Rational homotopy of non-connected spaces

Rational homotopy of non-connected spaces Urtzi Buijs 5th GeToPhyMa CIMPA summer school on Rational Homotopy Theory and its interactions Celebrating Jim Stashe and Dennis Sullivan for their respective 80th and 75th anniversaries. July 11-21, 2016 Rabat, Morocco http://algtop.net/geto2016 These notes are based on the papers \Algebraic models of non-connected spaces and homotopy theory of L algebras" [5] and \Lie models of simplicial sets and representability of the Quillen functor" [7] treated in the Sections 1 and 2 respectively. There is also an introductory section on rational homotopy theory to present the basic tools necessary for the rest of the notes. This introduction is mainly based on [11, 21]. 1 Introduction to rational homotopy theory In 1967 Daniel Quillen published \Homotopical algebra" [22] where the author presents a categorical framework in which make homotopy theory. Two years later, in 1969, he applies these ideas in the fundamental paper \Rational homotopy theory" [23]. 1.1 Homotopical algebra. Model categories Definition 1.1. A model category is a category C together with three distinguished classes of maps weak equivalences, brations,

1.1 Homotopical algebra. Model categories 2 cobrations, satisfying the following axioms: (1) (nite) limits and colimits exist in C. g (2) Given morphisms X f Y Z if two out of the maps f; g and g f are weak equivalences, then so is the third. (3) Suppose that f is a retract of g, i.e., there is a commutative diagram A X A ; f B Y B with the two horizontal compositions being identities; Then, if g is a weak equivalence, bration or cobration, then so is f. (4) The lifting problem A i B can be solved whenever i is a cobration, p is a bration, and at least one of i and p is a weak equivalence. (5) Every map f : X Y can be factored in two ways g X Y p f X i f Z p Y X j f W Y In order to dene the notion of homotopy in a model category we need to introduce some notation. Let C be a model category. If f : A C and g : B C are given maps, f + g : A B C is the unique map dened by the universal property of the coproduct. If h: C A and k : C B are given maps, (h; k): C A B is the unique map dened by the universal property of the product. Definition 1.2. Let be the initial object of C. An object X of C is cobrant if X is a cobration. Let e be the nal object in C. An object X of C is brant if X e is a bration. Definition 1.3. A cylinder (A I; @ 0 ; @ 1 ; ) of an object A is given by an object A I called a cylinder object and maps @ 0 A A I @1 A such that @ 0 = @ 1 = id A, is a weak equivalence and (@ 0 +@ 1 ) is a cobration.

1.1 Homotopical algebra. Model categories 3 Definition 1.4. A path structure (B I ; d 0 ; d 1 ; ) of an object B is given by an object B I called a path object and maps B d 0 B I B d 1 such that d 0 = @ 1 = id B, is a weak equivalence and (d 0 ; d 1 ) is a cobration. Definition 1.5. Let f; g : A B be two maps in C. A left homotopy from f to g is a map H : A I B such that H@ 0 = f, H@ 1 = g where (A I; @ 0 ; @ 1 ; ) is a cylinder on A. A @ 0 @ 1 A I A right homotopy from f to g is a map K : A B I such that d 0 K = f, d 1 K = g where (B I ; d 0 ; d 1 ; ) is a path structure on B. f g H B A f g B d 0 d 1 K B I If there exists a left (right) homotopy between f and g, we call f and g left (right)homotopic. Proposition 1.6. Let f; g : A B be two maps in C with A cobrant and B brant. If f and g are left (right) homotopic, a left (right) homotopy exists for any cylinder (path structure). Moreover, f and g are left homotopic if and only if they are right homotopic. Definition 1.7. We call this relation homotopy and write f g. equivalence relation on the set of maps on C from A to B. This is an Definition 1.8. The cstegory hoc has Objects: brant and cobrant objects of C. Maps: homotopy classes of maps of C. Quillen also introduce the homotopy category of a model category C, denoted by HoC, obtained from C by formally inverting all weak equivalences: Objects: Objects of C. Maps: Maps of C plus formal inverses of any weak equivalence. Theorem 1.9. The canonic functor C HoC induce an equivalence of categories hoc HoC:

1.2 Commutative differential graded algebras 4 Definition 1.10. A Quillen adjunction between two model categories C and D is an adjunction C F G (i.e., hom C (F Y; X) = hom D (Y; GY ) natural for all variables X in C and Y in D.) such that the left adjoint F preserves cobrations and the right adjoint G preserves brations. Every Quillen adjunction induces an adjunction between the associated homotopy categories HoC LF RG D HoD and the Quillen adjunction is called a Quillen equivalence if this induced adjunction is an equivalence of categories. 1.2 Commutative differential graded algebras Definition 1.11. A graded algebra consists of a Z-graded vector space A = p Z A p together with a bilinear product A p A q A p+q which is associative. A dierential graded algebra is a graded algebra A endowed with a linear derivation d of degree +1 such that d 2 = 0. A commutative dierential graded algebra (CDGA from now on) is a differential graded algebra such a b = ( 1) a b b a; for any a; b A: Here a denotes the degree of a. An augmentation of CDGA's is a morphism of commutative dierential graded algebras ": (A; d) (Q; 0). The augmentation ideal of (A; ") is the kernel Ker", denoted by A. A morphism f : (A; ") (A ; " ) of augmented CDGA's is a degree 0 morphim s of CDGA's such that " = " f. An augmented CDGA (A; d) is n-connected (cohomologically n-connected) if A p = 0 for p n. (H(A; d) = 0 for p n). We denote by CDGA n (CDGA cn ) the associated categories. Example 1.12. (1) (Tensor product of graded algebras) If A and B are graded algebras, then A B is a graded algebra with multiplication (a b)(a b ) = ( 1) b a aa bb : Exercise 1. (i) Show that the above multiplication is associative,

1.2 Commutative differential graded algebras 5 (ii) Prove that with the dierential d(a b) = da b + ( 1) a A B is a dierential graded algebra. (iii) Show that if A and B are commutative then A B is a CDGA. (2) (Tensor algebra) For any graded vector space V, the tensor algebra T V is dened by T V = T k V; T k V = V V ; }{{} k=0 k where T 0 V = Q. Multiplication is given by a b = a b. Any linear map of degree zero from V to a graded algebra A extends to a unique morphism of graded algebras, T V A. Any degree k linear map V T V extends to a unique derivation of T V. (3) (Free commutative graded algebras) Let V be graded vector space. The elements v w ( 1) v w w v, where v; w V generate an ideal I T V. The quotient graded algebra V = T V=I is called the free commutative graded algebra on V. Exercise 2. Show that (V W ) = V W. (4) (The simplicial de Rham algebra) Dene, the simplicial de Rham algebra, as the simplicial CDGA with n-simplices n = (t 0; : : : ; t n ; dt 0 ; : : : ; dt n ) ( i t i 1; i dt i) ; t i = 0; dt i = 1: The face and degeneracy morphisms are the unique cochain algebra morphisms @ i : n+1 n and s j : n n+1 satisfying: t k if k < i; @ i : t k 0 if k = i; t k 1 if k > i; and t k if k < j; s j : t k t k + t k+1 if k = j; t k+1 if k > j: Theorem 1.13. [3] The category of commutative cochain algebras CDGA admits the structure of a model category where:

1.2 Commutative differential graded algebras 6 Weak equivalences: quasi-isomorphisms, i.e., morphisms ': (A; d) (A ; d ) such that the induced map in cohomology is an isomorphism. H ('): H (A; d) = H(A ; d ) Fibrations: surjective morphisms, Cobrations: morphisms i: A X satisfying the extension problem A i X B C In [13], Steve Halperin works with the category "-CDGA c0 of augmented CDGA's cohomologically connected. The dierence with the Bouseld-Gugenheim approach is that the cobrations can be explicitely determined by KS-extensions [21, xii.3. (3)]. Definition 1.14. Let A be a commutative cochain algebra. The spatial realization of A is the simplicial set There exists an adjunction A = Hom CDGA (A; ): SimpSet ( ) CDGA : The contravariant functor ( ): SimpSet CDGA takes rational homotopy equivalences to quasi-isomorphisms, and it takes inclusions of simplicial sets to surjections of cochain algebras. The contravariant functor : CDGA SimpSet takes cobrations of cochain algebras to rational Kan brations and it takes trivial cobrations to trivial - brations. Theorem 1.15. The spatial realization functor induces an equivalence of categories Ho(CDGA cof;f.t. 1 ) Ho(SimpSet f.t. 1 ) The left hand side denotes the homotopy category of cobrant, simply connected, nite type, commutative cochain algebras. The right hand side denotes the homotopy category of simply connected rational Kan complexes of nite Q-type.

1.3 Differential graded Lie algebras 7 1.3 Differential graded Lie algebras Throughout this notes we assume that Q is the base eld. Definition 1.16. A graded Lie algebra consists of a Z-graded vector space L = p Z L p together with a bilinear product called the Lie bracket denoted by [ ; ] such that [x; y] = ( 1) x y [y; x] and ( 1) x z [ x; [y; z] ] + ( 1) y x [ y; [z; x] ] + ( 1) z y [ z; [x; y] ] = 0: Here x denotes the degree of x. A dierential graded Lie algebra is a graded Lie algebra L endowed with a linear derivation @ of degree 1 such that @ 2 = 0. A chain Lie algebra is a DGL such that L = L 0, i.e. L n = 0 for n < 0. A chain Lie algebra is connected if L = L 1. The category of dierential graded Lie algebras and its morphisms will be denoted by DGL and the subcategory of connected DGL's by DGL 1. Example 1.17. (1) Every chain complex L can be seen as a DGL with [ ; ] = 0. We call these DGL's abelian. (2) Every associative dierential graded algebra A has a DGL structure A Lie with the commutator Lie bracket [a; b] = ab ( 1) a b ba: (3) (Free Lie algebras) For any graded vector spave V, the free Lie algebra L(V ) is characterized by the following universal property: there is a natural inclusion of graded vector spaces V, L(V ) such that any morphism of graded vector spaces V L into a graded Lie algebra extends uniquely to a morphism of graded Lie algebras L(V ) L V L(V ) f f L: The free Lie algebra L(V ) may be constructed as follows. Consider the free associative tensor algebra on V T (V ) = k 0(V k ); V 0 = Q: T (V ) becomes a Lie algebra with the commutator Lie bracket. Then the free Lie algebra L(V ) is the smallest sub Lie algebra of T (V ) Lie that contains V T (V ). (4) (Products and coproducts) If L and M are DGL's, then the direct sum L M becomes a DGL with the coordinate-wise structure: (L M) n = L n M n ;

1.3 Differential graded Lie algebras 8 [ (x; y); (x ; y ) d(x; y) = (dx; dy); ] = The direct sum together with the projections L L M ([x; x ] L ; [y; y ] M ) : M represent the product of L and M in the category DGL. If L and M are DGL's the free product L M = L(L M)=J where J is the ideal generated by elements of the form [x; y] L [x; y] L ; x; y L; [z; w] L [z; w] M ; x; y M: The free product together with the inclusions L L M M represent the coproduct of L and M in the category DGL. Exercise 3. Show that if L = L(V ) and M = L(W ) are free as Lie algebras, then L M = L(V W ): Example 1.18. (Tensor product of a DGL and a commutative dierential graded algebra). Let (L; @) be a DGL and (A; d) be a commutative dierential graded algebra. Exercise 4. Show that the tensor product L A inherits a natural DGL structure with dierential and Lie bracket given by: D(x a) = @x a + ( 1) x x da; [x 1 a 1 ; x 2 a 2 ] = ( 1) a1 x2 [x 1 ; x 2 ] a 1 a 2 : Proposition 1.19. DGL 1 is a model category. Weak equivalences: quasi-isomorphisms, i.e., morphisms : (L; @) (L ; @ ) such that the induced map in homology H ( ): H(L; @) = H(L ; @ ) is an isomorphism. Fibrations: surjective morphisms Cobrations: morphisms i: A X satisfying the extension problem A i X B C

1.3 Differential graded Lie algebras 9 Every object in DGL 1 is brant and the cobrant objects are the free Lie algebras. We can characterize the cobrations more explicitely: Definition 1.20. A KQ-extension (Koszul-Quillen) is a sequence L L L(V ) = L L(V ) where the vertical map is an isomorphism of graded Lie algebras (no dierential!). Whenever L is a free Lie algebra L = L(W ) a KQ-extension is a sequence where ( ) L(W ); d ( ) L(W V ); D ( ) L(V ); @ Dw = dw; w W; and Dv = @v + ; v V; L + (W ) L(V ): Definition 1.21. An object (L; @) of DGL 1 is minimal if and only if: (i) L is isomorphic as Lie algebras to a free Lie algebra L = L(V ). (ii) The dierential of the generators of the free Lie algebra has no linear term @(V ) L 2 (V ). Definition 1.22. Let (L ; @ ) DGL 1. If : (L; @) (L ; @ ) is a quasiisomorphism we say that (L; @) is DGL-model of L. If L = L(V ) we say that it is a Quillen model and if it is minimal that it is a minimal Quillen model. Theorem 1.23. Every object in DGL 1 has a minimal Quillen model We can describe explicitely the homotopy in DGL 1 by means of a path structure and a cylinder. Path structure. Consider the free commutative dierential graded algebra (t; dt) where t = 0, dt = 1 with d(t) = dt. Let (L; @) be an object in DGL 1. Consider the DGL product described in example 1.3. Consider the projections (M; d) = (L; @) (t; dt) p 0 ; p 1 : (L; @) (t; dt) (L; @); characterized by p 0 (t) = 0; p 1 (t) = 1;

1.3 Differential graded Lie algebras 10 and the canonical injection `: (L; @) (L; @) (t; dt): Dene the DGL (L; @) I by M i if i > 0; (L I ) i = Z(M 0 ) if i = 0; 0 if i < 0; Then, ( ) (L; @) I ; p 0 ; p 1 ; ` is a path structure on (L; @). Then f; g : (L ; @ ) (L; @) are homotopic if there exists such that (L ; @ ) f g (L; @) (t; dt) p 0 p 1 (L; @) commutes. Definition 1.24. Let f : (L; @) (L ; @ ) be a DGL morphism, ) ((L(V ); d ); Quillen models of L and L. Then, ( (L(V ); d); ) and f : (L(V ); d) (L(V ); d ) is a Quillen model of f (relative to and ) if the following diagram is commutative up to homotopy (L; @) (L(V ); d) f f (L ; @ ) (L(V ); d ): If (L(V ); d) and (L(V ); d ) are minimal then f is a minimal Quillen model of f. ( ) Proposition 1.25. Let f : (L; @) (L ; partial ) be a map in DGL 1. (L(V ); d); ) and ((L(V ); d ); two Quillen models of (L; @) and (L ; @ ) respectively. Then: (i) There exists a Quillen model f of f relative to these models. (ii) Two Quillen models f and ĝ of f are homotopic. (iii) If f; g : (L(A); @) (L ; @ ) have as Quillen models f and ĝ respectively, then f g if and only if f ĝ.

1.3 Differential graded Lie algebras 11 Definition 1.26. (L; @) and (L ; @ ) have the same homotopy type if they have a common Quillen model (L; @) (L(V ); @) (L ; @ ): In [23] Quillen denes a sequence of adjunctions G SimpSet 1 SGp 1 W Q G SCHA 1 P Û SLA 1 N N DGL 1 We briey explain the elements of this composition following the main source [23]: (i) (ii) (iii) (iv) G SimpSet 1 SGp 1 W If K is a reduced simplicial set, GK is the simplicial group constructed by Kan [14] playing the role of the loop space of K. If G is a simplicial group W G is the simplicial set which acts as its \classifying space". [14, 1, 8] Q SGp 1 SCHA 1 G If G is a group then QG is the complete Hopf algebra obtained by completing the group ring QG by the powers of its augmentation ideal. If R is a complete Hopf algebra, then GR is its group of group-like elements. These functors are extended dimension-wise to simplicial groups and simplicial CHA's and denoted by the same letters. SCHA 1 P SLA 1 Û If G is a Lie algebra over Q, ÛG is the CHA obtained by completing the universal enveloping algebra Ug by powers of its augmentation ideal. If R is a CHA, then PR is its Lie algebra of primitive elements. These functors are applied dimension-wise to simplicial objects. N SLA 1 DGL 1 N If L is a simplicial Lie algebra, its complex of normalized chains NL is a DGL with bracket dened by means of the Eilenberg-Zilber map. N is the left adjoint of N.

1.4 The bridge between DGL s and CDGA s 12 The composition NP Q G is called the Quillen functor. Theorem 1.27. The above sequence of adjunctions induce an equivalence of categories Ho(DGL cof 1 ) Ho(SimpSet Q;1 ) The left hand side denotes the homotopy category of cobrant, 1-connected dierential graded Lie algebras. The right hand side denotes the homotopy category of simply connected rational Kan complexes. 1.4 The bridge between DGL s and CDGA s In the fundational paper, Quillen gave another adjunction between dierential graded Lie algebras and cocommutative dierential graded coalgebras. Let dene rst the necessary objects. A graded coalgebra C is a graded vector space C together with two linear maps of degree 0: A comultiplication : C C C and an augmentation : C Q such that ( id) = (id ) y (id ) = ( id) = id C, i.e., the following diagrams are commutative C C C C C id id C C C C C C C C id id C id C Q C Q C A morphism ': C C of graded coalgebras is a linear map of degree 0 such that (' ') = ' and = '. A graded coalgebra is cocommutative if =, where : C C C C is the involution a b ( 1) a b b a. A graded coalgebra is coaugmented by the choice of an element 1 C 0 such that (1) = 1 and (1) = 1 1. Given such a choice, the above relations imply that for a Ker, a (a 1 + 1 a) Ker Ker : An element, a, in a coaugmented coalgebra is called primitive if a Ker and a = a 1 + 1 a. The primitive elements constitute a graded subspace of Ker, and a morphism of augmented graded coalgebras send primitive elements to primitive elements. For a coaugmented graded coalgebra we write C = Ker and therefore C = Q C. A coderivacion of degree k in a graded coalgebra C is a linear map : C C of degree k such that = ( id + id ) y = 0.

1.4 The bridge between DGL s and CDGA s 13 A dierntial graded coalgebra (CDGC from now on) is a graded coalgebra C together with a dierential which is a coderivation on C. If C is a graded coalgebra, then C = Hom(C; Q) is a graded algebra with multiplication dened by (f g)(c) = (f g)( c); f; g Hom(C; Q); c C and with identity given by the map : C Q. If (C; d) is a CDGC, then C = Hom(C; Q) es un graded dierential algebra. Example 1.28. (1) (primitively cogenerated coalgebras) The reduced comultiplication : C C C is dened by c = c (c 1 + 1 c). Its kernel is the graded subspace generated by the primitive elements. Write now (0) = id C, (1) = and dene the n-th reduced diagonal (n) = ( id id) (n 1) : C C C (n + 1 factors). We say that C is primitively cogenerated if C = n Ker (n). (2) The main example for us of a cocommutative graded coalgebra primitively cogenerated is V which comultiplication id the unique morphism of algebras such that (v) = v 1 + 1 v, v V. It is augmented by : + V 0, 1 1 and coaugmented by Q = 0 V. It is trivially cocommutative. Exercise 5. Show that V is primitively cogenerated (3) Among the cocommutative graded coalgebras primitively cogenerated, V has an important universal property. Let : + V V be the surjective linear map dened by a a 2 V. Lemma 1.29. If C = Q C is a cocommutative graded coalgebra primitively cogenerated, then for any linear map of degree 0: f : C V lifts to a unique morphism of grade coalgebras ': C V such that ' C = f. Theorem 1.30. [23] There is an adjunction DGL C L CDGC Let (C; ; ) be a cocommutative dierential graded coalgebra. L is dened by L(C) = (L(s 1 C); @ = @ 1 + @ 2 ); where @ 1 (s 1 c) = s 1 c; @ 2 (s 1 c) = 1 ( 1) ai [s 1 a i ; s 1 b i ]; 2 where c C and c = i a i b i. i

2 L -algebras and the cochain functor 14 Let (L; [ ; ]; @) be a dierential graded Lie algebra. Eilenberg-Chevalley construction, is dened by C, the Cartan- C (L) = ( sl; = 1 + 2 ); where k 1 (sx 1 sx k ) = (1) n i sx 1 s@(x i ) sx k i=1 2 (sx 1 sx k ) = i<j (1) sxi +n ij s[x i ; x j ] sx 1 : : : ŝx i : : : ŝx j sx k where the signs are given by n i = j<i sx j, and sx 1 sx k = ( 1) n ij sx i sx j sx 1 : : : ŝx i : : : ŝx j sx k : In fact, with the appropriate restrictions we have an equivalence between the corresponding homotopy categories of DGL's and CDGC's. Moreover, if we consider nite type DGL's and nite type CDGC's we can connect DGL's with CDGA's C DGL f.t. C CDGC f.t. ( ) L ( ) CDGA f.t. where the composition C = ( ) C is called the cochain functor. The whole picture can be summarized in the following diagram: Necessary: 1-connected. Non necessary: nite type. G SimpSet 1 SGp 1 W ^K G SCHA 1 P ^U SLA 1 N N DGL 1 L C CDGC 1 ( ) ( ) SimpSet 1 A P L CDGA 1 ; Necessary: nite type. Non necessary: 1-connected (nilpotent is good enough). 2 L -algebras and the cochain functor The main goal of the present section is to extend the denition of the cochain functor of dierential graded Lie algebras to the more general setting of (nonbounded) L -algebras. Then, using the Sullivan realization functor dene a geometrical realization for L -algebras.

2 L -algebras and the cochain functor 15 Recall from the previous section that there is a functor DGL f.t. 1 C CDGA f.t. 1 : If (L; @) DGL f.t. 1, then C (L) = Hom(C (L); Q) = C (L) (1) This cochain CDGA turns out to be in fact a Sullivan algebra Lemma 2.1. If (L; @) is a connected chain Lie algebra and each L i is nite dimensional, then : (sl) = C (L) is an isomorphism of graded algebras which exhibits C (L) as a Sullivan algebra Proof. Exercise 6. check all the details of the proof given in ref In fact, we can describe the dierential of the cochain algebra C (L) in terms of the dierential and the Lie bracket of the DGL (L; @). Proposition 2.2. If (L; @) is a connected chain Lie algebra of nite type then: (1) C (L) = ( V; d) with V and sl dual graded vector spaces. (2) d = d 1 + d 2 is the sum of its linear and quadratic parts where Proof. d 1 v; sx = ( 1) v v; s@x ; x L; v V d 2 v; sx sy = ( 1) y +1 v; s[x; y] ; x; y L; v V: Exercise 7. Check carefully all the details of the proof given in The main goal of the present section is to extend the denition of the cochain functor to the setting of L -algebras. An L -algebra on a graded vector space L is a collection of degree k 2 linear maps `k : L k L, for k 1, satisfying the following two conditions: (i) For any permutation of k elements, `k(x (1) ; : : : ; x (k) ) = `k(x 1 ; : : : ; x k ); where is the signature of the permutation and is the sign given by the Koszul convention.

2 L -algebras and the cochain functor 16 (ii) The generalized Jacobi identity holds, that is ( 1) i(n i)`n i (`i(x (1) ; : : : ; x (i) ); x (i+1) ; : : : ; x (n) ) = 0; i+j=n+1 S(i;n i) where S(i; n i) denotes the set of (i; n i)-shues. Equivalently, and L -algebra structure on the graded vector space L can be seen as a dierential graded coalgebra structure on + sl, the cofree graded cocommutative coalgebra generated by the suspension of L. Exercise 8. [11, Lemma 22.2] Suppose g : V V is a linear map of some arbitrary degree, k 1. Dene g : V V by g (v 1 v n ) = ±g(v i1 v ik ) v 1 : : : î1 : : : îk v n : (2) 1 i 1< <i k n where means delete and the sign ± is given by v 1 v n = ±v i1 v ik v 1 : : : î1 : : : îk v n. Prove the following assertions: (1) g decreases wordlength by k 1. (2) g is a coderivation in V. (3) g is the unique coderivation that extends g and decreases wordlength by k 1. Exercise 9. Let : sl sl be a coderivation. Write = k 1 h k, where h k : k sl sl, k 1. Prove the following: (1) The \innite" sum = k 1 h k is well dened. (2) The collection of degree k 2 linear maps `k = s 1 h k s k : L k L; denes an L -algebra structure on the graded vector space L. (3) Reciprocally, if (L; {`k}) is an L -algebra. Then, the collection of linear maps h k = ( 1) k(k 1) 2 s `k (s 1 ) k : k sl sl denes a coderivation in the cofree coalgebra sl. If (L; {`k}) is an L algebra, we denote the dierential coalgebra structure on sl as C (L) by analogy with the Cartan-Eilenberg-Chevaley construction. Given two L -algebras L and L, a morphism of L -algebras is a cocommutative dierential graded coalgebra morphism f : C (L) C (L ):

2 L -algebras and the cochain functor 17 f is determined by f : sl sl which can be written as k 1 (f)(k). Note that, as before, the collection of linear maps {(f) (k) } k 1 is in one to one correspondence with a system {f (k) } of skew-symmetric maps of degree 1 k, where f (k) : L k L. Indeed, each f (k) is uniquely determined by (f) (k) as follows: f (k) = s 1 (f) (k) s k ; (f) (k) = ( 1) k(k 1) 2 s f (k) (s 1 ) k : Exercise 10. Write explicitely the equations that the linear maps f (1) f (2) must satisfy. and For which L -algebras (L; {`k} k 1 ) can we give an analogue of Proposition 2.2? Definition 2.3. An L -algebra (L; {`k} k 1 ) is mild if every bracket is locally nite, i.e. for any a L there are nite dimensional subspaces S k L k, k 1 which vanish for k 0 and such that ` 1 k a Ker`k S k. Definition 2.4. Given a mild L -algebra (L; {`k} k 1 ), choose a homogeneous basis {z i } of L and denote by V (sl), V = {v i }, where v i (sz r ) = r i. Then dene C (L) = ( V; d); where the dierential satises d = k 1 d k ; d k (V ) k V d k v; sx 1 sx k = ± v; s`k(x 1 ; : : : ; x k ) : (3) Exercise 11. (1) Show that the free commutative dierential graded algebra C (L) is well dened when L is a mild L -algebra. (2) Prove that (L; {`k} k 1 ) is a mild L -algebra if and only if for each v i (xed) v i ; s`k(z j1 ; : : : ; z jk ) = 0 for almost all z j1 z jk L k, k 1. (3) We say that a free commutative graded algebra ( V; d) is mild if for each z j1 z jk L k (xed) we have for almost all v i V. d k v i ; z j1 z jk = 0 Show that the L -algebra (L; {`k} k 1 ) is mild if and only if the free CDGA C (L) is mild. (4) Give an example of a non-mild free CDGA ( V; d) and show that it can not be of the form C (L) for an L -algebra L.

2 L -algebras and the cochain functor 18 C ( ) does not dene a functor unless we also restircts the class of L - morphisms. Definition 2.5. An L -morphism : ( sl; ) ( sm; ) is mild if every (k) : k sl sm is locally nite, i.e. for any a M there is a nite dimensional subspace S k L k, k 1 with S k = 0, k 0, such that ( (k) ) 1 a = Ker (k) S k : Definition 2.6. If is a mild L -morphism, dene C (): C (M) = ( W; d) ( V; d) = C (L); with C () = k 1 C () k where C () k : W k V is given by C () k ; sx 1 sx k = ± w; s (k) (x 1 x k ) : (4) Exercise 12. (1) Rewrite the denition of a mild L -morphism in an analogue way as in the previous exercise. (2) Show that if C (M) = ( W; d) and C (L) = ( V; d) are mild CDGA's not all CDGA morphism ': ( W; d) ( V; d) can be written as C () for a mild L -morphism. (3) Dene properly the concept of a mild CDGA morphism verifying that is a mild L -morphism if and only if ' = C () is a mild CDGA morphism. We denote by L mild the category of mild L -algebras and mild L -morphisms. The next remarks are a discussion with examples of the implications between mildness and other conditions classically used for L -algebras Remark 2.7. Exercise 13. L. FINITE TYPE + BOUNDED MILD (1). Find a non-mild, nite type and bounded L -algebra (2) Find a mild, non-nite type and non-bounded L -algebra L. Remark 2.8. In general WHY NOT DEFINE C (L) = C (L)? ( sl) (sl) unless very strict restrictions are assumed. Recall by Lemma 2.1 that if V is a graded vector space of nite type, bounded and with V 0 = 0 then ( V ) = V. Exercise* 14. Show that if V is a graded vector space of nite type, bounded and with V 0 0, then ( V ) V.

2.1 The connected components in the CDGA setting 19 Remark 2.9. NILPOTENT MILD. The lower central ltration on an L -algebra is dened inductively by F 1 L = L; F i L = `k(f i1 L; : : : ; F i k L); i > 1: i 1+ +i k =i L is nilpotent if F i L = 0 for i 0. Exercise 15. (1) Compute ( V; d), the Sullivan minimal model of S 3 S 3 (until generators of degree 11). (2) Show that the associated L -algebra L such that C (L) = ( V; d) is not nilpotent. SimpSet as the composi- Finally dene the realization functor : L mild tion: L mild SimpSet C CDGA S 2.1 The connected components in the CDGA setting Let K be a simplicial set. Given a 0-simplex x 0 K 0, we dene the connected component of K containing x 0 as the simplicial subset K x0 of K where (K x0 ) q = {x K q @ q i @ i x = x 0 for any i }: Denote also by x 0 the point in K identied with the 0-simplex x 0, and by K x0 the component which contains the point x 0. We have then Lemma 2.10. (1) If K is a Kan complex, there exists an homotopical equiv- alence K x0 S ( K x0 ) which makes the following diagram commutative K x0 S ( K x0 ) In particular K x0 K x0. K S K : (2) Given a homotopical equivalence h: K L between simplicial sets, and x 0 K 0, then the restriction K x0 L hx0 is a homotopical equivalence whose geometric realization corresponds with the equivalence K x0 L hx0.

2.1 The connected components in the CDGA setting 20 Let ( V; d) be a CDGA where V = p Z V p. Its realization V; d S X is a non-necessarily connected simplicial set. ( ) Consider a 0-simplex of this simplicial set ' V; d S = Hom CDGA ( V; d); (A P L ) 0 = ) Hom CDGA (( V; d); Q. This 0-simplex represents a map, X; Figure 1 or equivalently a connected component. We can describe algebraically in terms of Sullivan algebras, the dierent connected components of the (non-connected) simplicial set X. Associated to the CDGA morphism ': ( V; d) Q we have in ( V; d) the ideal K ' generated by A 1 A 2 A 3 where A 1 = ( V ) <0 ; A 2 = d( V ) 0 ; A 3 = { '() ( V ) 0 }: Lemma 2.11. The ideal K ' agrees with the ideal K ' generated by A 1 A 2 A 3, where A 1 = V <0 ; A 2 = d(v 0 ); A 3 = {v '(v) v V 0 }: Proof. The inclusion K ' K ' is trivial since A i A i, i = 1; 2; 3. Let's check the inclusion K ' K '. If A 1, then + (V <0 ) V K '. Consider = '() A 3, with ( V ) 0. Write = a + b where a + (V <0 ) ( V ) and b W 0. Then, '() = a + b '(a) '(b) = a+b '(b). Since a A 1 K ' it only remains to show that b '(b) K '. We will suppose that b n V 0 and proceed by induction. If n = 1, then b V 0 and b '(b) A 3 K '. Suppose that b = b 1 b n with b i V 0, i = 1; : : : ; n. Then ) ( ) b '(b) = b 1 ((b 2 b n ) '(b 2 b n ) + '(b 2 b n ) b 1 '(b 1 ) ; which obviously belongs to K ' by inductive hypothesis. Finally, consider an element = d A 3 with ( V ) 0, and write = a + b + c where a + (V < 1 ) V, b (V 0 ) and c V 1 V.

2.1 The connected components in the CDGA setting 21 Clearly, da; db K '. We can write c as a sum of terms of the form c 1 c 2 with c 1 = 1. Then dc = (dc 1 c 2 c 1 dc 2 ). Thus, on the one hand c 1 dc 2 K ' trivially. On the other hand '(dc 1 ) = d('c 1 ) = 0 and therefore ( ) dc 1 c 2 = dc 1 '(dc 1 ) c 2 K ' Another important property of K ' is the following Lemma 2.12. K ' is a dierential ideal of ( V; d). Proof. By Lemma 2.11, in order to prove that dk ' K ', it suces to show that da i K ' = K ', i = 1; 2; 3. Let v A 1 = V <0. If v V 1, then dv ( V ) <0 K '. If w V 1, since '(dv) = d'(v) = 0, we have dv = dv '(dv) K '. If v A 3, then v = x '(x) for some x V 0 and dv = dx d'(x) = dx K '. Finally, da 2 = 0. It turns out that ( V; d)=k ' is isomorphic to a free CDGA. To prove that consider the ideal K ' generated only by A 1 A 3. Following the proof of Lemma 2.11 it is easy to see that this ideal agrees with the one generated by A 1 A 3. Lemma 2.13. The map : V 1 ( V ) 1 ( V= K ' ) 1 ; is an isomorphism of vector spaces. Proof. The map is clearly injecitve by denition of K'. Let [ ] ( V= K ' ) 1 and write = 0 + 1 + 2 where 0 + V <0 ( V ); 1 ( + V 0 ) V 1 ; 2 V 1 : In order to simplify the notation we will write the elemt 1 as 1 = where ( + V 0 ) and V 1. Then '() + 1 V 1 and ( ) '() + 1 = ['() + 1] = [ ]: Consider now the linear map @ : V 0 d ( V ) 1 ( V= K ' ) 1 1 V 1 ; and denote by V 1 a complement of the image of @, i.e. V 1 = @V 0 V 1. In what follows, if we have an element of the form = + V 0 V we will denote by =' the element '(). With this notation if dv = 0+ 1+ 2 where 0 + V <0 ( V ); 1 ( + V 0 ) V 1 ; 2 V 1 ; then @(v) = 1=' + 2. Then we have:

2.1 The connected components in the CDGA setting 22 Proposition 2.14. There exists a dierential d ' such that we have an isomorphism of CDGA's ( V=K ' ; d) = ( (V 1 V 2 ); d ' ): Proof. We rst dene a morphism of graded algebras : V (V 1 V 2 ) by 0 if v V <0 @V 0 ; (v) = '(v) if v V 0 ; v otherwise. This morphism is clearly surjective and we will show that its kernel is precisely K ' : In order to show that (K ' ) = 0 it is enough to see (A i ) = 0, i = 1; 2; 3. It is immediate that (A 1) = 0. For an element v '(v) A 3, we have v V 0 and then ( ) v '(v) = (v) '(v) = '(v) '(v) (1) = '(v) '(v) = 0: Finally, if v = 0, then the element dv A 2 can be written as dv = 0 + 1 + 2; 0 + V <0 ( V ); 1 ( + V 0 ) V 1 ; 2 V 1 ; or equivalently, dw = 0 + 1 1=' + 1=' + 2 = 0 + 1 1=' + @v; and we have (sw) = ( 0) + ( 1 1=') + (@v) = 0. Indeed, ( 0) = (@v) = 0 by denition. Now, if we write 1 = w with + V 0 and w V 1, then ( 1 1=') = ( w '() w ) = ( '() ) w = 0: On the other hand we will show that any element ker belongs to K ' : Write = 0 + 1 + 2 where 0 + (V <0 @V 0 ) V, 1 + V 0 (V 1 V 2 ) and 2 (V 1 V 2 ). Then, 0 = ( ) = ( 0 + 1 + 2) = ( 1) + 2 = 1=' + 2: Take a basis { i } in (V 1 V 2 ) and write 1 = i i ; i + V 0 ; 2 = i i ; i Q:

2.2 Points, augmentations and Maurer-Cartan elements 23 Thus, 0 = 1=' + 2 = '( i ) i + i i = ( '( i ) i ) i ; so we conclude that '( i ) = i and then 1 + 2 = ( ) i '( i ) i K '. Since 0 K ' by denition, we obtain = 0 + 1 + 2 K '. Therefore, induces an isomorphism of graded algebras making the following diagram commutative: V V=K ' = (V 1 V 2 ) Finally, we endow (V 1 V 2 ) with the dierential d ' = d 1. Exercise 16. Write an explicit formula for the dierential d' = d 1. Theorem 2.15. The projection ( V; d) ( V=K ' ; d) induces an homotopical equivalence V=K ' V ' making the following diagram commutative V ' V V=K ' V ; where stands for the Sullivan realization S. In conclusion, we have that the Sullivan algebra ( V 1 V 2 ; d ' ) is a model of the component of X = V S ': V Q. which contains the 0-simplex 2.2 Points, augmentations and Maurer-Cartan elements We have seen that augmentations ': ( V; d) Q represent maps as in Figure 1. The problem to \model" this gure with DGL's (or L -algebras) is that the morphisms represent base-point preserving maps and then the unique inclusion would be

2.2 Points, augmentations and Maurer-Cartan elements 24 Figure 2 So if we want to describe algebraically in the DGL/L -algebra setting the map of Figure 1 we need to factor the map through the inclusion of the singleton in S 0 Figure 3 Then if we want to \model" this gure, we need rst a model for S 0. Definition 2.16. A Maurer-Cartan element of an L -algebra L is an element k z L 1 for which `k( z; : : : ; z) = 0 for k suciently large and 1 k z; : : : ; z) = 0: k!`k( k 1 Observe that, whenever (L; @) is a DGL, i.e., an L -algebra such that `k = 0 for k 3, then z L 1 is a Maurer-Cartan element if @z = 1 [z; z]: 2 We will denote the set of Maurer-Cartan elements in L by MC(L). Exercise 17. (The DGL model of S 0 ). Consider the free diferential graded Lie algebra L(u) with u = 1 and with a dierential that makes u a Maurer-Cartan element @u = 1 2 [u; u]. (1) Show that the cochain functor on this DGL is C ( ) ( ) L(u); @ = (x; y); d ;

2.2 Points, augmentations and Maurer-Cartan elements 25 where x and y are generators of degrees 0 and 1 respectively, dx = 0 and dy = 1 2 (x2 x). ( ) (2) Show that the geometric realization of (x; y); d has the homotopy type of S 0. In the CDGA setting we can model the map Figure 4 By the CDGA morphism ( ) : (x; y); d Q; (x) = 1; (y) = 0: Then, ( if we dene ) a based augmentation of a given CDGA A by a morphism A (x; y); d it is clear that the composition of any based augmentation ( ) with : (x; y); d Q gives rise to a classical augmentation A Q. Conversely, we have the following. Lemma 2.17. Let ( V; d) be a free CDGA and let + x such that ( ) = ( 1. Then, ) any augmentation f : ( V; d) Q has a unique lifting f to (x; y); d such that, for any v V 0, f (v) = f(v) : ( ) (x; y); d f Q ( V; d) f Figure 5 Proof. For degree reasons we set f to be zero in V 1 and V 2. Let w V 1 and write dw = +, where + V 0 and + V 0 ( V ). Then,

2.2 Points, augmentations and Maurer-Cartan elements 26 f(dw) = f(). Write = p(v 1 ; : : : ; v n ) as a polynomial without constant term in the generators of V 0, and set i = f(v i ), for i = 1; : : : ; n. Then, On the other hand, p( 1 ; : : : ; n ) = fd(w) = df(w) = 0: f (dw) = p ( 1 ; : : : ; n ) = P (x); which is a polynomial in x without constant term, and it satises P (1) = p( 1 ; : : : ; n ) = 0. Hence P (x) = x(x 1)r(x) and we dene f (w) = 2yr(x) so that df (w) = f (dw). Finally, we check that, for any generator u V 2, f (du) = 0. Indeed, write f (du) = yq(x) whose dierential 1 2 (x2 x)q(x) has to vanish. Thus Q(x) = 0 and the lemma holds. Following the same spirit of algebraically describe Figure 3 in the DGL/L setting we have the following. Lemma 2.18. Let L be an L -algebra. Then, for any z L 1, there exists a unique L morphism : (L(u); @) L such that (1) (u) = z; (k) (u : : : u) = 0; k 2: Moreover, z MC(L) if and only if (k) ([u; u] u : : : u) = 0 for k large enough. Proof. Since (L(u); @) is the vector space spanned only by u and [u; u], with @u = 1 2 [u; u], an L morphism : L(u) L is simply a CDGC morphism, : ( (su; s[u; u]); ) ( sl; ); which is completely determined by the elements (k) (u : : : u); (k) ([u; u] u : : : u); k 1; satisfying the system referred in Exercise 10. In this particular case, if we set (1) (u) = z; (k) (u : : : u) = 0; k 2; and since `i = 0, for i 3 in (L(u); @), a direct computation shows that is indeed an L morphism if the following identities hold for any k 1, ( ) k `k(z; : : : ; z) = (k 1) ([u; u] u u) k 2 2 (k) ([u; u] u u); k j=1 ( ) k 1 ( ) `j (k j+1) ([u; u] u u); z; j 1 : : :; z = 0: (5) j 1

2.2 Points, augmentations and Maurer-Cartan elements 27 We will show that (k) ([u; u] u : : : u), satisfying the above identities, are uniquely determined by the formula (k) ([u; u] u u) = 2(k 1)! k i=1 First of all, for k = 1, the rst identity in (5) is simply Thus, we are forced to dene `1z = 1 2 (1) [u; u]: (1) [u; u] = 2`1(z); 1 i!`i(z; : : :; z): (6) as in (6). The second identity in (5) for k = 1 reads `1 (1) [u; u] = 0 which is trivially satised: `1 (1) [u; u] = 2`21(z) = 0: Assume the identities in (5) are satised for k 1 by setting formula (6) for integers smaller than k. Again, from the rst identity in (5) for k, we are forced to dene (k) ([u; u] u ::: u) = (k 1) (k 1) ([u; u] u ::: u) 2 k `k(z; :::; z): (7) Now, by the inductive hypothesis for k 1, this expression becomes k 1 1 2(k 1)(k 2)! : : : ; z) i!`i(z; 2 k `k(z; : : : ; z) = 2(k 1)! k i=1 i=1 1 i!`i(z; : : : ; z); which is precisely the equation (6) for k. To nish, we must check that the second identity in (5) for k, k ( ( ) k 1 j=1 j 1)`j (k j+1) ([u; u] u u); z; j 1 : : :; z = 0 holds. For it, replace in this equation (k j+1) ([u; u] u u) by its value on equation (7) above for k j + 1. This yields the following, in which we have avoid the sign for simplicity: k j=1 ( k 1 j 1)`j ( ((k j) (k j) ([u; u]u:::u) 2 k j+1`k j+1(z; :::; z) ) ; z; j 1 ::: ; z Then, this expression splits as k 1 ( ) k 2 ( ) (k 1) `j (k j) ([u; u]u u); z; j 1 : : :; z j 1 j=1 ) :

2.2 Points, augmentations and Maurer-Cartan elements 28 2 k k j=1 ( ) k `j(`k j+1 (z; : : : ; z); z; j 1 : : :; z)): j 1 By induction hypothesis the rst summand is zero as it is the second identity in (5) for (k 1). The second summand is also zero by the k th higher Jacobi identity on L. Now we prove the second assertion. If z MC(L), then there is an integer N such that `k(z; : : : ; z) = 0 for k N. Therefore, via equation (6), and for k N, (k) ([u; u] u u) = 2(k 1)! i=1 The converse is also trivially satised in light of (6). 1 i!`i(z; i : : :; z) = 0: Remark 2.19. Note that by the previous Lemma, any element z of degree 1 of a given L -algebra L can be written as i 1 1 i! (i) (u u) and thus, independently of any nitness or mildness assumption, Maurer-Cartan elements are not preserved in the standard fashion by L morphisms. Note also that, even for mild, the condition (k) ([u; u] u ) = 0 for k large enough is not automatically satised. Lemmas 2.17 and 2.18 are related by the following diagram: ) C (L(u) L(u) z L 1 ( ) (x; y); d f C () Q ( V; d) f L C (L); where the lower left corner is Lemma 2.17 and the upper right corner is Lemma 2.18. The hypothesis required to and f just translate into the fact that C () = f. In order to detect Maurer-Cartan elements at the cochain level, let L be a mild L -algebra and let {z j } j J and {v j } j J be basis of L 1 and V 0 respectively (see Denition 2.4). Then, any z L 1, written as z = j jz j, is obviously identied with the linear map V 0 Q sending v j to j for all j J. However, Maurer-Cartan elements of L are not, in general, those z for which this map can be extended as an augmentation of the cochains, i.e., as a CDGA morphism C (L) Q. The following exercises corroborates this assertion. Exercise 18. Show that if L is an innite-dimensional abelian L -algebra concentrated in degree 1, the sets MC(L) and ( ) Aug C (L) = Hom CDGA (C (L); Q) can not be one-to-one correspondant.

2.2 Points, augmentations and Maurer-Cartan elements 29 Exercise 19. Let L be the mild L algebra generated by B = {! i ; } i 2, with! i = 2, = 1, and where the only non zero brackets on generators are: `1() =! 2 ; `k(; : : : ; ) = k!(! k! k+1 ); k 2: (1) Show that C (L) = ( V; d) in which V is generated by {v; u i } i 2, with v = 0, u i = 1, dv = 0 and du i = v i v i 1 for i 2. (2) Show that the morphism C (L) Q sending u i to 0 for all i and v to 1 is a well dened augmentation, but is not a Maurer-Cartan element. Remark 2.20. In light of previous exercises, it is important to note that, if one considers non-nite type mild L algebras, the Maurer-Cartan set can not be identied with the set of augmentations from the cochain algebra and very special and technical restrictions are needed to have this identication. In the same way, in view of Lemma 2.18 and Remark 2.19, Maurer-Cartan elements are not preserved by mild L morphisms unless either nite type is assumed, or again, special restrictions are applied. Thus, hereafter, and again for the sake of clearness, we restrict the categroy L to the class of mild, nite type L -algebras, denoted by L f.t.. Corollary 2.21. Let L be an L -algebra of nite type. Then, an element z L 1 is Maurer-Cartan if and only if there exists a mild L morphism : (L(u); @) L such that (1) (u) = z and (k) (u : : : u) = 0 for k 2. Proof. If z MC(L) the morphism of Lemma 2.18 is obviously mild as (k) ([u; u] u : : : u) = 0 for k large enough. Conversely, if is a mild L morphism and L is of nite type, then (k) ([u; u] u : : : u) necessarily vanishes for k large. Definition 2.22. Let g : L L be a morphism in L f.t. the map MC(g): MC(L) MC(L ) by MC(g)(z) = k 1 1 k! g(k) (z z): and z MC(L). Dene In the next result we see that MC(g) is well dened. Moreover, with the niteness type assumptions in the above remark, we identify the Maurer-Cartan elements of L L f.t. in a functorial way with the set AugC (L) of augmentations of C (L). We stress here that, to our knowledge, the following result and the Corollary 2.24 that follows are not straightforward and do not follow at once by simply generalizing their classical DGL counterpart of [6, Remark 16] or [12, Proposition 1.1] (compare to [2, Lemma 2.3] or [10, Proposition 2.2]). Proposition 2.23. Let g : L L be a morphism in L f.t. and z MC(L). Then, MC(g)(z) is indeed a Maurer-Cartan element in L. That is, k 1 1 k! g(k) (z z) MC(L ):

2.2 Points, augmentations and Maurer-Cartan elements 30 Moreover, the functor is naturally equivalent to the functor MC: L f.t. Set Aug: L f.t. Set which assigns to g : L L the map Aug(g): AugC (L) AugC (L ) given by composition, Aug(g)(") = "C (g). Proof. We rst show that there is a natural bijection MC(L) = AugC (L): Choose a basis {z j } m j=1 of L 1, set C (L) = ( V; d) with V = (sl) ], and for each j denote by v j the element (sz j ) ] of V 0. Given z MC(L), write z = m j=1 jz j and apply Corollary 2.21 (recall that L is assumed to be of nite type) to obtain the mild L morphism : L(u) L for which (1) (u) = z, (n) (u u) = 0 for n 2. Then, since is mild, we can construct the based augmentation C (): ( V; d) ( (x; y); d) which sends each v j to j x. Therefore, the composition C (): C (L) Q is an augmentation denoted by " z. Conversely, consider any augmentation ": ( V; d) Q and set "(v j ) = j. Lift " via Lemma 2.17 to a based augmentation " x : ( V; d) ( (x; y); d). Then, observe that " x = C () for a mild L morphism : L(u) L in which (1) (u) = m j=1 jz j and (n) (u u) = 0 for n 2. Since L is of nite type, again by Corollary 2.21, the element z = m j=1 jz j is a Maurer-Cartan element of L. Thus, the correspondence z " z establishes the asserted bijection. Next, we prove the rst assertion of the proposition by showing that, given g : L L a morphism in L f.t., then is identied with MC(g): MC(L) MC(L ) Aug(g): AugC (L) AugC (L ): For it, let z MC(L). By the bijection MC(L) = AugC (L), the Maurer- Cartan element z corresponds to the augmentation in AugC (L) given by C () where : L(u) L is the mild L morphism, obtained via Corollary 2.21, corresponding to z MC(L). Applying Aug(g) to this augmentation we obtain, Aug(g) ( C () ) = C ()C (g) = C (g) Aug(L ):

2.2 Points, augmentations and Maurer-Cartan elements 31 We will prove that this augmentation corresponds, via again the bijection MC(L ) = AugC (L ), with the element 1 k! k g(k) (z z) L 1 which must be then a Maurer-Cartan element in L as stated. For it, we need to lift this augmentation C (g), via Lemma??, to a based augmentation " x : C (L ) ( (x; y); d). Observe that " x is, in general, far from being C (g) = C ()C (g). Indeed, although the image of C () on degree zero elements is linear on x, the image of C (g) may not be linear on degree zero elements. Let us then describe explicitly " x. Choose nite basis {z j } j J, {z i } i I of L 1 and L 1 respectively and write C (L) = W, C (L ) = V. Observe that W 0 and V 0 are generated by {w j } j J and {v i } i I where w j = (sz j ) ] and v i = (sz i )] for each i I and j J. If z = j jz j, then C (): C (L) ( (x; y); d) is dened on W 0 by C ()w j = j x. On the other hand, write C (g) = k 1 C (g) k with C (g) k V k W and set Then, C (g) k (v i ) = P ik + Q ik ; with P ik k W 0 and Q ik + W 0 W: C ()C (g) k (v i ) = C ()(P ik ) = P ik ( j ); where P ik ( j ) is the scalar obtained by evaluating the \polynomial" P ik on the j 's. Thus, " x is dened on V 0 as, " x (v i ) = k 1 P ik ( j )x; being this a nite sum due to the mildness assumption. Now that we have explicitly precised the lifting " x of the augmentation C (g), we need to identify the Maurer-Cartan element z that it represents. By the rst part of the present proof, this element is precisely, z = ( P ik ( j ) ) z i: i k On the other hand, an easy computation shows that C (g) k v i ; sz; : : : ; sz = k!p ik ( j ) which, in light of (3) of Section 1, let us conclude that Therefore, P ik ( j ) = 1 k! v i; sg (k) (z z) : z = i;k P ik ( j )z i = i;k 1 k! v i; sg (k) (z z) z i = k 1 k! g(k) (z z) and the proposition is proved.

2.3 The connected components in the DGL/L -algebra setting 32 Exercise 20. Show that, given an L algebra L and a commutative dierential graded algebra A, the tensor product L A inherits a natural L structure with brackets: `1(x a) = @x a + ( 1) x x da; `k(x 1 a 1 ; : : : ; x k a k ) = "`k(x 1 ; : : : ; x k ) a 1 : : : a k ; k 2; where " = ( 1) i>j x i a j is the sign provided by the Koszul convention. Corollary 2.24. Let L L f.t. Then, there is a bijection and A CDGA such that L A is of nite type. MC(L A) = Hom CDGA (C (L); A): Proof. Since L A is mild and of nite type, apply Proposition 2.23, to identify a given Maurer-Cartan element z of L A with an augmentation " z : C (L A) = (sl A) Q: This produces a degree zero linear map (sl) A which is extended to an algebra morphism C (L) A. A straightforward computation shows that it commutes with dierential since " z does. Conversely, any CDGA morphism C (L) A gives rise, by the procedure above, to an augmentation C (L A) Q. It is important also to observe that if L A fails to be of nite type, and even if L and A are, MC(L A) is no longer identied with the set of morphisms Hom CDGA (C (L); A) as shown in the following exercise. In the general case, as in Remark 2.20, it is necessary to impose technical niteness restrictions in the class of morphisms. Exercise 21. Let L = n<0 L 2n+1 be an abelian L -algebra (i.e., all brackets are zero) concentrated in odd negative degrees, with L 2n+1 of dimension 1 for all n, and let A = ( x; 0) be the polynomial algebra on a single generator of degree 2, without constant terms. (1) Show that MC(L A) = (L A) 1 and is of innite countable dimension. (2) Show that, C (L) = ( (y 0 ; y 2 ; y 4 ; : : :); 0) and that Hom CDGA (C (L); A) is of innite, uncountable dimension. 2.3 The connected components in the DGL/L -algebra setting Definition 2.25. Given an L -algebra L and z MC(L), dene the perturbation of `k by z as `zk(x 1 ; : : : ; x k ) = [x 1 ; : : : ; x k ] z = i=0 1 i!`i+k(z; i : : : ; z; x 1 ; : : : ; x k ):

3 Complete DGL s and the representable realization functor 33 Exercise 22. Show that whenever the above sum is always nite, (L; {`zi }) is again an L algebra which will be denoted by L z. We can truncate L z to produce a non-negatively graded L algebra L (z) whose underlying graded vector space is L (z) i = L z i = L i if i > 0; Ker`z1 if i = 0; 0 if i < 0; and with brackets induced by `zk for any k 1. Theorem 2.26. [2, Corollary 1.2] [?, Theorem 1.1] Let ': C (L) Q be the augmentation corresponding to the Maurer-Cartan element z of a given mild L -algebra. Then L ' and L (z) are homotopy equivalent simplicial sets. Proof. First, observe that, for a given augmentation f : ( V; d) Q of a free CDGA, the quotient ( V; d)=k f is again a free CDGA ( (V 1 V 2 ); d f ) in which V 1 is the coker of the map d: V 0 V 1 resulting by applying the dierential d and then projecting over the ideal generated by V <0 and {v f(v); v V 0 }. In particular, if ( V; d) = C (L), we write, C (L)=K ' = ( (V 1 V 2 ); d ' ): A straightforward computation shows that ( (V 1 V 2 ); d ' ) is precisely C (L (z) ). Then, L (z) = C (L (z) ) = C (L)=K ' C (L) ' = L ' : Exercise 23. Check carefully the step in the previous proof. ( (V 1 V 2 ); d ' ) = C (L (z) ) 3 Complete DGL s and the representable realization functor The second part of this course deals also with realization functors. Recall that Quillen's adjoint pair is the composition of a sequence of adjoint pairs: { Arc-connected, simply connected spaces } G W SGp 1 ^K G SCHA 1 P ^U N SLA 1 DGL 1 ; N which induce an equivalence between the corresponding homotopical categories [18].