≃ f φ . Axiom (EA1) implies that φ has a right inverse, hence from Prop. 3.3(1) we deduce that f g ˜ f ≃ f . On the other hand, Prop. 3.3(1)combined with the equivalence (3.3) implies that f g ˜ f ≃ ˜ f . Hence, f ≃ ˜ f , since right homotopy equivalenceis an equivalence relation.Finally, we apply Prop. 3.3(2) again. This time we set ν = g : Y ∼ −→ X , α = f and β = ˜ f . We obtain gf ψ ≃ g ˜ f ψ , for some acyclic fibration ψ . We compose this with a right inverse of ψ to obtain gf ≃ g ˜ f , andtherefore we conclude that gf ≃ id X . The standard way to compare categories of fibrant objects is via left exact functors, which are the analog ofright Quillen functors between model categories. We recall the definition:
Definition 3.7 (Def. 2.3.3 [1]) . A functor F : C → D between categories of fibrant objects is a (left) exactfunctor if and only if1. F preserves the terminal object, fibrations, and acyclic fibrations.2. Any pullback square in C of the form P XZ Y f in which f : X → Y is a fibration in C , is mapped by F to a pullback square in D .Note that the above axioms imply that an exact functor preserves finite products and weak equivalences.The latter statement follows from the factorization discussed in Sec. 3.1 which implies that a weak equivalenceis the composition of an acyclic fibration with a right inverse of another acyclic fibration. For a category C , we denote by tow ( C ) the category of towers in C , i.e. diagrams in C of the form · · · → X n q n −→ X n − q n − −−−→ · · · q −→ X q −→ X p −→ X . A morphism between towers X and Y is a sequence of morphisms { f i : X i → Y i } i ≥ in C such that theobvious diagram commutes.Also, for a category C with terminal object ∗ ∈ C , we denote by tow ( C ) the category of “1-reduced”towers, i.e., the full subcategory of tow ( C ) whose objects are satisfy the property X = X = ∗ .We will use the next proposition a few times in this paper, for the particular cases when M = Ch ∗ proj and M = s Set . Proposition 3.8.
Let M be a model category and C ⊆ M the full subcategory of fibrant objects of M . Let f : X → Y ∈ tow ( C ) a morphism between towers in C . . Suppose for all i ≥ , the maps p X ( i +1) : X i +1 → X i and p Y ( i +1) : Y i +1 → Y i are fibrations in C . If each f i : X i ∼ −→ Y i is a weak equivalence in C , then the morphism lim ←− f : lim ←− X → lim ←− Y is also a weak equivalence in C .2. If f : X ։ Y is a (acyclic) fibration in C , and the unique morphism X n +1 → X n × Y n Y n +1 in thefollowing commutative diagram X n +1 X n × Y n Y n +1 Y n +1 X n Y np Y ( n +1) f n f n +1 p X ( n +1) (3.4) is a (acyclic) fibration in C for all i ≥ , then the morphism lim ←− f : lim ←− X → lim ←− Y is a (acyclic) fibration in C .Proof. Apply Prop. 2.5 and Prop. 2.6 of [5] to the model category M op We begin by recalling some basic facts concerning filtered and complete filtered vector spaces. We refer thereader to [7, Sec. 2] and [11, Sec. 7.3.1 – 7.3.4] for additional background material.
We denote by
FiltVect the additive category whose objects are Z -graded k -vector spaces V equipped with adecreasing filtration of subspaces beginning in filtration degree 1 V = F V ⊇ F V ⊇ F V ⊇ · · · Morphisms in
FiltVect are degree 0 linear maps f : V → V ′ that are compatible with the filtrations: f (cid:0) F n V (cid:1) ⊆ F n V ′ ∀ n ≥ . The following notation will be used constantly throughout the paper. Let V ∈ FiltVect . For each n ≥ , wehave the canonical surjections and quotient maps p ( n ) : V → V / F n V p ( n ) : V / F n +1 V → V / F n V such that the diagram VV / F n +1 V V / F n V p ( n +1) p ( n ) p ( n ) ˆ p ( n ) : lim ←− k V / F k V → V / F n V ∀ n ≥ the canonical map out of the projective limit, and we let p V : V → lim ←− k V / F k V denote the unique map satisfying ˆ p ( n ) ◦ p V = p ( n ) for all n ≥ .Given a morphism f : V → V ′ ∈ FiltVect , we denote its restriction to each piece of the filtration as F n f := f | F n V : F n V → F n V ′ . For each n ≥ , the compatibility of f with the filtrations gives two commuting diagrams of short exactsequences in Vect which appear repeatedly throughout the paper: F n V V V / F n V F n V ′ V ′ V ′ / F n V ′ F n f p ( n ) f f ( n ) p ′ ( n ) (D1)and F n V / F n +1 V V / F n +1 V V / F n V F n V ′ / F n +1 V ′ V ′ / F n +1 V ′ V ′ / F n V ′ p ( n ) p ′ ( n ) F n f ( n +1) f ( n +1) f ( n ) (D2)In the above diagrams, the morphisms f ( n +1) , f ( n ) , and F n f ( n +1) are the obvious ones induced by f and theuniversal property for quotients. Remark . The commutativity of diagram (D1) implies that a morphism between filtered vector spaces f : V → V ′ is an isomorphism in FiltVect if and only if f is an isomorphism in Vect and f ( n ) is an isomor-phism in Vect for each n ≥ . We next record notation and conventions for completions of filtered vector spaces. Given a filtered vectorspace V ∈ FiltVect , the projective limit b V := lim ←− k V / F k V is also an object in FiltVect when equipped withfiltration given by the subspaces F n b V := ker (cid:0) ˆ p ( n ) : b V → V / F n V (cid:1) . The space b V is complete with respect to the topology induced by this filtration, and the universal map p V : V → b V extends to a filtered morphism in FiltVect . We denote by d Vect ⊆ FiltVect the full subcategory of complete filtered vector spaces whose objects are those filtered vector spaces V ∈ FiltVect such that p V : V ∼ = −→ b V is an isomorphism in FiltVect .12hroughout we represent elements of b V as coherent sequences in V or, equivalently, as convergent seriesin V , i.e. ∞ X i =0 x i ∈ b V ⇔ ∀ n ≥ ∃ m n ∈ N such that x i ∈ F n V for all i ≥ m n . See [7, Lemma 2.3] for further details.We denote by tow : FiltVect → tow ( Vect ) the functor which assigns to a filtered vector space V the 1-reduced tower tow( V ) := · · · → V / F n +1 V p ( n ) −−→ V / F n V p ( n − −−−−→ · · · p (2) −−→ V / F V → → (4.1)We express the completion of a filtered vector space V ∈ FiltVect as a composition of functors lim ←− tow : FiltVect → d Vect V f −→ V ′ b V b f −→ c V ′ (4.2)where b V := lim ←− n V / F k V and b f := lim ←− n f ( n ) . Note that if f : V → V ′ is a morphism of filtered vectorspaces, then the diagram in FiltVect
V V ′ b V c V ′ fp V p V ′ b f commutes. This gives us the following basic results, which we record for later reference. Lemma 4.2.
For each
V, V ′ ∈ FiltVect , there is a natural isomorphism of abelian groups hom tow ( Vect ) (cid:0) tow( V ) , tow( V ′ ) (cid:1) ∼ = hom FiltVect ( V, c V ′ ) . (4.3) In particular, for each
V, V ′ ∈ d Vect hom tow ( Vect ) (cid:0) tow( V ) , tow( V ′ ) (cid:1) ∼ = hom d Vect ( V, V ′ ) . (4.4)The next lemma will be useful in analyzing fibrations in Sec. 4.3 and 6.1. Lemma 4.3. If f : V → V ′ is a morphism in d Vect such that the restriction F n f : F n V → F n V ′ is surjectivefor all n ≥ , then there exists a morphism σ : V ′ → V in d Vect such that f σ = id V ′ .Proof. In the notation of diagram (D2), we will inductively construct linear maps σ n : V ′ / F n V ′ → V / F n V for each n ≥ such that f ( n ) σ n = id V ′ / F n V ′ , and such that for all n ≥ σ n − p ′ ( n − = p ( n − σ n (4.5)This will give a morphism between towers tow ( V ) → tow ( V ′ ) , and then Lemma 4.2 will provide the desiredright inverse σ = lim ←− σ n .Set σ = 0 , and let σ : V ′ / F V ′ → V / F V be a section of the surjective linear map f (2) . Since V / F V = V ′ / F V ′ = 0 , Eq. 4.5 is satisfied trivially. This establishes the base case.13et n ≥ and suppose σ n : V ′ / F n V ′ → V / F n V is a section of f ( n ) satisfying Eq. 4.5. Consider thecommutative diagram (D2). Choose a linear map s : V / F n V → V / F n +1 V such that p ( n ) s = id V/ F n V , andlet ˜ σ n +1 : V ′ / F n +1 V ′ → V / F n +1 V be ˜ σ n +1 := s ◦ σ n ◦ p ′ ( n ) . Then the image of the map θ : V ′ / F n +1 V ′ → V ′ / F n +1 V ′ defined as θ := f n +1 ˜ σ n +1 − id V ′ / F n +1 V ′ lies in ker p ′ ( n ) = F n V ′ / F n +1 V ′ . Finally, let τ : F n V ′ / F n +1 V ′ → F n V / F n +1 V be a section of the surjec-tive map F n f ( n +1) , and define σ n +1 : V ′ / F n +1 V ′ → V / F n +1 V, σ n +1 := ˜ σ n +1 − τ θ. A direct calculation then shows that f ( n +1) σ n +1 = id V ′ / F n +1 V ′ , and σ n p ′ ( n ) = p ( n ) σ n +1 . d Vect
We recall a special case of the tensor product b ⊗ in d Vect . Let V ∈ FiltVect be a filtered vector space and A ∈ Vect a non-filtered graded vector space. The filtration on V induces a canonical filtration on the tensorproduct V ⊗ A : given by the subspaces F n ( V ⊗ A ) := F n V ⊗ A . Since in our case, − ⊗ k A : Vect → Vect is an exact functor, there are natural identifications for each n ≥ V ⊗ A/ F n ( V ⊗ A ) ∼ = ( V / F n V ) ⊗ A. Therefore, the maps p ( k ) : V / F k +1 V → V / F k V induce the inverse system of surjections p ( k ) ⊗ id A : V / F k +1 V ⊗ A → V / F k V ⊗ A. We define the completed tensor product of V and A as V b ⊗ A := lim ←− tow( V ⊗ A ) = lim ←− n (cid:0) V / F n V ⊗ A (cid:1) . If f : A → A ′ is a linear map in Vect , then the morphism id V ⊗ f : V ⊗ A → V ⊗ A ′ is compatible with therespective filtrations. Hence, for a fixed V ∈ d Vect we obtain a functor V b ⊗− : Vect → d Vect f : A → A ′ id V b ⊗ f : V b ⊗ A → V b ⊗ A ′ , (4.6)where id V b ⊗ f := lim ←− id V/ F n V ⊗ f . We denote by d Ch ∗ the category of complete filtered cochain complexes. An object ( V, d ) of d Ch ∗ consists ofa complete vector space V ∈ d Vect equipped with a degree differential d : V → V such that each piece ofthe filtration on V is a sub-cochain complex ( F n V, d ) . Morphisms in d Ch ∗ are linear maps in d Vect which arecompatible with the differential. 14 otation 4.4.
Given ( V, d ) ∈ d Ch ∗ , for each n ≥ we have the following short exact sequences in Ch ∗ → ( F n V, d ) → ( V, d ) → ( V / F n V, d ( n ) ) → , → (cid:0) F n V / F n +1 V, d ( n +1) (cid:1) → (cid:0) V / F n +1 V, d ( n +1) (cid:1) → (cid:0) V / F n V, d ( n ) (cid:1) → Above the differentials d ( n ) and d ( n +1) are the usual ones on the quotient complexes. Given a morphism f : ( V, d ) → ( V ′ , d ′ ) in d Ch ∗ , the diagrams (D1) and (D2) lift to the category Ch ∗ in the obvious way.Note that the category d Ch ∗ has finite products. In particular, if ( V, d V ) , ( W, d W ) ∈ d Ch ∗ , then the usualdirect sum of complexes ( V ⊕ W, d ⊕ ) is complete with respect to the filtration given by the subcomplexes F n ( V ⊕ W ) := F n V ⊕ F n W .Let us return briefly to the tensor product from Sec. 4.1.3. Given a complete filtered cochain complex ( V, d ) ∈ d Ch ∗ and a unfiltered complex ( A, δ ) ∈ Ch ∗ , let ( V ⊗ A, d A ) denote the usual tensor product ofcomplexes. The functor (4.6) extends to cochain complexes V b ⊗− : Ch ∗ → d Ch ∗ and sends f : ( A, δ ) → ( A ′ , δ ′ ) in Ch ∗ to the filtered chain map id V b ⊗ f : ( V b ⊗ A, d A ) → ( V b ⊗ A ′ , d A ′ ) . d Ch ∗ as a category of fibrant objects In this section, we first give a fairly explicit proof that d Ch ∗ forms a category of fibrant objects. Then we showthat this CFO structure satisfies the additional axioms (EA1) and (EA2) from Sec. 3.2. We will use theseresults later in Sec. 6 to construct the CFO structure on complete L ∞ -algebras. Definition 4.5.
Let f : ( V, d ) → ( V ′ , d ′ ) be a morphism in d Ch ∗ .1. We say f is a weak equivalence if and only if for all n ≥ the restrictions F n f : ( F n V, d ) ∼ −→ ( F n V ′ , d ′ ) are quasi-isomorphisms of cochain complexes.2. We say f is a fibration if and only if for all n ≥ the restrictions F n f : ( F n V, d ) → ( F n V ′ , d ′ ) are degree-wise surjective maps of cochain complexes. Remark .
1. If f : ( V, d ) ∼ −→ ( V ′ , d ′ ) is a weak equivalence in d Ch ∗ , then the long exact sequence in cohomologyapplied to diagram (D1) implies that f ( n ) : ( V / F n V, d ( n ) ) ∼ −→ ( V ′ / F n V ′ , d ′ ( n ) ) is a quasi-isomorphism for each n ≥ . By applying the same argument to diagram (D2), we thenconclude that F n f ( n +1) : ( F n V / F n +1 V, d ( n ) ) ∼ −→ ( F n V ′ / F n +1 V ′ ) is a quasi-isomorphism for each n ≥ , as well.15. If f : ( V, d ) ։ ( V ′ , d ′ ) is a fibration, then the surjectivity of the restriction F n f implies that theinclusion ker f / F n ker f ֒ → ker f ( n ) is an isomorphism for each n ≥ . Theorem 4.7.
The category d Ch ∗ of complete filtered cochain complexes equipped with the weak equivalencesand fibrations introduced in Def. 4.5 is a category of fibrant objects with a functorial path object.Proof. It is easy to see that axioms 1,2,3, and 7 of Def. 3.1 are satisfied. We verify the remaining axioms viathe propositions given below. d Ch ∗ (axiom 6) Let us consider the commutative dg algebra ( k [ z, dz ] , d ) ∈ cdga (4.7)whose underlying graded algebra is freely generated by degree 0 and degree 1 indeterminates z and dz ,respectively. The differential is the unique derivation sending z dz and dz . As a graded vectorspace, k [ z, dz ] is concentrated in degrees 0 and 1: k [ z, dz ] = k [ z ] ⊕ k [ z ] dz , where k [ z ] denotes the usualpolynomial algebra. As a cdga, we have ( k [ z, dz ] , d ) ∼ = (Ω , d dR ) where, as usual, Ω is the polynomial de Rham forms on the geometric 1-simplex.Since char k = 0 , the unit η : k ∼ −→ k [ z, dz ] (4.8)is a quasi-isomorphism of cdgas, and hence a weak equivalence in Ch ∗ proj . We also have the algebra mapscorresponding to evaluation: ev , ev : k [ z, dz ] → k , ev (cid:0) f ( z ) + g ( z ) dz (cid:1) := f (0) , ev (cid:0) f ( z ) + g ( z ) dz (cid:1) := f (1) . The induced algebra morphism into the product ρ := (ev , ev ) : k [ z, dz ] ։ k × k (4.9)is surjective and hence a fibration in Ch ∗ proj . Proposition 4.8.
The assignment ( V, d ) (cid:0) V b ⊗ Ω , d Ω (cid:1) to each ( V, d ) ∈ d Ch ∗ is a functorial path object forthe category d Ch ∗ . The proof of the proposition will follow almost immediately from the next lemma.
Lemma 4.9.
Let ( V, d V ) ∈ d Ch ∗ be a complete cochain complex.1. If f : ( A, δ ) ∼ −→ ( A ′ , δ ′ ) is a quasi-isomorphism in Ch ∗ , then id V b ⊗ f : V b ⊗ A → V b ⊗ A ′ is a weakequivalence in d Ch ∗ .2. If f : A ։ A ′ is an epimorphism in Ch ∗ , then id V b ⊗ f : V b ⊗ A → V b ⊗ A ′ is a fibration in d Ch ∗ .Proof.
16. We need to verify that id V b ⊗ f : V b ⊗ A → V b ⊗ A ′ is a quasi-isomorphism and that F n (id V b ⊗ f ) : F n ( V b ⊗ A ) → F n ( V b ⊗ A ′ ) is a quasi-isomorphism for each n ≥ . The functor V / F n V ⊗ k − : Ch ∗ → Ch ∗ is exact, and thereforepreserves quasi-isomorphisms. This gives a commutative ladder of cochain complexes: · · · V / F n +1 V ⊗ A V / F n V ⊗ A · · ·· · · V / F n +1 V ⊗ A ′ V / F n V ⊗ A ′ · · · id V/ F n +1 V ⊗ f ∼ id V/ F n V ⊗ f ∼ in which every horizontal map is an epimorphism, and every vertical map is a weak equivalence. Byapplying Prop. 3.8 to the model category Ch ∗ proj , we then deduce that id V b ⊗ f = lim ←− (id V/ F n V ⊗ f ) isa quasi-isomorphism. As a consequence, for each n ≥ , we obtain a morphism between short exactsequences F n ( V b ⊗ A ) V b ⊗ A V / F n V ⊗ A F n ( V b ⊗ A ′ ) V b ⊗ A ′ V / F n V ⊗ A ′ ˆ q ( n ) ˆ q ′ ( n ) F n (id V b ⊗ f ) id V b ⊗ f ∼ id V/ F n V ⊗ f ∼ in which two of the three vertical maps are quasi-isomorphisms. Hence, the third map is a quasi-isomorphismas well.2. Suppose f : ( A, δ ) → ( A ′ , δ ′ ) is an epimorphism. We need to verify that id V b ⊗ f : V b ⊗ A → V b ⊗ A ′ issurjective, and that F n (id V b ⊗ f ) : F n ( V b ⊗ A ) → F n ( V b ⊗ A ′ ) is surjective for all n ≥ . It suffices to verifythese properties in the categories d Vect and
Vect , respectively. Let σ : A ′ → A in Vect be a right inverseto f as a map between graded vector spaces. Then id V ⊗ σ : A ′ ⊗ V → A ⊗ V is a filtration preservingright inverse to id V ⊗ f . Therefore ˆ σ := lim ←− n id V/ F n V ⊗ σ is a right inverse to id V b ⊗ f , and so the lattermap is surjective. For each n ≥ , recall that F n ( V b ⊗ A ) := ker ˆ q ( n ) and F n ( V b ⊗ A ′ ) := ker ˆ q ′ ( n ) , where ˆ q ( n ) and ˆ q ′ ( n ) are the canonical surjections to V / F n V ⊗ A and V / F n V ⊗ A ′ , respectively. Hence, thecommutativity of the diagram V b ⊗ A ′ V / F n V ⊗ A ′ V b ⊗ A V / F n V ⊗ A ˆ σ id V/ F n V ⊗ σ ˆ q ′ ( n ) ˆ q ( n ) implies that the restriction σ | F n ( V b ⊗ A ′ ) : F n ( V b ⊗ A ′ ) → F n ( V b ⊗ A ) is a right inverse to F n (id V b ⊗ f ) . Proof of Prop. 4.11.
Let ( V, d ) ∈ d Ch ∗ . There are canonical isomorphisms of complete cochain complexes: V b ⊗ k = lim ←− n (cid:16) V / F n V ⊗ k (cid:17) ∼ = VV b ⊗ ( k × k ) = lim ←− n (cid:16) V / F n V ⊗ ( k ⊕ k ) (cid:17) ∼ = V ⊕ V. Therefore, by applying Lemma 4.9 to the cdga morphisms η and ρ from (4.8) and (4.9), respectively, weobtain the desired factorization of the diagonal map V → V ⊕ V in d Ch ∗ : V id V b ⊗ η −−−−→ k [ z, dz ] id V b ⊗ ρ −−−−→ V ⊕ V. .3.2 Pullbacks of (acyclic) fibrations in d Ch ∗ (axioms 4 & 5) We recall a basic fact about kernels of morphisms in the additive category d Vect . Lemma 4.10.
Let f : V → W be a morphism between complete filtered graded vector spaces. Let ker f ∈ Vect be the usual kernel of the underlying linear map. Then ker f , equipped with the induced filtration F n ker f := F n V ∩ ker f is a complete filtered vector space, and is the kernel object of the map f : V → W in the category d Vect .Proof.
See [11, Sec. 7.3.4(b)]The next proposition is straightforward, but we give an explicit proof following [31, Thm. 4.1] in prepa-ration for the analogous result in Prop. 6.5.
Proposition 4.11.
1. The pullback of a fibration exists in d Ch ∗ , and is a fibration.2. The pullback of an acyclic fibration in d Ch ∗ is an acyclic fibration.Proof.
1. Consider a diagram of the form ( W, d W ) g −→ ( U, d U ) f ←− ( V, d V ) , where f is a fibration in d Ch ∗ . Since d Ch ∗ is an additive category, the pullback object of f along g is a kernel object. Hence Lemma 4.10 impliesthat the usual pullback P ⊆ W ⊕ V in Vect is complete with respect to the induced subspace filtration F n P := P ∩ F n W ⊕ F n V , and moreover, P is the pullback object of f along g in d Vect .Since f is a fibration, Lemma 4.3 implies that there exists a filtered linear map σ : U → V in d Vect suchthat f σ = id U . Let ker f ∈ d Vect denote the kernel of f as in Lemma 4.10. The linear map h : W ⊕ ker f → P, h ( w, x ) := ( w, σg ( w ) + x ) is a filtration preserving isomorphism in d Vect with filtered inverse j : P → W ⊕ ker f, j ( w, v ) := (cid:0) w, v − σg ( w ) (cid:1) . It then follows that (cid:0) W ⊕ ker f, ˜ d (cid:1) ( V, d V )( W, d W ) ( U, d U ) pr V h pr W h fg (4.10)is a pullback diagram in d Ch ∗ , where ˜ d := j ◦ ( d W ⊕ d V ) h , and clearly F n (pr W h ) = pr W h | F n ( W ⊕ ker f ) : F n W ⊕ F n (ker f ) → F n W is a surjection for all n ≥ . Hence, pr W h is a fibration.18. Suppose that f : ( V, d V ) ∼ ։ ( U, d U ) is an acyclic fibration. Clearly, pr W h in the pullback diagram (4.10)is a surjective quasi-isomorphism. (Indeed, as a chain map f is an acyclic fibration in the model cate-gory Ch ∗ proj .) Let n ≥ . By hypothesis, F n f is a surjective quasi-isomorphism. Hence, the complex (cid:0) ker F n f, d V (cid:1) is acyclic. It follows from the definition of the differential ˜ d that we have a short exactsequence of cochain complexes (cid:0) F n ker f, d V (cid:1) ֒ → (cid:0) F n W ⊕ F n ker f, ˜ d (cid:1) pr W h −−−→ (cid:0) F n W, d W (cid:1) . Therefore, since F n ker f = ker F n f , we deduce from the long exact sequence in cohomology that pr W h | F n ( W ⊕ ker f ) is a quasi-isomorphism. d Ch ∗ Proposition 4.8 obviously implies that the CFO structure on d Ch ∗ satisfies the additional Axiom (EA2) fromSec. 3.2. We now verify that Axiom (EA1) is also satisfied by proving that every acyclic fibration in d Ch ∗ isa deformation retraction. Proposition 4.12.
Let f : ( V, d ) ∼ ։ ( V ′ , d ′ ) be an acyclic fibration in d Ch ∗ . Then1. There exists a morphism τ : ( V ′ , d ′ ) → ( V, d ) in d Ch ∗ such that f τ = id ′ V .2. Given a morphism τ as above, there exists a degree − linear map h : V → ker f such that h ( F n V ) ⊆ ker f ∩ F n V for all n ≥ , and such that id − τ f = dh + hd .Proof.
1. We proceed by induction. Set τ := 0 . Since f (2) : V / F V → V ′ / F V ′ is a surjective quasi-isomorphism,we may choose a right inverse τ : V ′ / F V ′ → V / F V in Ch ∗ . Now assume n ≥ , and suppose wehave a chain map τ n : ( V ′ / F n V ′ , d ′ ( n ) ) → ( V / F n V, d ( n ) ) satisfying f ( n ) τ n = id V ′ / F n V ′ , τ n − p ′ ( n − = p ( n − τ n . (4.11)As in the proof of Lemma 4.3, we use the inductive hypothesis to produce a linear map σ n +1 : V ′ / F n +1 V ′ → V / F n +1 V such that f ( n +1) σ n +1 = id , and τ n ◦ p ′ ( n ) = p ( n ) ◦ σ n +1 .Let θ : V ′ / F n +1 V ′ → V / F n +1 V be the degree linear map θ := d ( n +1) σ n +1 − σ n +1 d ′ ( n +1) . Then θ is adegree coboundary in the hom-complex (cid:0) Hom k ( V ′ / F n +1 V ′ , V / F n +1 V ) , ∂ (cid:1) . Since τ n is a chain map,we have the equalities p ( n ) θ = p ( n ) ◦ ∂σ n +1 = ( d ( n ) τ n − τ n d ′ ( n ) ) p ′ ( n ) = 0 . Hence im θ ⊆ ker p ( n ) = F n V / F n +1 V . Moreover, since f ( n +1) is a chain map and σ n +1 is a linear right inverse of f ( n +1) , wededuce that im θ ⊆ ker p ( n ) ∩ ker f ( n +1) = ker F n f ( n +1) .It follows from Remark 4.6 that F n f ( n +1) is a quasi-isomorphism, and since F n f ( n +1) is also surjec-tive, we conclude that ker F n f ( n +1) is contractible. Hence, θ is a degree cocycle in the acyclic com-plex Hom k ( V ′ / F n +1 V ′ , ker F n f ( n +1) ) , so there exists a degree zero linear map η : V ′ / F n +1 V ′ → ker F n f ( n +1) such that θ = d ( n +1) η − η ◦ d ′ ( n +1) . Finally, let τ n +1 := σ n +1 − iη , where i : F n V / F n +1 V ֒ → V / F n +1 V is the inclusion. By construction, τ n +1 is a chain map satisfying the equalities f ( n +1) τ n +1 = id , and τ n ◦ p ′ ( n ) = p ( n ) ◦ τ n +1 . This completesthe inductive step, and therefore τ := lim ←− τ k is a right inverse to f in d Ch ∗ .19. First, since f is a fibration, Remark 4.6(2) implies that we have an isomorphism of complexes ker f / F k ker f ∼ = ker f ( k ) for each k ≥ . So we proceed as in part (1), and inductively construct for each k ≥ a degree − linearmap h k : V / F k V → ker f ( k ) such that p ( k − h k = h k − p ( k − , d ( k ) h k + h k d ( k ) = g ( k ) , (4.12)where g ( k ) : V / F k V → ker f ( k ) is the chain map g ( k ) := id V/ F k V − τ k f ( k ) , and τ k is the chain mapsatisfying (4.11) from part (1).Set h := 0 . Since f (2) is a surjective quasi-isomorphism, the complex ker f (2) is acyclic. Since we workover a field, we may choose a null homotopy h : ker f (2) → ker f (2) . Let h : V / F V → ker f (2) be thedegree − linear map h := hg (2) . Then, by construction, h satisfies the equalities (4.12) with k = 2 .Assume n ≥ , and suppose h n : V / F n V → ker f ( n ) satisfies the equalities (4.12) for k = n . Considerthe commutative diagram of cochain complexes F n V / F n +1 V V / F n +1 V V / F n V ker F n f ( n +1) ker f ( n +1) ker f ( n ) i F n g ( n +1) g ( n +1) g ( n ) i p ( n ) | ker p ( n ) (4.13)The hypotheses on f imply that F n f ( n +1) , f ( n +1) , and f ( n ) are all surjective quasi-isomorphisms. There-fore each complex in the bottom row of diagram (4.13) is acyclic. Since the top row of (4.13) splits inthe category of graded vector spaces, the bottom row does as well. Hence, p ( n ) | ker is a surjective quasi-isomorphism of cochain complexes, and therefore there exists a chain map η : (ker f ( n ) , d ( n ) ) → (ker f ( n +1) , d ( n +1) ) such that p ( n ) | ker ◦ η = id . Define κ : V / F n +1 V → ker f ( n +1) be the degree − linear map κ := ηh n ◦ p ( n ) . By construction, we have p ( n ) κ = h n p ( n ) .Next, we modify κ to obtain the desired homotopy satisfying (4.12). Let λ : V / F n +1 V → ker f ( n +1) bethe degree linear map λ := d ( n +1) κ + κd ( n +1) − g ( n +1) = ηg ( n ) ◦ p ( n ) − g ( n +1) . Note that the last line follows from the commutativity of the diagram (4.13). A direct computation showsthat p ( n ) λ = 0 . Hence, λ is a degree cocycle in the acyclic complex (cid:0) Hom k ( V / F n +1 V, ker F n f ( n +1) ) , ∂ (cid:1) . Let ρ : V / F n +1 V → ker F n f ( n +1) be a degree − map satisfying λ = ∂ρ . Finally, define h n +1 := κ − iρ ,where i is the inclusion map in diagram (4.13). By construction, we have p ( n ) h n +1 = p ( n ) κ = h n p ( n ) ,and ∂h n +1 = λ + g ( n +1) − ∂ρ = g ( n +1) . This completes the inductive step. Applying a degree − analog of Lemma 4.2, we then conclude that h := lim ←− h n is the sought after filtered homotopy satisfying id − τ f = dh + hd .20ote that the above proposition 4.12 implies that d Ch ∗ satisfies Axiom A1 from Sec. 3.2. Hence, fromProp. 3.6 we have the following corollary Corollary 4.13. If f : ( V, d ) ∼ −→ ( V ′ , d ′ ) is a weak equivalence in d Ch ∗ , then there exists a weak equivalence g : ( V ′ , d ′ ) ∼ −→ ( V, d ) and homotopies h V : ( V, d ) → ( V b ⊗ Ω , d Ω ) , h V ′ : ( V ′ , d ′ ) → ( V ′ b ⊗ Ω , d ′ Ω ) , which induce equivalences gf ≃ id V and f g ≃ id V ′ L ∞ -algebras For a graded vector space V ∈ Vect , we let ¯ S ( V ) denote the reduced cofree conilpotent cocommutativecoalgebra generated by V . Let Φ : ¯ S ( V ) → ¯ S ( V ′ ) be any k -linear map. For p, m ≥ the notation Φ pm isreserved for the restriction-projections Φ pm : ¯ S m ( V ) → ¯ S p ( V ′ ) Φ pm := pr ¯ S p ( V ′ ) ◦ Φ | ¯ S m ( V ) (5.1)Furthermore, we denote by Φ : ¯ S ( V ) → V ′ the linear map Φ := pr V ′ Φ . In particular, we recall that coderivations Q : ¯ S ( V ) → ¯ S ( V ) are in one to one correspondence with linearmaps Q : ¯ S ( V ) → V via the formula Q ( v v · · · v n ) = Q n ( v v · · · v n ) + n X t ≥ Q tn ( v v · · · v n ) , where Q tn ( v · · · v n ) = X σ ∈ Sh( m − t +1 ,t − ǫ ( σ ) Q n − t +1 ( v σ (1) , . . . , v σ ( n − t +1) ) v σ ( n − t +2) · · · v σ ( n ) . (5.2)Similarly, a coalgebra morphism Φ : ¯ S ( V ) → ¯ S ( V ′ ) is uniquely determined by the degree linear map Φ : ¯ S ( V ) → V ′ via the formula Φ( v v . . . v n ) = X t ≥ X k + ··· + k t = nk j ≥ X σ ∈ Sh >k ,k ,...,kt Φ ( v σ (1) . . . v σ ( k ) )Φ ( v σ ( k +1) . . . v σ ( k + k ) ) . . .. . . Φ ( v σ ( n − k t +1) . . . v σ ( n ) ) , (5.3)where Sh >k ,k ,...,k t is the subset of permutations in Sh k ,k ,...,k t satisfying the condition σ (1) < σ ( k + 1) < σ ( k + k + 1) < · · · < σ ( n − k t + 1) . Let Q and Q ′ be coderivations on ¯ S ( V ) and ¯ S ( V ′ ) , respectively, and let ¯ S ( V ) Φ −→ ¯ S ( V ′ ) Ψ −→ ¯ S ( V ′′ ) becoalgebra morphisms. We note that the above formulas imply that the various compositions allowed between Q , Q ′ , Φ , and Ψ are uniquely determined by linear maps (Φ Q ) : ¯ S ( V ) → V ′ , ( Q ′ Φ) : ¯ S ( V ) → V ′ , and (ΨΦ) : ¯ S ( V ) → V ′′ , where (Φ Q ) n = n X t =1 Φ t Q tn , ( Q ′ Φ) n = n X t =1 Q ′ t Φ tn , (ΨΦ) n = n X t =1 Ψ t Φ tn . (5.4)21 .1.1 Filtered coderivations and coalgebra morphisms If V ∈ FiltVect is filtered, then, for each k ≥ , the filtration V = F V = F V ⊇ F V ⊇ · · · induces acanonical bounded decreasing filtration on the vector space V ⊗ k : V ⊗ k = F V ⊗ k = F V ⊗ k ⊇ F V ⊗ k ⊇ · · ·F n V ⊗ k := M i + i + ··· + i k = n F i V ⊗ F i V ⊗ · · · ⊗ F i k V, (5.5)which in turn induces a filtration on ¯ T ( V ) : ¯ T ( V ) = F ¯ T ( V ) ⊇ F ¯ T ( V ) ⊇ · · · , F n ¯ T ( V ) := M k ≥ F n V ⊗ k . And finally, this filtration induces a canonical bounded decreasing filtration on the subspace ¯ S ( V ) : ¯ S ( V ) = F ¯ S ( V ) ⊇ F ¯ S ( V ) ⊇ · · · , F n ¯ S ( V ) := ¯ S ( V ) ∩ F n ¯ T ( V ) . (5.6)Combining this with the discussion in Sec. 5.1 on coderivations and coalgebra morphisms gives the followinglemma, whose proof is elementary. Lemma 5.1.
Let
V, V ′ , V ′′ ∈ FiltVect be filtered graded vector spaces.1. Let Φ : ¯ S ( V ) → V ′ be a degree linear map, and Q : ¯ S ( V ) → V be a linear map of arbitrarydegree. Let Φ : ¯ S ( V ) → ¯ S ( V ′ ) and Q : ¯ S ( V ) → ¯ S ( V ) denote the corresponding coalgebra morphism (5.3) and coderivation (5.2) , respectively. If for all m ≥ and i , . . . , i m ≥ we have Φ ( F i V ⊗ F i V ⊗ · · · ⊗ F i m V ) ⊆ F i + i + ··· + i m V ′ ,Q ( F i V ⊗ F i V ⊗ · · · ⊗ F i m V ) ⊆ F i + i + ··· + i m V, (5.7) then Φ( F m ¯ S ( V )) ⊆ F m ¯ S ( V ′ ) , Q ( F m ¯ S ( V )) ⊆ F m ¯ S ( V ) . That is, Φ and Q are compatible with the induced filtrations (5.6) .2. Suppose Q and Q ′ are coderivations on ¯ S ( V ) and ¯ S ( V ′ ) , respectively, and ¯ S ( V ) Φ −→ ¯ S ( V ′ ) Ψ −→ ¯ S ( V ′′ ) are coalgebra morphisms such that such that the linear maps Q , Q ′ , Φ , and Ψ satisfy (5.7) above.Then the compositions Φ ◦ Q , Q ′ ◦ Φ , and Ψ ◦ Φ are compatible with the induced filtrations (5.6) . L ∞ -algebras In order to match conventions in our previous work [9], we define an L ∞ -algebra ( L, d, Q ) to be a cochaincomplex ( L, d ) ∈ Ch ∗ for which the reduced cocommutative coalgebra ¯ S ( L ) is equipped with a degree 1coderivation Q such that Qx = dx for all x ∈ L and Q = 0 . This structure is equivalent to specifyinga sequence of degree one brackets {· , · , . . . , ·} m : ¯ S m ( L ) → L for each m ≥ satisfying compatibilityconditions which can be thought of as higher–order Jacobi and Leibniz identities. (See Eq. 2.5. in [9].) Ashifted L ∞ -structure on L ias equivalent to a traditional L ∞ -structure on s L , the suspension of L .A (weak L ∞ ) morphism , or “ ∞ -morphism” Φ : (
L, d, Q ) → ( L ′ , d ′ , Q ′ ) between two L ∞ -algebras is adg coalgebra morphism Φ : (cid:0) ¯ S ( L ) , Q (cid:1) → (cid:0) ¯ S ( L ′ ) , Q ′ (cid:1) . (5.8)22e denote by Lie [1] ∞ the category of shifted L ∞ -algebras and weak L ∞ -morphisms. For the sake of brevity,from here on we drop the adjective “shifted”, and an L ∞ -algebra will mean “shifted L ∞ -algebra.The compatibility of Φ with the codifferentials Q and Q ′ can be expressed as a sequence of equations interms of the brackets {· · · } m and linear maps Φ k . (See Eq. 2.8 in [10].) In particular, Φ Q = Q ′ Φ impliesthat the “linear term” of dg coalgebra morphism Φ induces a map of cochain complexes: Φ = pr ˜ L Φ | L : ( L, d ) → ( L ′ , d ′ ) . (5.9)We recall some standard terminology for morphisms. Definition 5.2.
A morphism
Φ : (
L, d, Q ) → ( L ′ , d ′ , Q ′ ) of L ∞ -algebras is an L ∞ -quasi-isomorphism if and only if Φ : ( L, d ) → ( L ′ , d ′ ) is a quasi–isomorphism of cochain complexes. Simiiarly, Φ is an L ∞ -epimorphism if and only if Φ : ( L, d ) → ( L ′ , d ′ ) is a degreewise surjective map between cochain complexes.Finally we say Φ : (
L, d, Q ) → ( L ′ , d ′ , Q ′ ) is strict if and only if it consists only of a linear term, i.e. Φ ( x , . . . , x m ) = 0 ∀ m ≥ . (5.10)Note that in the strict case, we have Φ( x , x , . . . , x m ) = Φ ( x )Φ ( x ) · · · Φ ( x m ) ∈ ¯ S m ( L ′ ) and Φ Q m ( x x · · · x m ) = Q ′ m (cid:0) Φ ( x )Φ ( x ) · · · Φ ( x m ) (cid:1) ∀ m ≥ . Notation 5.3.
We will often write a strict L ∞ -morphism Φ : (
L, d, Q ) → ( L ′ , d ′ , Q ′ ) as Φ = Φ in order to emphasize the equalities (5.10). In doing so, we tacitly identify Φ with the chain map Φ : ( L, d ) → ( L ′ , d ′ ) L ∞ -algebras Definition 5.4. A filtered L ∞ -algebra ( L, d, Q ) is a filtered cochain complex ( L, d ) ∈ FiltCh ∗ whose re-duced cocommutative coalgebra ¯ S ( L ) is equipped with a degree 1 coderivation Q such that • Q | L = d and Q ◦ Q = 0 , • for all m ≥ and i , . . . , i m ≥ Q (cid:0) F i L, F i L, . . . , F i m L (cid:1) ⊆ F i + i + ··· + i m L. A filtered L ∞ -algebra ( L, d, Q ) is complete iff ( L, d ) ∈ d Ch ∗ . Remark . A complete filtered L ∞ -algebra in our sense is a shifted analog of a complete L ∞ -algebra, inthe sense of A. Berglund [3, Def. 5.1]. See also [21, Ch. 6]. Definition 5.6.
FiltLie [1] ∞ denotes the category whose objects are filtered L ∞ -algebras and whose mor-phisms are filtered (weak) L ∞ -morphisms , i.e., those morphisms Φ : (
L, d, Q ) → ( L ′ , d ′ , Q ′ ) in Lie [1] ∞ between filtered L ∞ -algebra that satisfy Φ ( F i L ⊗ F i L ⊗ · · · ⊗ F i m L ) ⊆ F i + i + ··· + i m L ′ . (5.11)We also denote by c Lie [1] ∞ ⊆ FiltLie [1] ∞ the full subcategory of filtered L ∞ -algebras whose objects are complete L ∞ -algebras.23 emark . Recall that every cochain complex ( V, d ) can be given the structure of an abelian L ∞ -algebra by defining Q := d and Q k ≥ := 0 . (This gives a faithful, but not full, functor Ch ∗ → Lie [1] ∞ .) Let ( L, d, Q ) ∈ Lie [1] ∞ be a L ∞ -algebra and give L the trivial or discrete filtration [7]: F L := L , and F k L = 0 ∀ k ≥ . Then this filtration is compatible with the L ∞ -structure Q if and only if ( L, d, Q ) isabelian. In particular, equipping a cochain complex ( V, d ) ∈ Ch ∗ with the discrete filtration, gives a functor disc : Ch ∗ → c Lie [1] ∞ to complete filtered L ∞ -algebras. Note that functor disc is both full and faithful.We end this subsection by noting that the familiar characterization of L ∞ -isomorphisms extends to iso-morphisms in c Lie [1] ∞ . Proposition 5.8.
A morphism
Φ : (
L, Q ) → ( L ′ , Q ′ ) in c Lie [1] ∞ with linear term Φ : ( L, d ) → ( L ′ , d ′ ) ∈ d Ch ∗ is an isomorphism if and only if there exists a morphism of filtered graded vector spaces ψ : L ′ → L ∈ FiltVect such that ψ Φ = id L and Φ ψ = id L ′ .Proof. Clearly, if Φ is an isomorphism in FiltLie [1] ∞ , then such a map ψ exists. Conversely, assume ψ : L ′ → L is inverse to Φ in d Vect . One constructs, using ψ and the structure maps Φ m , an inverse Ψ : ¯ S ( L ′ ) → ¯ S ( L ) recursively in the usual way (e.g., see the formulas in the proof of [21, Prop. 7.5]). Since ψ and Φ arecompatible with the filtrations on L ′ and ¯ S ( L ) , it then follows that Ψ is a morphism in c Lie [1] ∞ . c Lie [1] ∞ The product of complete L ∞ -algebras ( L, d, Q ) and ( L ′ , d ′ , Q ′ ) ∈ c Lie [1] ∞ is denoted by ( L ⊕ L ′ , d ⊕ , Q ⊕ ) ,where ( L ⊕ L ′ , d ⊕ ) is the direct sum of complete cochain complexes, and Q ⊕ is the degree one codifferentialon ¯ S ( L ⊕ L ′ ) whose structure maps are ( Q ⊕ ) k (cid:0) ( x , x ′ ) , ( x , x ′ ) , . . . , ( x k , x ′ k ) (cid:1) := (cid:16) Q k ( x , x , . . . , x k ) , Q ′ k ( x ′ , x ′ , . . . , x ′ k ) (cid:17) . (5.12)The usual projections pr : L ⊕ L ′ → L , pr ′ : L ⊕ L ′ → L ′ lift to strict L ∞ -epimorphisms ( L, d, Q ) Pr ←− ( L ⊕ L ′ , d ⊕ , Q ⊕ ) Pr ′ −−→ ( L ′ , d ′ , Q ′ ) We will frequently identify the (counital) cocommutative coalgebra S ( L ⊕ L ′ ) with the categorical product S ( L ) ⊗ k S ( L ′ ) , and we will be consistent with our convention for writing L ∞ -morphisms as dg coalgebra mor-phisms. In particular, given L ∞ -morphisms Φ : ( L , d L , Q L ) → ( L ′ , d L ′ , Q L ′ ) and Φ : ( L , d L , Q L ) → ( L ′ , d L ′ , Q L ′ ) , we denote by Φ ⊗ Φ : ( L ⊕ L , d ⊕ Q ⊕ ) → ( L ′ ⊕ L ′ , d ′⊕ , Q ′⊕ ) (5.13)the unique morphism in c Lie [1] ∞ between the corresponding products that satisfies the usual universal prop-erty.The next proposition is elementary. We will use it in Sec. 6.2 to analyze acyclic fibrations in c Lie [1] ∞ . Proposition 5.9.
Let ( L, d, Q ) and ( L ′ , d ′ , Q ′ ) ∈ c Lie [1] ∞ be complete L ∞ -algebras. The inclusion map ofcomplete vector spaces L ֒ → L ⊕ L ′ extends to a strict L ∞ -morphism in c Lie [1] ∞ ( L, d, Q ) → ( L ⊕ L ′ , d ⊕ , Q ⊕ ) . .3.2 Nilpotent L ∞ -algebras Let ( L, d, Q ) ∈ Lie [1] ∞ be an L ∞ -algebra. Following Berglund [3], let { Γ k L } k ≥ denote the level-wiseintersection of all filtrations on L compatible with the L ∞ -structure Q . Then { Γ k L } is also a compatiblefiltration called the lower central series of ( L, Q ) . The explicit inductive formula [21, Def. 6.10] for Γ k L is Γ L := L , and for k ≥ k L := X m ≥ X i + i + ··· + i m ≥ ki ,i ,...,i m Let ( L, d, Q ) and ( L ′ , d ′ , Q ′ ) be nilpotent L ∞ -algebras. Let Φ = Φ : ( L, d, Q ) → ( L ′ , d ′ , Q ′ ) be any strict L ∞ -morphism in Lie [1] ∞ . Then Φ is compatible with the lower central series on L and L ′ , and therefore is a morphism in c Lie [1] nil ∞ .Proof. Since Φ is strict, we have Q ′ m ◦ (Φ ) ⊗ m = Φ Q m for all m ≥ . It then follows from the explicitdescription (5.14) of the lower central series that Φ (cid:0) Γ k L (cid:1) ⊆ Γ k L ′ for all k ≥ .We denote by Lie [1] nil/str ∞ ⊆ Lie [1] ∞ the category whose objects are nilpotent L ∞ -algebras and whosemorphisms are strict L ∞ -morphisms. In light of Prop. 5.10, we identify Lie [1] nil/str ∞ with its image under theinclusion functor Lie [1] nil/str ∞ → c Lie [1] ∞ (5.15)which equips each nilpotent L ∞ -algebra with its lower central series. L ∞ -algebras The nilpotent L ∞ -algebras considered in this paper arise as filtered L ∞ -algebras equipped with a specifiedbounded descending filtration, which is not necessarily the one induced by its lower central series. It will beconvenient at times to keep track of the depth of the bound. Definition 5.11. A filtered L ∞ -algebra ( L, d, Q ) ∈ FiltLie [1] ∞ with filtration {F k L } k ≥ is bounded fil-tered (at N ) if and only if there exists N ≥ such that F N L = 0 . Remark . A bounded filtered L ∞ -algebra ( L, d, Q ) is clearly nilpotent and hence complete. Obviously,an infinite series P ∞ i =1 x i in L converges with respect to the bounded filtration {F k L } k ≥ if and only if itconverges with respect to the lower central series { Γ k L } k ≥ if and only if the series is finite. This should not be confused with the property that the underlying cochain complex ( L, d ) is bounded, which is often requiredwhen working with so-called “Lie N -algebras” 25e denote by c Lie [1] bdflt ∞ the full subcategory of c Lie [1] ∞ whose objects are bounded filtered L ∞ -algebras.Moreover, the above observations imply that we have a forgetful functor c Lie [1] bdflt/str ∞ str −−→ Lie [1] nil/str ∞ (5.16)from the category of bounded filtered L ∞ -algebras and strict filtered L ∞ -morphisms to the category of nilpo-tent L ∞ -algebras and strict L ∞ -morphisms. Note that every abelian L ∞ -algebra equipped with the discretefiltration from Remark 5.7 is an object in c Lie [1] bdflt ∞ .Here is our main example of a bounded filtered L ∞ -algebra. Let ( L, d, Q ) ∈ FiltLie [1] ∞ be a filtered L ∞ -algebra. For each n ≥ , the filtration on L induces a filtration on the quotient vector space L/ F n LL/ F n L = F (cid:0) L/ F n L (cid:1) ⊇ F (cid:0) L/ F n L (cid:1) ⊇ · · · , and the canonical surjections p ( n ) : L → L/ F n L and p ( n ) : L/ F n +1 L → L/ F n L from Sec. 4.1.1 upgradeto morphisms between filtered vector spaces. We define the degree linear maps d ( n ) : L/ F n L → L/ F N L ,and ( Q ( n ) ) k : ¯ S k ( L/ F n L ) → L/ F n L for k ≥ : d ( n ) (¯ x ) := dx ( Q ( n ) ) k (¯ x ¯ x · · · ¯ x k ) := Q k ( x x · · · x k ) , (5.17)where ¯ x = p ( n ) ( x ) and ¯ x i = p ( n ) ( x i ) for all i = 1 , . . . , k .The statements in the next proposition follow directly from the fact that the L ∞ -structures and morphismsin FiltLie [1] ∞ are compatible with the filtrations in the sense of Def. 5.4 and (5.11), respectively. Proposition 5.13. Let ( L, d, Q ) ∈ FiltLie [1] ∞ be a filtered L ∞ -algebra with filtration {F k L } k ≥ .1. For each n ≥ , the linear maps (5.17) give the quotient L/ F n L the structure of an L ∞ -algebra ( L/ F n L, d ( n ) , Q ( n ) ) that is bounded filtered at N = n .2. The canonical surjections from Sec. 4.1.1 induce strict filtered morphisms of L ∞ -algebras p ( n ) : ( L, d, Q ) → ( L/ F n L, d ( n ) , Q ( n ) ) ,p ( n ) : ( L/ F n +1 L, d ( n +1) , Q ( n +1) ) → ( L/ F n L, d ( n ) , Q ( n ) ) 3. For each n ≥ , the inclusion of the subcomplex ker p ( n ) = ( F n − L/ F n L, d ( n ) ) ⊆ ( L/ F n , d ( n ) ) extends to a strict filtered L ∞ -morphism ( F n − L/ F n L, d ( n ) ) ֒ → ( L/ F n L, d ( n ) , Q ( n ) ) which embeds ker p ( n ) as an abelian sub L ∞ -algebra of ( L/ F n L, d ( n ) , Q ( n ) ) .4. If Φ : ( L, d, Q ) → ( L ′ , d ′ , Q ′ ) is a morphism in FiltLie [1] ∞ , then for each n ≥ , the linear maps (Φ ( n ) ) k ≥ : ¯ S k ≥ ( L/ F n L ) → L ′ / F n L ′ , (Φ ( n ) ) k (¯ x ¯ x · · · ¯ x k ) := Φ k ( x x · · · x k ) , induce a unique morphism Φ ( n ) : ( L/ F n L, d ( n ) , Q ( n ) ) → ( L ′ / F n L ′ , d ′ ( n ) , Q ′ ( n ) ) (5.18) of filtered L ∞ -algebras satisfying p ′ ( n ) Φ = Φ ( n ) p ( n ) . .3.4 Towers of L ∞ -algebras Let Φ : ( L, d, Q ) → ( L, d ′ , Q ′ ) be a morphism in FiltLie [1] ∞ . Then statement 4 of Prop. 5.13 above impliesthat the following diagram in c Lie [1] bdflt ∞ commutes: ( L/ F n +1 L, d ( n +1) , Q ( n +1) ) ( L ′ / F n +1 L ′ , d ′ ( n +1) , Q ′ ( n +1) )( L/ F n L, d ( n ) , Q ( n ) ) ( L ′ / F n L ′ , d ′ ( n ) , Q ′ ( n ) ) Φ ( n +1) p ( n ) p ′ ( n ) Φ ( n ) (5.19)Hence, the functor tow : FiltVect → tow ( Vect ) defined in (4.1) extends to a functor tow : FiltLie [1] ∞ → tow ( c Lie [1] bdflt ∞ ) . (5.20)Moreover, given a filtered L ∞ -algebra ( L, d, Q ) ∈ FiltLie [1] ∞ , the level-wise L ∞ -structures on the thecorresponding tower tow ( L, d, Q ) induces a unique L ∞ -structure on the completion b L = lim ←− n L/ F n L suchthat the linear map p L : L → b L from Sec. 4.1.1 extends to a strict morphism of filtered L ∞ -algebras p L : ( L, d, Q ) → ( b L, b d, b Q ) . It is straightforward to check that this assignment of a filtered L ∞ -algebra to its completion defines a functor lim ←− tow : FiltLie [1] ∞ → c Lie [1] ∞ . (5.21) Here we recall extensions of the results from Sec. 4.1.3 to completed tensor products between L ∞ -algebrasand commutative dg algebras. Let ( L, d, Q ) ∈ Lie [1] ∞ be an L ∞ -algebra and ( B, δ ) ∈ cdga . Denote by ( L ⊗ B, d B , Q B ) the L ∞ -algebra whose underlying cochain complex is the usual tensor product of complexes ( L ⊗ B, d B ) , where d B := d ⊗ id + id ⊗ δ , and whose higher brackets are defined as: ( Q B ) k (cid:0) x ⊗ b · · · x k ⊗ b k (cid:1) := ( − ε Q k ( x , . . . , x k ) ⊗ b b · · · b k (5.22)where ε := P ≤ i L, d, Q ) → ( L ′ , d ′ , Q ′ ) is a mor-phism in Lie [1] ∞ , then the maps (Φ B ) k : ¯ S k ( L ⊗ B ) → L ′ ⊗ B defined as (Φ B ) k (cid:0) x ⊗ b · · · x k ⊗ b k (cid:1) := ( − ε Φ k ( x , . . . , x k ) ⊗ b b · · · b k . (5.23)assemble together to give a L ∞ -morphism Φ B : ( L ⊗ B, d B , Q B ) → ( L ′ ⊗ B, d ′ B , Q ′ B ) .Now let ( L, d, Q ) ∈ FiltLie [1] ∞ be a filtered L ∞ -algebra. The vector space L ⊗ B is equipped with theinduced filtration from Section 4.1.3, and it is easy to see that it is compatible with the L ∞ -structure Q B .Hence, ( L ⊗ B, d B , Q B ) ∈ FiltLie [1] ∞ , and so for any B ∈ cdga we obtain a functor: − b ⊗ B : c Lie [1] ∞ → c Lie [1] ∞ (5.24)which sends ( L, d, Q ) to the completion ( L b ⊗ B, b d B , b Q B ) of the filtered L ∞ -algebra ( L ⊗ B, d B , Q B ) via thefunctor (5.21). The proof of the following proposition is straightforward, and follows from the definition ofthe filtration on the tensor product. Proposition 5.14. Let ( B, δ ) ∈ cdga be a cdga. If ( L, d, Q ) ∈ c Lie [1] bdflt ∞ is bounded filtered at N , then the L ∞ -algebra ( L ⊗ B, d B , Q B ) is also bounded filtered at N , and therefore ( L b ⊗ B, b d B , b Q B ) ∼ = ( L ⊗ B, d B , Q B ) .4 Maurer–Cartan elements and twisting Our reference for this section is Section 2 of [10]. We refer the reader there for details. Let ( L, d, Q ) ∈ c Lie [1] ∞ . The curvature curv : L → L is the set-theoretic function curv ( a ) = da + X m ≥ m ! Q m ( a · · · a | {z } m times ) ∀ a ∈ L , (5.25)which is well-defined due to: (1) the compatibility of the L ∞ -structure with the filtrations, (2) the fact that L = F L , and (3) the fact that L is complete. Elements of the set MC( L ) := { α ∈ L | curv ( α ) = 0 } are called the Maurer–Cartan (MC) elements of L . Note that MC elements of L are elements of degree 0.Similarly, let Φ : ( L, d, Q ) → ( L ′ , d ′ , Q ′ ) be a morphism in c Lie [1] ∞ . Such a morphism induces a function Φ ∗ : L → L ′ : Φ ∗ ( a ) := X m ≥ m ! Φ ( a · · · a | {z } m times ) ∀ a ∈ L . (5.26)The curvature function satisfies the following useful identities (which are proved in [10, Prop. 2.2]): d ( curv ( a )) + ∞ X m =1 m ! Q m (cid:0) a · · · a | {z } m times · curv ( a ) (cid:1) = 0 ∀ a ∈ L , (5.27)and curv (cid:0) Φ ∗ ( a ) (cid:1) = X m ≥ m ! Φ m +1 (cid:0) a · · · a | {z } m times · curv ( a ) (cid:1) ∀ a ∈ L . (5.28)If F : ( L, d, Q ) → ( L ′ , d ′ , Q ′ ) is a morphism in c Lie [1] ∞ , and α ∈ MC( L ) is a MC element of L , then Eq.5.28 implies that Φ ∗ ( α ) is a MC element of L ′ . Using the composition formulas (5.4), it is straightforwardto check that the assignment L MC( L ) a functor MC : c Lie [1] ∞ → Set . (5.29) Remark . 1. Eq. 5.27 above is known as the Bianchi identity [13]2. Let ( L, d ) ∈ Ch ∗ be a cochain complex considers as an abelian L ∞ -algebra. In this case, the set ofMaurer-Cartan elements MC( L ) is equal to the subspace of degree 0 cocycles Z ( L ) .3. It is easy to show that for any ( L, Q ) ∈ c Lie [1] ∞ , there is a natural isomorphism MC( L ) ∼ = lim ←− n MC( L/ F n L ) . See, for example, [21, Prop. 3.17]. 28 .4.1 Twisting Elements of degree 0 in a complete filtered L ∞ -algebra ( L, d, Q ) can be used to “twist” the differential andmulti-brackets on L . Let a ∈ L be any element of degree . We define the degree 1 maps d a : L → L , and ( Q a ) k : ¯ S k ( L ) → L as d a ( x ) := dx + ∞ X k =1 k ! Q k +1 ( a · · · a | {z } k times x ) ∀ x ∈ L, ( Q a ) m ( x x · · · x m ) := ∞ X k =0 k ! Q k + m ( a · · · a | {z } k times · x x · · · x m ) ∀ x , . . . , x m ∈ L. (5.30)Then the following equalities hold [10, Prop. 2.2]: d a ◦ d a ( x ) = − pr L Q a ( curv ( a ) · x ) ∀ a ∈ L ∀ x ∈ L, curv ( a + b ) = curv ( a ) + d a ( b ) + ∞ X m =2 m ! ( Q a ) m ( b · · · b | {z } m times ) ∀ a, b ∈ L . (5.31)In particular, given an MC element α ∈ MC( L ) , we can construct a new complete filtered L ∞ -algebra usingthe α -twisted structures defined above: ( L α , d, α , Q α ) ∈ c Lie [1] ∞ , (5.32)where L α := L as a complete filtered graded vector space. We also recall that if Φ : ( L, d, Q ) → ( L ′ , d ′ , Q ′ ) is a morphism in c Lie [1] ∞ and α ∈ MC( L ) , then Φ ∗ ( α ) ∈ MC( L ′ ) , and we can twist Φ to form a new L ∞ -morphism Φ α : ( L α , d α , Q α ) → ( L ′ Φ ∗ ( α ) , d ′ Φ ∗ ( α ) , Q ′ Φ ∗ ( α ) ) whose structure maps are given by: (Φ α ) m ( x x . . . x m ) := ∞ X k =0 k ! Φ m + k ( α · · · α | {z } k times x x · · · x m ) ∀ x , . . . , x m ∈ L α ∀ m ≥ . (5.33) Remark . 1. In the proof of [10, Prop. 2.2]), the following fact is used for verifying some of the above identitiesconcerning curvature and twisting: For any a ∈ L , the expression exp( a ) − is a well-defined elementof the usual completion of ¯ S ( L ) defined by the corresponding formal power series. By extending Q inthe natural way, a straightforward calculation shows that Q (cid:0) exp( a ) − (cid:1) = exp( a ) curv ( a ) (5.34)and hence curv ( a ) = pr L ◦ Q (cid:0) exp( a ) − (cid:1) . (5.35)2. Similarly, if Φ : ( L, d, Q ) → ( L ′ , d ′ , Q ′ ) is a morphism in c Lie [1] ∞ , then we may extend Φ to thecompletions of ¯ S ( L ) and ¯ S ( L ′ ) by formal power series, and a straightforward calculation gives us theequality Φ (cid:0) exp( a ) − (cid:1) = exp(Φ ∗ ( a )) − . (5.36)3. Suppose ( L, d, Q ) and ( L ′ , d ′ , Q ′ ) are complete L ∞ -algebras, and Φ : ¯ S ( L ) → ¯ S ( L ′ ) is a coalgebramorphism that is compatible with the filtrations, but not necessarily compatible with the differentials.Then we note that the function Φ ∗ defined in Eq. 5.26 is still well-defined, and the equality (5.36) stillholds. 29 .4.2 Maurer-Cartan elements of bounded filtered L ∞ -algebras Suppose ( L, d, Q ) ∈ c Lie [1] bdflt ∞ is bounded filtered at N (Def. 5.11). Then clearly the codifferential Q isnecessarily truncated at N , i.e., Q k = 0 for k ≥ N . However, the compatibility of the L ∞ structure withthe filtration also implies a bit more. The following technical results concerning the curvature function of ( L, d, Q ) will be used in the proof of Thm. 7.4. Lemma 5.17. Let ( L, d, Q ) ∈ c Lie [1] bdflt ∞ be an L ∞ -algebra that is bounded filtered at N .1. If a ∈ L and y ∈ F N − L , then curv ( a + y ) = curv ( a ) + dy. (5.37) 2. If a ∈ L and curv ( a ) ∈ F N − L , then d curv ( a ) = 0 . (5.38) Proof. 1. The identities (5.31) imply that curv ( a + y ) = curv ( a ) + d a ( y ) + ∞ X m =2 m ! ( Q a ) m ( y · · · y | {z } m times ) . (5.39)Recall L = F L , therefore for all k ≥ and m ≥ , we have Q k + m ( a · · · a | {z } k times y · · · y | {z } m times ) ∈ F m ( N − k L ⊆ F N L = 0 . It then follows from the definition of ( Q a ) m in (5.30) that the infinite summation on the right-hand sideof (5.39) above vanishes. Now consider the remaining term d a ( y ) = dy + ∞ X k =1 k ! Q k +1 ( a · · · a | {z } k times y ) . Using again the hypothesis on the filtration degree of y , we deduce that for all k ≥ Q k +1 ( a · · · a | {z } k times y ) ∈ F N + k − L ⊆ F N L = 0 . Hence, d a ( y ) = dy , and the desired equality then follows.2. We proceed using the same idea as in the proof of part 1. The hypothesis on the filtration degree of curv ( a ) , along with the compatibility of Q with the filtration, implies that infinite sum on the left-handside of the Bianchi identity (5.27) vanishes. Therefore, the identity implies that d curv ( a ) = 0 .30 .5 An aside on obstruction theory We point out that Lemma 5.17 gives a quick proof of a fact well known to experts: Every complete L ∞ -algebra ( L, d, Q ) encodes a canonical “obstruction theory” for its Maurer-Cartan elements. This idea (alongwith an appropriate choice for L ) is behind many of the familiar obstruction-theoretic arguments used indeformation theory and rational homotopy theory. One example is the obstruction theory for ∞ -morphismsbetween homotopy P -algebras developed in [31, Appendix A].Let ( L, d, Q ) ∈ c Lie [1] ∞ . Recall that the L ∞ -algebra (cid:0) L/ F L, d (2) , Q (2) (cid:1) is abelian and is just the thecochain complex ( L/ F L, d (2) ) . Hence, the set of MC elements in L/ F L is the subspace of degree 0 co-cycles Z ( L/ F L ) . Let β be such a 0-cocycle, and let p (2) : ( L, d, Q ) ։ ( L/ F L, d (2) ) be the fibration in c Lie [1] ∞ corresponding to the canonical surjection, as usual. A question that frequently arises in the afore-mentioned applications is:Does β lift through p (2) to a Maurer-Cartan element α ∈ MC( L ) ?The next proposition – which we claim no originality for – characterizes the existence and uniqueness ofsuch Maurer-Cartan elements α ∈ MC( L ) via the associate graded cochain complex Gr( L ) . In particular,the answer to the above question is yes, if and only if an infinite sequence of cohomology classes vanishes in H (Gr( L )) . Proposition 5.18. Let n ≥ . Let β ∈ MC( L/ F n − L ) be a Maurer-Cartan element of the L ∞ -algebra ( L/ F n − L, d ( n − , Q ( n − ) , and suppose a ∈ ( L/ F n L ) is any degree 0 vector satisfying p ( n − ( a ) = β ∈ ( L/ F n − L ) . Then:1. The element curv ( n ) ( a ) is a 1-cocycle in the subcomplex ker p ( n − := ( F n − L/ F n L, d ( n ) ) , where curv ( n ) is the curvature function of the L ∞ -algebra ( L/ F n L, d ( n ) , Q ( n ) ) .2. The class (cid:2) curv ( n ) ( a ) (cid:3) ∈ H ( F n − L/ F n L ) is independent of the choice of a ∈ ( L/ F n L ) lifting the degree 0 vector β .3. There exists a Maurer-Cartan element α ∈ MC( L/ F n L ) lifting β ∈ MC( L/ F n − L ) through the function p ( n − ∗ : MC( L/ F n L ) → MC( L/ F n − L ) if and only if the class [ curv ( n ) ( a )] is trivial.4. The fiber p − n − ∗ (cid:0) { β } (cid:1) ⊆ MC( L/ F n L ) is either empty or a torsor over the abelian group MC( F n − L/ F n L ) = Z ( F n − L/ F n L ) . Proof. 1. Since p ( n − : ( L/ F n L, d ( n ) , Q ( n ) ) → ( L/ F n − L, d ( n − , Q ( n − ) is a strict L ∞ -morphism, we have p ( n − (cid:0) curv ( n ) ( a ) (cid:1) = curv ( n − (cid:0) p ( n − ( a ) (cid:1) = 0 . Therefore, curv ( n ) ( a ) ∈ F n − L/ F n L . Proposition5.13 implies that ( L/ F n L, d ( n ) , Q ( n ) ) is bounded filtered at n , and since curv ( n ) ( a ) has filtration degree n − , it follows from Lemma 5.17 that d ( n ) curv ( n ) ( a ) = 0 .31. Suppose a ′ ∈ ( L/ F n L ) is another element satisfying p ( n − ( a ′ ) = β . Then the vector η := a ′ − a ∈ ( F n − L/ F n L ) has filtration degree n − , and so Lemma 5.17 implies that curv ( n ) ( a ′ ) = curv ( n ) ( a + η ) = curv ( n ) ( a ) + d ( n ) η. 3. If there exists α ∈ MC( L/ F n L ) lifting β , then statement (2) implies that the obstruction class is trivial.Conversely, suppose there exists η ∈ ( F n − L/ F n L ) such that curv ( n ) ( a ) = d ( n ) η . Let α := a − η .Then Lemma 5.17 implies that curv ( n ) ( α ) = curv ( n ) ( a ) − d ( n ) η = 0 .4. Suppose α, α ′ ∈ MC( L/ F n L ) satisfy p ( n − ∗ ( α ) = p ( n − ∗ ( α ′ ) = β . Since p ( n − ∗ ( α ) is a strict L ∞ -morphism, it follows that η := α ′ − α ∈ ( F n − L/ F n L ) . Lemma 5.17 implies that curv ( n ) ( α ′ ) = curv ( n ) ( α ) + d ( n ) η . Since α and α ′ are Maurer-Cartan, it then follows that η ∈ Z (( F n − L/ F n L )) =MC( F n − L/ F n L ) . Conversely, the same argument implies that α ′ := α + η is a Maurer-Cartan el-ement of L/ F n L satisfying p ( n − ∗ ( α ′ ) = β whenever α ∈ MC( L/ F n L ) is as well, provided η ∈ Z ( F n − L/ F n L ) . The MC elements of a complete filtered L ∞ -algebra are the vertices of a simplicial set which we now recall.Let Ω n denote the de Rham-Sullivan algebra of polynomial differential forms on the geometric simplex ∆ n with coefficients in k . These assemble together into a simplicial cdga { Ω n } n ≥ . The simplicial Maurer–Cartan functor MC : c Lie [1] ∞ → s Set (5.40)assigns to each complete filtered L ∞ -algebra ( L, d, Q ) the simplicial set whose n -simplices are MC n ( L ) := MC (cid:0) L b ⊗ Ω n , b d Ω n , b Q Ω n (cid:1) (5.41)and to each L ∞ -morphism Φ : ( L, d, Q ) → ( L ′ , d ′ , Q ′ ) ∈ c Lie [1] ∞ the simplical map MC n (Φ) := ( b Φ Ω n ) ∗ : MC( L b ⊗ Ω n ) → MC( L ′ b ⊗ Ω n ) . (5.42)Above, ( b Φ Ω n ) ∗ is the function obtained from Eq. 5.23 followed by Eq. 5.26. c Lie [1] ∞ We first give a fairly explicit proof that c Lie [1] ∞ forms a category of fibrant objects. Then we show that thisCFO structure satisfies the additional axioms (EA1) and (EA2) from Sec. 3.2, which then implies that everyweak equivalence in c Lie [1] ∞ has a homotopy inverse. c Lie [1] ∞ as a category of fibrant objects From the point of view of most applications , reasonable notions of weak equivalence and fibration in c Lie [1] ∞ are the following: Definition 6.1. Let Φ : ( L, d, Q ) → ( L ′ , d ′ , Q ′ ) be a morphism in c Lie [1] ∞ . See, however, Remark 7.7. 32. We say Φ is a weak equivalence if its linear term Φ : ( L, d ) → ( L ′ , d ′ ) induces for each n ≥ aquasi–isomorphism of cochain complexes F n Φ : ( F n L, d ) ∼ −→ ( F n L ′ , d ′ ) . 2. We say Φ is a fibration if its linear term Φ : ( L, d ) → ( L ′ , d ′ ) induces for each n ≥ an epimorphismof cochain complexes F n Φ : ( F n L, d ) ։ ( F n L ′ , d ′ ) . Theorem 6.2. The category c Lie [1] ∞ of complete filtered L ∞ -algebras has the structure of a category offibrant objects with functorial path objects in which the weak equivalences and fibrations are those morphismsthat satisfy the defining criteria given in Def. 6.1Proof. As in the proof of Thm. 4.7, we will proceed by directly verifying axioms 1 through 7 of Def. 3.1.Finite products in c Lie [1] ∞ were verified in Sec. 5.3.1, and axioms 1,2,3, and 7 are trivial. We verify theremaining axioms by building on the results of Sec. 4.3 in the subsequent propositions below. c Lie [1] ∞ (axiom 6) From our definition of weak equivalences and fibration in c Lie [1] ∞ , Prop. 4.8 immediate implies Proposition 6.3. To each complete L ∞ -algebra ( L, d, Q ) , the assignment ( L, d, Q ) (cid:0) L b ⊗ Ω , d Ω , Q Ω (cid:1) is a functorial path object for the category c Lie [1] ∞ . c Lie [1] ∞ (axioms 4 & 5) We first record a filtered analog of a useful simplification due to B. Vallette [31, Prop. 4.3] which allows usto “strictify” any fibration in c Lie [1] ∞ without altering the linear term. Proposition 6.4. Let Φ : ( L, Q ) ։ ( L ′ , Q ′ ) be a fibration in c Lie [1] ∞ Then there exists a complete L ∞ -algebra ( ˘ L, ˘ Q ) and an isomorphism Ψ : ( ˘ L, ˘ Q ) ∼ = −→ ( L, Q ) in c Lie [1] ∞ such that ΦΨ : ( ˘ L, ˘ Q ) → ( L ′ , Q ′ ) is a strict fibration with linear term (ΦΨ) = Φ .Proof. Consider the linear term Φ : L → L ′ as a morphism in d Vect . By definition, the restriction F n Φ : F n L → F n L ′ is a surjection for all n ≥ . Therefore, Lemma 4.3 implies that there exists a filtration preserving linear map σ : L ′ → L such that φ ◦ σ = id L .Define ˘ L := L and Ψ := id ˘ L . Let m ≥ and suppose we have defined a sequence of degree 0 filtration-preserving linear maps { Ψ k : ¯ S k ( ˘ L ) → L } m − k ≥ . It follows from Eq. 5.3 that this sequence gives us welldefined linear maps Ψ km : ¯ S m ( ˘ L ) → ¯ S k ( L ) for ≤ k ≤ m . Now define Ψ m : ¯ S m ( ˘ L ) → L as: Ψ m = − m X k ≥ σ Φ k Ψ km . Ψ m is filtration preserving, since it is the composition of filtration preserving maps. This con-struction inductively yields a coalgebra isomorphism Ψ : ¯ S ( ˘ L ) ∼ = −→ ¯ S ( L ) . By define a new codifferential ˘ Q := Ψ − Q Ψ , we promote Ψ to an isomorphism Ψ : (cid:0) ˘ L, ˘ Q (cid:1) ∼ = −→ (cid:0) L, Q (cid:1) in c Lie [1] ∞ . And since Ψ = id , itfollows from Eq. 5.4 that we have (ΦΨ) = Φ : L → L ′ . Finally, to show that ΦΨ is strict, i.e. (ΦΨ) m = 0 for all m ≥ , we observe that Eq. 5.4 and the construction of Ψ imply that (ΦΨ) m = − Φ (cid:0) m X k ≥ σ Φ k Ψ km (cid:1) + m X k ≥ Φ k Ψ km . This completes the proof.Next we show that strict fibrations and acyclic fibrations are preserved under pullbacks in c Lie [1] ∞ . Thegeneral case for weak (acyclic) fibrations will then follow as a corollary.In the proof of Thm. 4.1 [31], B. Vallette provides a useful construction of pullbacks along strict L ∞ -epimorphisms in Lie [1] ∞ . The construction easily extends to c Lie [1] ∞ in the following way. Suppose Φ =Φ : ( L, d, Q ) ։ ( L ′′ , d ′′ , Q ′′ ) is a strict fibration in c Lie [1] ∞ , and Θ : ( L ′ , d ′ , Q ′ ) → ( L ′′ , d ′′ , Q ′′ ) is anarbitrary morphism in c Lie [1] ∞ . Since Φ : ( L, d ) ։ ( L ′ , d ′ ) is a fibration in d Ch ∗ , Prop. 4.11 implies that wehave a pullback square of the following form (cid:0) ˜ L, ˜ d (cid:1) ( L, d )( L ′ , d ′ ) ( L ′′ , d ′′ ) pr ◦ h pr ′ Φ Θ with ˜ L = L ′ ⊕ ker Φ , and ˜ d = j ◦ ( d ′ ⊕ d ) ◦ h . Recall that h : L ′ ⊕ L ∼ = −→ L ′ ⊕ L and j = h − : L ′ ⊕ L ∼ = −→ L ′ ⊕ L are the filtered linear maps h ( v ′ , v ) := ( v ′ , σ Θ ( v ′ ) + v ) j ( v ′ , v ) := ( v ′ , v − σ Θ ( v ′ )) , (6.1)where σ : L ′′ → L is a section in FiltVect of of the surjection Φ .We denote elements of the direct sum L ′ ⊕ L as vectors ~v := ( v ′ , v ) . In order to lift the above pullbackdiagram to c Lie [1] ∞ , we follow Vallette and define the linear maps H k : ¯ S k ( L ′ ⊕ L ) → L ′ ⊕ L to be H ( ~v ) := h ( ~v ) , H k ( ~v , . . . , ~v k ) := (cid:0) , σ Θ k ( v ′ , . . . , v ′ k ) (cid:1) . (6.2)Similarly, let J k : ¯ S k ( L ′ ⊕ L ) → L ′ ⊕ L denote the linear maps J ( ~v ) := j ( ~v ) , J k ( ~v , . . . , ~v k ) := (cid:0) , − σ Θ k ( v ′ , . . . , v ′ k ) (cid:1) . (6.3)A direct calculation shows that HJ = J H = id ¯ S ( L ′ ⊕ L ) .Finally, since ¯ S ( ˜ L ) ⊆ ¯ S ( L ′ ⊕ L ) , we can define a degree 1 codifferential ˜ Q : ¯ S ( ˜ L ) → ¯ S ( ˜ L ) via ˜ Q := J Q ⊕ H | ¯ S (˜ L ) (6.4)where Q ⊕ is the L ∞ -structure on L ′ ⊕ L induced by Q ′ and Q . By construction, ˜ Q = ˜ d and ˜ Q = 0 , and adirect calculation shows that indeed ˜ Q is well-defined, i.e., im ˜ Q ⊆ ¯ S ( L ⊕ L ′ ) . (See, for example, [26, Sec.4.1] for details of an almost identical calculation.) 34ince H , J , and ˜ Q are constructed from filtration preserving maps, we conclude that ( ˜ L, ˜ Q ) ∈ Lie [1] ∞ ,and H | ¯ S (˜ L ) : ( ˜ L, ˜ d, ˜ Q ) → ( L ′ ⊕ L, d ⊕ , Q ⊕ ) (6.5)is a morphism in c Lie [1] ∞ . Let Pr ′ and Pr denote the canonical projections out of the product ( L ′ ⊕ L, d ⊕ , Q ⊕ ) .We then have: Proposition 6.5. If Φ = Φ : ( L, d, Q ) ։ ( L ′′ , d ′′ , Q ′′ ) is a strict fibration (acyclic fibration) in c Lie [1] ∞ and Θ : ( L ′ , d ′ , Q ′ ) → ( L ′′ , d ′′ , Q ′′ ) is an arbitrary morphism in c Lie [1] ∞ , then ( ˜ L, ˜ Q ) (cid:0) L, Q ) (cid:0) L ′ , Q ′ ) (cid:0) L ′′ , Q ′′ ) Pr H Pr ′ H ΦΘ (6.6) is a pullback diagram in the category c Lie [1] ∞ . Moreover, the morphism Pr ′ H : ( ˜ L, ˜ d, ˜ Q ) → ( L ′ , d ′ , Q ′ ) is afibration (acyclic fibration) in c Lie [1] ∞ .Proof. Theorem 4.1 of [31] implies that the diagram (6.6) is the pullback in the category Lie [1] ∞ . Indeed, if ( V, d V , Q V ) Ψ −→ ( L, d, Q ) and ( V, d V , Q V ) Ψ ′ −→ ( L ′ , d ′ , Q ′ ) are morphisms in Lie [1] ∞ such that ΦΨ = ΘΨ ′ ,then the L ∞ -morphism J ◦ (Ψ ⊗ Ψ ′ ) : ( V, d V , Q V ) → ( ˜ L, ˜ d, ˜ Q ) (6.7)is the unique map which makes the relevant diagrams commute. (See Sec. 5.3.1 for a reminder of the notation Ψ ⊗ Ψ ′ .)It is then clear that if Ψ and Ψ ′ are morphism in c Lie [1] ∞ , then so is J ◦ (Ψ ⊗ Ψ ′ ) . Hence, diagram (6.6)lifts to a pullback diagram in c Lie [1] ∞ . Finally, it follows from Prop. 4.11 that Pr ′ H is a fibration (acyclicfibration).Now we remove the assumption in Prop. 6.5 that the fibration Φ is strict. Corollary 6.6. Let Φ : ( L, d, Q ) ։ ( L ′′ , d ′′ , Q ′′ ) be a fibration (acyclic fibration) in c Lie [1] ∞ .Let Θ : ( L ′ , d ′ , Q ′ ) → ( L ′′ , d, ′′ , Q ′′ ) be an arbitrary morphism in c Lie [1] ∞ . Then the pullback of the diagram ( L ′ , d ′ , Q ′ ) Θ −→ ( L ′′ , d ′′ , Q ′′ ) Φ ←− ( L, d, Q ) exists in c Lie [1] ∞ . Furthermore, the canonical projection from the pullback to ( L ′ , d ′ , Q ′ ) is a (acyclic) fibra-tion.Proof. Proposition 6.4 implies that there exists an isomorphism Ψ : ( ˘ L, ˘ Q ) ∼ = −→ ( L, Q ) in c Lie [1] ∞ such that ˘Φ := ΦΨ : ( ˘ L, ˘ d, ˘ Q ) → ( L ′′ , d ′′ , Q ′′ ) is a strict fibration with linear term ( ˘Φ) = Φ . Now we apply Prop.6.5 to the strict fibration ˘Φ and Θ and obtain the pullback diagram ( ˜ L, ˜ d, ˜ Q ) (cid:0) ˘ L, ˘ d, ˘ Q ) (cid:0) L ′ , d ′ , Q ′ ) (cid:0) L ′′ , d ′′ , Q ′′ ) Pr H Pr ′ H ˘ΦΘ 35n which Pr ′ H is a (acyclic) fibration. Then, since Ψ is an isomorphism in c Lie [1] ∞ , the diagram ( ˜ L, ˜ Q ) (cid:0) L, Q ) (cid:0) L ′ , Q ′ ) (cid:0) L ′′ , Q ′′ ) Ψ ◦ (Pr H )Pr ′ H FG (6.8)is also a pullback in c Lie [1] ∞ . c Lie [1] ∞ We begin with a decomposition lemma for acyclic fibrations in c Lie [1] ∞ . Lemma 6.7. Let Φ : ( L, d, Q ) ∼ ։ ( L ′ , d ′ , Q ′ ) be an acyclic fibration in c Lie [1] ∞ . Let (ker Φ , d ) denote thekernel of the chain map Φ : ( L, d ) → ( L ′ , d ′ ) considered as an abelian L ∞ -algebra. Then there exists a L ∞ -morphism in c Lie [1] ∞ Ψ : ( L, d, Q ) → (ker Φ , d ) such that the morphism induced via the universal property of the product: (cid:0) Φ , Ψ (cid:1) : ( L, d, Q ) ∼ = −→ ( L ′ ⊕ ker Φ , d ⊕ , Q ⊕ ) is an isomorphism of complete L ∞ -algebras.Proof. Since Φ is an acyclic fibration, the chain map Φ : ( L, d ) → ( L ′ , d ′ ) is an acyclic fibration in d Ch ∗ .Proposition 4.12 implies that there exists a filtered chain map τ : ( L ′ , d ′ ) → ( L, d ) in d Ch ∗ and a filtered chainhomotopy h : V → V [ − such that Φ ◦ τ = id L ′ , Ψ = dh + hd. (6.9)where Ψ : ( L, d ) → (ker Φ , d ) is the chain map Ψ := id L − τ ◦ Φ . For each n ≥ , let Ψ n : ¯ S n ( L ) → ker Φ be the linear map Ψ n ≥ := Ψ ◦ h ◦ Q n . (6.10)Since τ , h and Q k are filtration preserving, so is Ψ : ¯ S ( L ) → ker Φ . A direct calculation using (6.9) andthe fact that Q = 0 , shows that for each n ≥ d | ker ◦ Ψ n = Ψ d + n X k =2 Ψ k Q kn . Hence Ψ defines a L ∞ -morphism in c Lie [1] ∞ . Finally, we note that the linear map θ : L ′ ⊕ ker Φ → L, θ ( x ′ , z ) := τ ( x ′ ) + z is filtration preserving and inverse to (Φ , Ψ ) in d Vect . Hence, the L ∞ -morphism (Ψ , Φ) is an isomorphismin c Lie [1] ∞ .The above lemma implies that every acyclic fibration in c Lie [1] ∞ is a retraction. Corollary 6.8. Let Φ : ( L, d, Q ) ∼ ։ ( L ′ , d ′ , Q ′ ) be an acyclic fibration in c Lie [1] ∞ . Then there exists an L ∞ -morphism χ : ( L ′ , d ′ , Q ′ ) → ( L, d, Q ) in c Lie [1] ∞ such that Φ ◦ χ = id L ′ . roof. Let (cid:0) Φ , Ψ (cid:1) : ( L, d, Q ) ∼ = −→ ( L ′ ⊕ ker Φ , d ⊕ , Q ⊕ ) denote the isomorphism from Lemma 6.7. FromProp. 5.9 we have the injective L ∞ -morphism i L ′ : ( L ′ , d ′ , Q ′ ) → ( L ′ ⊕ ker Φ , d ⊕ , Q ⊕ ) . Let χ := (Φ , Ψ) − ◦ i L ′ . Then Φ ◦ χ = Pr L ′ (Φ , Ψ) − ◦ (Φ , Ψ) ◦ i L ′ = id L ′ . It now follows that every weak equivalence in c Lie [1] ∞ has a homotopy inverse via any choice of pathobject. In particular Theorem 6.9. If Φ : ( L, d, Q ) ∼ −→ ( L ′ , d ′ , Q ′ ) is a weak equivalence in c Lie [1] ∞ , then there exists a weakequivalence Ψ : ( L ′ , d ′ , Q ′ ) ∼ −→ ( L, d, Q ) and homotopies H L : ( L, d, Q ) → ( L b ⊗ Ω , d Ω , Q Ω ) , H L ′ : ( L ′ , d ′ , Q ′ ) → ( L ′ b ⊗ Ω , d ′ Ω , Q ′ Ω ) , which induce equivalences ΨΦ ≃ id L and ΦΨ ≃ id L ′ .Proof. Cor. 6.8 implies that c Lie [1] ∞ satisfies Axiom A1 from Sec. 3.2. Hence, the result follows from Prop.3.6. MC : c Lie [1] ∞ → KanCplx MC ( L ) Let us first consider the bounded filtered case. We recall a theorem of Getzler concerning the simplicialMaurer-Cartan set for nilpotent L ∞ -algebras. Theorem 7.1 (Prop. 4.7 [13]) . Let Φ = Φ : ( L, d, Q ) → ( L ′ , d ′ , Q ′ ) be a strict L ∞ -epimorphism in Lie [1] nil/str ∞ . Then MC (Φ) : MC ( L ) → MC ( L ′ ) is a Kan fibration. In particular, MC ( L ) is a Kan simpli-cial set for all ( L, d, Q ) ∈ Lie [1] nil/str ∞ . Recall from (5.16) the discussion of the forgetful functor c Lie [1] bdflt/str ∞ str −−→ Lie [1] nil/str ∞ which sends abounded filtered L ∞ -algebra to its underlying nilpotent L ∞ -algebra. Remark 5.12 implies that MC( L ) =MC (cid:0) str ( L ) (cid:1) . Therefore, from Prop. 5.14, we have MC( L b ⊗ B ) = MC (cid:0) str ( L ) ⊗ B (cid:1) for every ( L, d, Q ) ∈ c Lie [1] bdflt/str ∞ and every ( B, δ ) ∈ cdga . This leads us to the next proposition, which isjust for the purposes of bookkeeping. It follows immediately from Thm. 7.1. Proposition 7.2. Let Φ = Φ : ( L, d, Q ) ։ ( L ′ , d ′ , Q ′ ) be a (strict) fibration in c Lie [1] bdflt/str ∞ . Then MC (Φ) : MC ( L ) → MC ( L ′ ) is a Kan fibration. In particular, MC ( L ) is a Kan simplicial set for all bounded filtered L ∞ -algebra.Proof. Every strict fibration in FiltLie [1] ∞ is obviously a strict L ∞ -epimorphism.Now consider arbitrary complete filtered L ∞ -algebras. The next result is well known and follows fromProp. 7.2 and the fact that MC ( L ) ∼ = lim ←− MC ( L/ F n L ) for all ( L, d, Q ) ∈ c Lie [1] ∞ . See, for example, [10,Prop. 4.1] and [21, Thm. 6.9]. Corollary 7.3. Let ( L, d, Q ) ∈ c Lie [1] ∞ be a complete filtered L ∞ -algebra. Then MC ( L ) is a Kan simplicialset, and the functor (5.40) factors as MC : c Lie [1] ∞ → KanCplx .2 MC as an exact functor The goal of this section is to prove Theorem 7.4. The simplicial Maurer-Cartan functor MC : c Lie [1] ∞ → KanCplx is an exact functor betweencategories of fibrant objects. Our proof of Thm. 7.4 will proceed by directly verifying the axioms of Def. 3.7 via the following collectionof propositions. Proposition 7.5. Let ( L, d, Q ) , ( L ′ , d ′ , Q ′ ) ∈ c Lie [1] ∞ . Then MC ( L ⊕ L ′ ) ∼ = MC ( L ) × MC ( L ′ ) . Proof. It follows from the definition (5.12) for the product in c Lie [1] ∞ that curv ⊕ = curv L × curv L ′ , where curv ⊕ is the curvature function for ( L ⊕ L ′ , d ⊕ , Q ⊕ ) . It is then a straightforward exercise to verify that thereis a natural isomorphism of sets MC (cid:0) ( L ′ ⊕ L ) ⊗ B (cid:1) ∼ = MC( L ⊗ B ) × MC( L ′ ⊗ B ) for any L, L ′ ∈ c Lie [1] ∞ and B ∈ cdga . See, for example, [21, Ex. 6.17].Let us next address weak equivalences. In previous work [9] with V. Dolgushev, the following general-ization of the classical Goldman-Millson theorem from deformation theory was established. Proposition 7.6 (Theorem 1.1 [9]) . Let Φ : ( L, d, Q ) ∼ −→ ( L ′ , d ′ , Q ′ ) be a weak equivalence in c Lie [1] ∞ . Then MC (Φ) : MC ( L ) → MC ( L ′ ) is a homotopy equivalence between simplicial sets.Remark . 1. The proof of the above given in [9] builds on Getzler’s proof of an analogous statement in [13] for strict L ∞ quasi-isomorphims between nilpotent L ∞ -algebra.2. It is not true that MC reflects weak equivalences. See [29] for a description of a larger class of mor-phisms in c Lie [1] ∞ , properly containing weak equivalences, which are sent to weak homotopy equiva-lences by MC .It remains to verify that MC preserves fibrations and pullbacks of fibrations. We split the work acrossthe next two subsections. MC preserves fibrationsProposition 7.8. Let Φ : ( L, d, Q ) ։ ( L ′ , d ′ , Q ′ ) be a fibration in c Lie [1] ∞ . Then the simplicial map MC (Φ) : MC ( L ) → MC ( L ′ ) is a Kan fibration.Proof. By Prop. 6.4, it suffices to consider the case when Φ = Φ is a strict fibration in c Lie [1] ∞ . Applyingthe functor tow : c Lie [1] ∞ → tow ( c Lie [1] bdflt ∞ ) gives us a morphism, as in (5.19), between towers of boundedfiltered L ∞ -algebras, in which all horizontal and all vertical morphisms are fibrations in c Lie [1] bdflt/str ∞ . Itthen follows from Prop. 7.2 that we obtain a map of towers in KanCplx in which all horizontal and all verticalmaps are Kan fibrations · · · MC ( L/ F n L ) MC ( L/ F n − L ) · · ·· · · MC ( L ′ / F n L ′ ) MC ( L ′ / F n − L ′ ) · · · φ ( n ) φ ( n − ρ ( n ) ρ ′ ( n ) ρ ( n − ρ ′ ( n − ρ ( n − ρ ′ ( n − (7.1)38bove, for the sake of brevity, we have used the notation φ ( k ) := MC (Φ ( k ) ) and ρ ( k ) := MC ( p ( k ) ) . Sinceall L ∞ -morphisms involved are strict, we have φ ( k ) = (Φ ( k ) ) ⊗ id Ω • , ρ ( k ) = p ( k ) ⊗ id Ω • , ρ ′ ( k ) = p ′ ( k ) ⊗ id Ω • . (7.2)We will use Prop. 3.8 to show that lim ←− φ ( k ) : MC ( L ) → MC ( L ′ ) is a Kan fibration. Let n ≥ . Fromdiagram 3.4 from Prop. 3.8, we see that it suffices to prove that the map ( ρ ( n ) , φ ( n +1) ) : MC ( L/ F n +1 L ) → MC ( L/ F n L ) × MC ( L ′ / F n L ′ ) MC ( L ′ / F n +1 L ′ ) (7.3)is a fibration. So suppose we have a horn γ : Λ mk → MC ( L/ F n +1 L ) and commuting diagrams of simplicalsets Λ mk MC ( L/ F n +1 L )∆ m MC ( L/ F n L ) γβ ρ ( n ) Λ mk MC ( L/ F n +1 L )∆ m MC ( L ′ / F n +1 L ′ ) γβ ′ φ ( n +1) ∆ m MC ( L ′ / F n +1 L ′ ) MC ( L/ F n L ) MC ( L ′ / F n L ′ ) β ′ β φ ( n ) ρ ′ ( n ) (7.4)To complete the proof, we will construct an m -simplex ˜ α : ∆ m → MC ( L/ F n +1 L ) filling γ and satisfying ρ ( n ) ˜ α = β, φ ( n +1) ˜ α = β ′ . (7.5)To begin with, since ρ ( n ) is a fibration, there exists α : ∆ m → MC ( L/ F n +1 L ) that fills γ and satisfies ρ ( n ) α = β . But, in general, φ ( n +1) α = β ′ . By definition, α ∈ MC( L/ F n +1 L b ⊗ Ω m ) . Proposition 5.13implies that ( L/ F n +1 L, d ( n +1) , Q ( n +1) ) is bounded filtered at n + 1 . Hence, by Prop. 5.14, the L ∞ -algebra ( L/ F n +1 L ⊗ Ω m , d ( n +1)Ω , Q ( n +1)Ω ) is also bounded filtered at n + 1 , and therefore already complete. So we may consider α as a vector in ( L/ F n +1 L ⊗ Ω m ) . Similarly, we may consider φ ( n +1) ( α ) and β ′ as vectors in ( L ′ / F n +1 L ′ ⊗ Ω m ) . Let η ∈ ( L ′ / F n +1 L ′ ⊗ Ω m ) denote the vector η := β ′ − φ ( n +1) ( α ) . From diagrams (7.1) and (7.4) we deduce that η ∈ ker( p ′ ( n ) ⊗ id Ω m ) ∼ = F n L ′ / F n +1 L ′ ⊗ Ω m . Next, recall from Prop. 5.13, that the cochain complex ( F n L ′ / F n +1 L ′ , d ′ ( n +1) ) is an abelian sub- L ∞ -algebra of ( L ′ / F n +1 L ′ , d ′ ( n +1) , Q ′ ( n +1) ) . We claim that η ∈ MC m (cid:0) F n L ′ / F n +1 L ′ (cid:1) ∼ = Z (cid:16) F n L ′ / F n +1 L ′ ⊗ Ω m , d ′ ( n +1)Ω (cid:17) . Indeed, ( L ′ / F n +1 L ′ ⊗ Ω m , d ′ ( n +1)Ω , Q ′ ( n +1)Ω ) is an L ∞ -algebra bounded filtered at n + 1 and η has filtrationdegree n . Therefore Lemma 5.17 implies that curv ′ ( n +1)Ω ( β ′ ) = curv ′ ( n +1)Ω (cid:0) φ ( n +1) ( α ) + η (cid:1) = curv ′ ( n +1)Ω (cid:0) φ ( n +1) ( α ) (cid:1) + d ′ ( n +1)Ω η. Hence, d ′ ( n +1)Ω η = 0 , since β ′ and φ ( n +1) ( α ) are Maurer-Cartan elements.39urthermore, since the diagrams from (7.4) imply that both β ′ and φ ( n +1) ( α ) fill the horn φ ( n +1) γ , the m -simplex η fits into the commutative diagram of Kan complexes Λ mk MC ( F n L/ F n +1 L )∆ m MC ( F n L ′ / F n +1 L ′ ) η φ ( n +1) | F n L (7.6)In the above diagram, the restriction φ ( n +1) | F n L of φ ( n +1) to the fiber MC ( F n L/ F n +1 L ) is, equivalently,the Maurer-Cartan functor applied to the strict morphism of abelian L ∞ -algebra F n (Φ ) ( n +1) : (cid:0) F n L/ F n +1 L, d ( n +1) (cid:1) → (cid:0) F n L ′ / F n +1 L ′ , d ′ ( n +1) (cid:1) . Since Φ = Φ is a fibration, the linear map F n Φ : F n L → F n L ′ is surjective. Therefore F n (Φ ) ( n +1) is afibration in c Lie [1] bdflt/str ∞ , and so, by Prop. 7.2 we conclude that φ ( n +1) | F n L is a Kan fibration.Finally, let θ : ∆ m → MC ( F n L/ F n +1 L ) be a lift of η through the fibration φ ( n +1) | F n L in diagram (7.6)such that θ | Λ mk = 0 . Define ˜ α ∈ ( L/ F n +1 L ⊗ Ω m ) to be the vector ˜ α := α + θ. We claim that ˜ α ∈ MC( L/ F n +1 L ⊗ Ω m ) . Indeed, since α is a Maurer-Cartan element and since θ ∈ Z ( F n L/ F n +1 L ⊗ Ω m , d ( n +1)Ω ) has filtration degree n in the filtered bounded L ∞ -algebra (cid:0) L/ F n +1 L ⊗ Ω m , d ( n +1)Ω , Q ( n +1)Ω (cid:1) , Lemma5.17 implies that curv ( n +1)Ω ( ˜ α ) = curv ( n +1)Ω ( α ) + d ( n +1)Ω θ = 0 . Hence, we have an m -simplex ˜ α : ∆ m → MC ( L/ F n +1 L ) which, by construction, fills γ and satisfies theequalities (7.5). This completes the proof. MC preserves pullbacks of fibrations Let F : ( L, Q ) ։ ( L ′′ , Q ′′ ) be a fibration in c Lie [1] ∞ and G : ( L ′ , Q ′ ) → ( L ′′ , Q ′′ ) be an arbitrary morphism.Let ( ˜ L, ˜ Q ) be the pullback of the diagram ( L ′ , Q ′ ) G −→ ( L ′′ , Q ′′ ) F ←− ( L ′ , Q ′ ) in c Lie [1] ∞ . The goal of thissection is to verify that MC ( ˜ L ) is the pullback of the diagram MC ( L ′ ) MC ( G ) −−−−→ MC ( L ′′ ) MC ( F ) ←−−−− MC ( L ′ ) in KanCplx .We consider the special case when F is a strict fibration, since, as in the proof of Cor. 6.6, the moregeneral case will follow almost automatically. So let F = F : ( L, Q ) ։ ( L ′′ , Q ′′ ) be a strict fibration in c Lie [1] ∞ . We have the pullback diagram (6.6) from Prop. 6.5: ( ˜ L, ˜ Q ) (cid:0) L, Q ) (cid:0) L ′ , Q ′ ) (cid:0) L ′′ , Q ′′ ) Pr HG Pr ′ H F We wish to show that for all m ≥ MC m ( ˜ L ) MC m ( L ) MC m ( L ′ ) MC m ( L ′′ ) MC m (Pr H ) MC m ( G ) MC m (Pr ′ H ) MC m ( F ) 40s a pullback diagram of sets. As mentioned in Remark 5.15, there is a natural isomorphism MC m ( L ) = MC( L b ⊗ Ω m ) ∼ = lim ←− n MC( L/ F n L ⊗ Ω m ) . Since projective limits commute with pullbacks, it suffices to prove the following: Proposition 7.9. Suppose F : ( L, Q ) ։ ( L ′′ , Q ′′ ) is a strict fibration in c Lie [1] bdflt ∞ and G : ( L ′ , Q ′ ) → ( L ′′ , Q ′′ ) is an arbitrary morphism. Let ( ˜ L, ˜ Q ) be the pullback of the diagram ( L ′ , Q ′ ) G −→ ( L ′′ , Q ′′ ) F ←− ( L, Q ) in c Lie [1] ∞ as constructed in Prop. 6.5. Then for any B ∈ cdga , the commutative diagram of sets MC( ˜ L ⊗ B ) MC( L ⊗ B )MC( L ′ ⊗ B ) MC( L ′′ ⊗ B ) (Pr H ) B ∗ G B ∗ (Pr ′ H ) B ∗ F B ∗ is a pullback diagram.Proof. We first fix some notation. Let P ⊆ ( L ′ ⊗ B ) × ( L ⊗ B ) denote the pullback of the diagram ( L ′ ⊗ B ) G B ∗ −−−→ ( L ′′ ⊗ B ) F B ∗ ←−− ( L ⊗ B ) in the category of sets, and let P MC denote the pullback of the diagram MC( L ′ ⊗ B ) G B ∗ −−−→ MC( L ′′ ⊗ B ) F B ∗ ←−− MC( L ⊗ B ) . We have P MC ⊆ MC (cid:0) ( L ′ ⊗ B ) ⊕ ( L ⊗ B ) (cid:1) ∼ = MC (cid:0) ( L ′ ⊕ L ) ⊗ B (cid:1) . From the L ∞ -morphism(6.5), we obtain the function H B ∗ : MC( ˜ L ⊗ B ) → MC (cid:0) ( L ′ ⊕ L ) ⊗ B (cid:1) . The commutativity of diagram(6.6) from Prop. 6.5, we deduce that H B ∗ factors through the pullback: H B ∗ : MC( ˜ L ⊗ B ) → P MC . To complete the proof, we will construct an inverse to H B ∗ . Let ϕ : P → ( L ′ ⊗ B ) ⊕ ( L ⊗ B ) denote thefunction ϕ ( a ′ , a ) := (cid:0) a ′ , a − ( s ⊗ id B ) ◦ G B ∗ ( a ′ ) (cid:1) . Hence, ϕ is the restriction of J B ∗ to P ⊆ ( L ′ ⊕ L ) ⊗ B , where J B ∗ is the polynomial function induced fromthe coalgebra map J : ¯ S ( L ′ ⊕ L ) → ¯ S ( L ′ ⊕ L ) defined in (6.3). Since J is the inverse to H , in order to showthat ϕ | P MC is the inverse to H B ∗ , it is sufficient to verify that im ϕ | P MC ⊆ MC( ˜ L ⊗ B ) .We first observe that im ϕ ⊆ ( ˜ L ⊗ B ) . Indeed, since F is strict, we have F B ∗ = F ⊗ id B . Therefore, if ( a ′ , a ) ∈ ( L ′ ⊗ B ) × ( L ⊗ B ) such that F B ∗ ( a ) = G B ∗ ( a ′ ) , then it follows that a − ( s ⊗ id B ) ◦ G B ∗ ( a ′ ) ∈ ker( F ⊗ id B ) = ker( F ) ⊗ B . Hence ϕ ( a ′ , a ) ∈ ( ˜ L ⊗ B ) .To finish the proof, suppose ( α ′ , α ) ∈ P MC . We will show that g curv B ( ϕ ( α ′ , α )) = 0 , where g curv B is thecurvature function of ( ˜ L ⊗ B, e Q B ) . From Eq. 5.35 we see that it is sufficient to prove that e Q B (cid:0) exp( ϕ ( α ′ , α )) − (cid:1) = 0 . As mentioned above, by definition, we have ϕ ( α ′ , α ) = J B ∗ ( α ′ , α ) . Therefore Eq. 5.36 in Remark5.36 implies that exp( ϕ ( α ′ , α )) − J B (exp( α ′ , α ) − . On the other hand, from the definition of e Q inEq. 6.4, we have e Q B = J B ◦ Q ⊗ B ◦ H B , where Q ⊗ B is the L ∞ -structure on the product ( L ′ ⊕ L ) . Therefore,since HJ = id ¯ S ( L ′ ⊕ L ) , we obtain e Q B (cid:0) exp( ϕ ( α ′ , α )) − (cid:1) = e Q B J B (exp( α ′ , α ) − 1) = J B Q ⊗ B (exp( α ′ , α ) − Applying Eq. 5.35 to Q ⊗ B (exp( α ′ , α ) − then gives us e Q B (cid:0) exp( ϕ ( α ′ , α )) − (cid:1) = J B (cid:16) exp( α ′ , α ) curv ⊗ B ( α ′ , α ) (cid:17) where curv ⊗ B is the curvature function for ( L ′ ⊕ L ) ⊗ B . And finally, since ( α ′ , α ) ∈ P MC , we have curv ′ B ( α ′ ) = curv B ( α ) = 0 , and hence curv ⊗ B ( α ′ , α ) = 0 , where curv ′ B , curv B , and curv ⊗ B are thecurvature functions for L ′ ⊗ B, L ⊗ B , respectively. Therefore, we conclude that e Q B (cid:0) exp( ϕ ( α ′ , α )) − (cid:1) = 0 ,and this completes the proof. 41 eferences [1] I. Barnea, Y. Harpaz, and G. Horel, Pro-categories in homotopy theory, Algebr. Geom. Topol. (2017),179–189.[2] R. Bandiera, Descent of Deligne-Getzler ∞ -groupoids. Available as arXiv:1705.02880[3] A. Berglund, Rational homotopy theory of mapping spaces via Lie theory for L ∞ -algebras, HomologyHomotopy Appl. (2), 343–369 (2015)[4] K. S. Brown, Abstract homotopy theory and generalized sheaf cohomology, Trans. Amer. Math. Soc. (1973), 419–458.[5] W. Chachólski and J. Scherer, Homotopy theory of diagrams, Mem. Amer. Math. Soc. (2002),no. 736, x+90 pp.[6] C. Di Natale, Derived moduli of complexes and derived Grassmannians, Appl. Categ. Structures (2017), no. 5, 809–861.[7] V. Dotsenko, S. Shadrin and B. Vallette, The twisting procedure. Available as arXiv:1810.02941.[8] V. Dolgushev, A. Hoffnung, and C. Rogers, What do homotopy algebras form? Adv. Math. , 562–605(2015).[9] V. Dolgushev and C. Rogers, A version of the Goldman-Millson theorem for filtered L ∞ -algebras. J.Algebra , 260–302 (2015).[10] V. A. Dolgushev and C. L. Rogers, On an enhancement of the category of shifted L ∞ -algebras, Appl.Categ. Structures (2017), no. 4, 489–503.[11] B. Fresse, Homotopy of operads and Grothendieck-Teichmüller groups. Part 1 , Mathematical Surveysand Monographs, 217, American Mathematical Society, Providence, RI, 2017.[12] B. Fresse, V. Turchin, and T. Willwacher, The rational homotopy of mapping spaces of En operads.Available as arXiv:1703.06123[13] E. Getzler, Lie theory for nilpotent L ∞ -algebras. Ann. of Math. (2) (1), 271–301 (2009).[14] P.G. Goerss and J.F. Jardine, Simplicial homotopy theory , Progress in Mathematics, vol. 174. BirkhäuserVerlag, Basel (1999).[15] W.M. Goldman and J.J. Millson, The deformation theory of representations of fundamental groups ofcompact Kähler manifolds, Inst. Hautes Etudes Sci. Publ. Math. (1988) 43–96.[16] V. Hinich, DG coalgebras as formal stacks. J. Pure Appl. Algebra , (2001), 209–250.[17] V. Hinich, Descent of Deligne groupoids. Internat. Math. Res. Notices. (1997), 223–239.[18] M. Hovey, Model categories , Mathematical Surveys and Monographs, 63, American Mathematical So-ciety, Providence, RI, 1999.[19] A. Lazarev, Maurer-Cartan moduli and models for function spaces, Adv. Math. , 296–320 (2013).[20] J. Lurie, Derived Algebraic Geometry X: Formal Moduli Problems. Available at Deformation theory of algebras and their diagrams , CBMS Regional Conference Series inMathematics, 116, Published for the Conference Board of the Mathematical Sciences, Washington, DC,2012.[22] J.P. Pridham, Unifying derived deformation theories, Adv. Math. (2010), no. 3, 772–826.[23] D. G. Quillen, Homotopical algebra , Lecture Notes in Mathematics, No. 43, Springer-Verlag, Berlin,1967.[24] D. Quillen, Rational homotopy theory, Ann. of Math. (2) (1969) 205–295.[25] D. Robert-Nicoud, Representing the deformation ∞ -groupoid, Algebr. Geom. Topol. (2019), no. 3,1453–1476.[26] C. Rogers, An explicit model for the homotopy theory of finite-type Lie n -algebras, Algebr. Geom.Topol. (2020), no. 3, 1371–1429.[27] C. Rogers, Homotopical properties of the simplicial Maurer-Cartan functor, in ,3–15, MATRIX Book Ser., 1, Springer, Cham.[28] C. Rogers and C. Zhu, On the homotopy theory for Lie ∞ -groupoids, with an application to integrating L ∞ -algebras, Algebr. Geom. Topol. (2020), no. 3, 1127–1219.[29] S. Schwarz, A Variation of the Goldman-Millson Theorem for filtered L ∞ - algebras (2019). Availableas arXiv:1909.06665.[30] N. Strickland, The model structure for chain complexes. Available ashttps://arxiv.org/abs/2001.089552001.08955.[31] B. Vallette, Homotopy theory of homotopy algebras. Ann. Inst. Fourier (2020), 683–738.[32] S, Yalin, Maurer–Cartan spaces of filtered L ∞ -algebras. J. Homotopy Relat. Struct11