Journal of Algebra. Postnikov towers, k-invariants and obstruction theory for dg categories

Transcription

1 Journal of lgebra 321 (2009) Contents lists available at ScienceDirect Journal of lgebra wwwelseviercom/locate/jalgebra Postnikov towers, k-invariants and obstruction theory for dg categories Gonçalo Tabuada Departamento de Matemática e CM, FCT-UNL, Quinta da Torre, Caparica, Portugal article info abstract rticle history: Received 3 June 2008 vailableonline27march2009 Communicated by Michel Van den Bergh Keywords: Dg category Drinfeld s DG quotient Postnikov tower k-invariants Obstruction theory Non-commutative algebraic geometry By inspiring ourselves in Drinfeld s DG quotient, we develop Postnikov towers, k-invariants and an obstruction theory for dg categories s an application, we obtain the following rigidification theorem: let be a homologically connective dg category and F 0 : B H 0 () a dg functor to its homotopy category If the inductive family {ω n (F n )} n 0 of obstruction classes vanishes, then aliftf : B for F 0 exists 2009 Elsevier Inc ll rights reserved Contents 1 Introduction Preliminaries Postnikovtowers Smallmodel Bigmodel Uniqueness and homotopy type k-invariants Obstructiontheory 3873 cknowledgments 3877 References address: tabuada@fctunlpt /$ see front matter 2009 Elsevier Inc ll rights reserved doi:101016/jjalgebra

2 G Tabuada / Journal of lgebra 321 (2009) Introduction differential graded (= dg) category is a category enriched in the category of complexes of modules over some commutative base ring R Throughout the article, we consider homological notation, ie the differential decreases the degree Dg categories provide a framework for homological geometry and for non-commutative algebraic geometry in the sense of Bondal, Drinfeld, Kapranov, Kontsevich, Toën, Van den Bergh, etc [2,3,7,8,14,15,22] They are considered as (enriched) derived categories of quasi-coherent sheaves on a hypothetical non-commutative space (see Keller s ICM-talk survey [13]) In [18], the homotopy theory of dg categories was constructed This theory has allowed several developments such as: the creation by Toën of a derived Morita theory [22]; the construction of a category of non-commutative motives [18]; the first conceptual characterization [19] of Quillen Waldhausen s K -theory [17,23] since its definition in the early 70 s, etc In this article, we develop new ingredients in this homotopy theory: Postnikov towers, k-invariants and an obstruction theory for homologically connective dg categories Homologically connective dg categories dgcategory is homologically connective if for all objects x, y, the homology R-modules H i ((x, y)) are zero for i < 0 Example 11 To any dg category, we can (functorialy) associate a homologically connective dg category τ 0 (), obtained by applying the intelligent truncation functor τ 0 ( ) to every complex of morphisms in See [21, 315] for details Moreover, we have a natural dg functor τ 0 () which induces isomorphisms ( H i τ 0 ()(x, y) ) ( ) Hi (x, y), i 0, for all objects x, y and so we obtain an equivalence H 0 (τ 0 ()) H 0 () between the homotopy categories For instance, given a differential graded R-algebra, we can apply this procedure to the dg category = D dg () of (cofibrant) complexes of right -modules We obtain then a dg category whose homotopy category is the derived category of, but which contains all the positive homological information of D dg () To any simplicial category B (see [1]), we can (functorialy) associate a homologically connective dg category N(R[B]), obtained by normalizing the R-linearization of each simplicial set of morphisms in B See [21, 6] for details For instance, given a category C and a class of morphisms W in C, we can apply this procedure to the Dwyer Kan simplicial localization L(C, W) [9,10] s L(C, W) carries non-trivial information about the localized category C[W 1 ], so it does the homologically connected dg category N(R[L(C, W)]) The purpose of this article is to develop a general technology that allows us to characterize precisely which are the obstructions appearing when one tries to lift information from the homotopy category to the differential graded one Postnikov towers Postnikov tower ( n ) n 0 for a homologically connective dg category is a commutative diagram in the category dgcat of dg categories 2 P 2 P 1 1 P 0 0

3 3852 G Tabuada / Journal of lgebra 321 (2009) such that: ) The dg functor P n : n satisfies the following conditions: 1) for all objects x, y, the induced map on the homology R-modules ( ) ( H i (x, y) Hi n (P n x, P n y) ), is an isomorphism for i n, and 2) the dg functor P n induces an equivalence of categories H 0 () H 0 ( n ) B) For all objects x, y n, the homology R-modules H i ( n (x, y)) are zero for i > n By inspiring ourselves in the description of the Hom-complexes in Drinfeld s DG quotient (see [7, 31]), we construct in Section 32 a Big (functorial) Postnikov model P() for We then use it to prove the following uniqueness theorem Theorem 12 (318) Given two objects in the category Post() of Postnikov towers for, thereexists a zig-zag of weak equivalences relating the two For many purposes, a dg category can be replaced by any of its Postnikov sections n For example if one is only interested in its homotopy category H 0 () or if one is only interested in its homology R-modules in a finite range of dimensions On the other hand, using a small Postnikov model P() for (see 31), we prove that the full homotopy type of can be recovered from any of its Postnikov towers by a homotopy limit procedure (see Proposition 320) k-invariants Having seen how to decompose a homologically connective dg category into its Postnikov sections n, n 0, we consider the inverse problem of building a Postnikov tower for, starting with 0 and inductively constructing n+1 from n In order to solve this problem, we construct (see 49) a dg functor γ n : H n+1 ()[n + 2], from the nth Big Postnikov section of to a square zero extension (see 47) of The image of γ n in the homotopy category Ho(dgcat ) of dg categories over is called the nth k- invariant α n () of (see 412) We show that α n () corresponds to a derived derivation of with values in the -bimodule H n+1 ()[n + 2] (see 413) Then we prove our main theorem, which shows how the full homotopy type of P n+1 () in dgcat can be entirely recovered from α n () Theorem 13 (416)Wehaveahomotopyfibersequence P n+1 () γ n H n+1 ()[n + 2] in Ho(dgcat ) Obstruction theory By inspiring ourselves in Example 11, we formulate the following general rigidification problem: let be a homologically connective dg category and F 0 : B H 0 () a dg functor

4 G Tabuada / Journal of lgebra 321 (2009) with values in its homotopy category, with B a cofibrant dg category Is there a lift F : B making the diagram B F F 0 τ 0 H 0 () commute? Intuitively the dg functor F 0 represents the up-to-homotopy information that one would like to rigidify, ie lift to the dg category In order to solve this problem, we consider a Postnikov tower for (eg its Big Postnikov model) P 2 () F 2 P 1 () F 1 B F 0 H 0 () P 0 () and we try to lift F 0 to dg functors F n : B for n = 1, 2 in succession If we are able to find all these lifts, there will be no difficulty in constructing the desired lift In the inductive step, we have a solid diagram P n+1 () F n+1 B F n The image of the composed dg functor B F n γ n H n+1 ()[n + 2] in the homotopy category Ho(dgcat ) is called the obstruction class ω n (F n ) of F n (see 52) We prove (see Proposition 55) that if the obstruction class ω n (F n ) vanishes, then there exists a lift F n+1 of F n (which is not uniquely determined, see Remark 56) In conclusion, we obtain the following result: Theorem 14 (57) If the inductive family {ω n (F n )} n 0 of obstruction classes vanishes, then the rigidification problem has a solution

5 3854 G Tabuada / Journal of lgebra 321 (2009) Finally notice that the results of this article can be generalized to other homotopical contexts, such as simplicial categories [1] or even spectral ones [20] The main point is that we are considering categories enriched over a base symmetric monoidal model category, which has a natural notion of Postnikov towers This allows us to define Postnikov towers in a hybrid way: conditions 1) and B) correspond to the base monoidal category, while condition 2) corresponds to dg categories Therefore, we can use the generating cofibrations of the base model category to built (Big) Postnikov models, which play a key role in the construction of k-invariants and in the proof of our main results 2 Preliminaries In what follows, R will denote a commutative ring with unit The tensor product will denote the tensor product over R Let Ch be the category of complexes of R-modules and Ch 0 the full subcategory of positively graded complexes (we consider homological notation, ie the differential decreases the degree) Recall from [12, 2311], that Ch carries a projective model structure, whose weak equivalences are the quasi-isomorphisms and whose fibrations are the degreewise surjective maps We denote by dgcat the category of small dg categories, see [7,13,18] Definition 21 Let be a small dg category the opposite dg category op of has the same objects as and its complexes on morphisms are defined by op (x, y) = (y, x) an -bimodule M is a dg functor M : op Ch Recall from [18, 18] that dgcat carries a cofibrantly generated Quillen model structure whose weak equivalences are defined as follows: Definition 22 dg functor F : B is a quasi-equivalence if: (i) for all objects x, y, the induced morphism F (x, y) : (x, y) B(Fx, Fy) is a quasi-isomorphism in Ch, and (ii) the induced functor H 0 (F ) : H 0 () H 0 (B) is an equivalence of categories Remark 23 Notice that if condition (i) is verified, condition (ii) is equivalent to: (ii) the induced functor is essentially surjective H 0 (F ) : H 0 () H 0 (B) Let us now recall from [18, 113], the following characterization of the fibrations in dgcat Proposition 24 dg functor F : B is a fibration if and only if: F1) for all objects x, y, theinducedmorphism F (x, y) : (x, y) B(Fx, Fy) is a fibration in Ch, and

6 G Tabuada / Journal of lgebra 321 (2009) F2) for every object a 1 and every morphism v Z 0 (B)(F (a 1 ), b) which becomes invertible in H 0 (B), there exists a morphism u Z 0 ()(a 1, a 2 ) such that F (u) = v and which become invertible in H 0 () Remark 25 Since the terminal object in dgcat is the zero category 0 (one object and trivial dg algebra of endomorphisms), every object in dgcat is fibrant Corollary 26 Let F : B be a dg functor such that: it induces a surjective map on the set of objects, for all objects x, y, the induced morphism F (x, y) : (x, y) B(Fx, Fy) is a fibration in Ch, and the induced functor H 0 (F ) : H 0 () H 0 (B) is an equivalence of categories Then F is a fibration in dgcat Definition 27 Let be a small dg category We say that is homologically connective if for all objects x, y, the homology R-modules H i ((x, y)) are zero for i < 0 We say that is positively graded if for all objects x, y, ther-modules (x, y) i are zero for i < 0 Notation 28 We denote by dgcat 0 the category of small positively graded dg categories Recall from [21, 416] that we have an adjunction dgcat i τ 0 dgcat 0, where τ 0 denotes the intelligent truncation functor Remark 29 Notice that for a homologically connective dg category, the co-unit of the previous adjunction furnishes a natural quasi-equivalence η : τ 0 (), which induces the identity map on set of objects This (functorial) procedure will allow us to extended several constructions from positively graded to homologically connective dg categories We finish these preliminaries with some homotopical algebra results and the notion of lax monoidal functor Let M be a Quillen model category and X an object of M Notation 210 We denote by M X the category of objects of M over X, see [11, 762] Notice that its terminal object is the identity morphism on X

7 3856 G Tabuada / Journal of lgebra 321 (2009) Remark 211 Recall from [11, 765] that M X carries a natural Quillen model structure induced by the one on M InparticularanobjectY X in M X is cofibrant if and only if Y is cofibrant in M and is fibrant if and only if the morphism Y X is a fibration in M Notice also that if f : X X is a morphism in M, wehaveaquillenadjunction M X f! f! M X, where f! associates to an object Y X in M X the object X Y X in M X and f X! associates to an object Z X in M X the object Z X f X in M X We have also a natural forgetful functor U : M X M, which preserves cofibrations, fibrations and weak equivalences This implies that U descends to the homotopy categories U : Ho(M X) Ho(M) and so we obtain the following lemma Lemma 212 Let f and f be two morphisms in M X If they become equal in Ho(M X),thenU( f ) and U( f ) become equal in Ho(M) Lemma 213 Let M be a Quillen model category Suppose we have a (non-commutative) diagram f X p Z f Y, where Z is cofibrant, Y is fibrant, p is a fibration in M and the composition p f becomes equal to f in the homotopy category Ho(M)Then,thereexistsalift f : Z X of f which makes the diagram f X p Z f Y commute Proof Notice that since Z is cofibrant and Y is fibrant, the composition p f becomes equal to f in Ho(M), if and only if p f and f are left homotopic This allows us to construct a (solid) commutative square Z f X i 0 H p I(Z) H Y,

8 G Tabuada / Journal of lgebra 321 (2009) where I(Z) is a cylinder object for Z and H is a homotopy between p f and f Finally, p has the right lifting property with respect to i 0 and so we obtain a desired morphism H i 1 f : Z I(Z) X, such that p f = f Definition 214 Let (C,, I C ) and (D,, I D ) be two symmetric monoidal categories lax monoidal functor is a functor F : C D equipped with: a morphism η : I D F (I C ), and natural morphisms ψ X,Y : F (X) F (Y ) F (X Y ), X, Y C, which are coherently associative and unital (see diagrams 627 and 628 in [4]) lax monoidal functor is strong monoidal if the morphisms η and ψ X,Y are isomorphisms Throughout this article the adjunctions are displayed vertically with the left, resp right, adjoint on the left side, resp right side 3 Postnikov towers In this section, we construct (functorial) Postnikov towers for homologically connective dg categories We prove that they are essentially unique (see Theorem 318) and that the full homotopy type of a homologically connective dg category can be recovered from any of its Postnikov towers (see Proposition 320) Definition 31 Postnikov tower ( n ) n 0 diagram in dgcat for a positively graded dg category is a commutative 2 P 2 P 1 1 P 0 0 such that: ) The dg functor P n : n satisfies the following conditions: 1) for all objects x, y, the induced map on the homology R-modules ( ) ( H i (x, y) Hi n (P n x, P n y) ) is an isomorphism for i n, and 2) it induces an equivalence of categories H 0 () H 0 ( n )

9 3858 G Tabuada / Journal of lgebra 321 (2009) B) For all objects x, y n, the homology R-modules H i ( n (x, y)) are zero for i > n The dg functor P n : n is called the nth Postnikov section of Remark 32 By the 2 out of 3 property, the dg functors n+1 n categories H 0 ( n+1 ) H 0 ( n ) induce an equivalence of Definition 33 morphism M : ( n ) n 0 ( n ) n 0 between two Postnikov towers for is a family of dg functors M n : n n which makes the obvious diagrams commute Notation 34 We denote by Post() the category of Postnikov towers for Remark 35 Let M : ( n ) n 0 ( n ) n 0 be a morphism between Postnikov towers for Bythe 2 out of 3 property, its Postnikov sections M n : n n are all quasi-equivalences Remark 36 Observe that in a Postnikov tower ( n ) n 0 for, we can replace each dg functor n+1 n by a fibration F ( n+1 ) F ( n ), starting with 1 0 and then going upward For the inductive step, we factor the composition n+1 n F (n ) by a trivial cofibration followed by a fibration F ( n+1 ) F ( n ) We obtain then a morphism ( n ) n 0 F ( n ) n 0 between Postnikov towers 2 F ( 2 ) P 2 1 F ( 1 ) P 1 P Definition 37 Let be a homologically connective dg category By a Postnikov tower for, we mean a Postnikov tower for τ 0 (), see Remark 29 We now present two functorial Postnikov tower models 31 Small model Let n 0 Consider the intelligent truncation functor τ n : Ch 0 Ch 0 which associates to a complex M : 0 M 0 M n 1 M n M n+1,

10 G Tabuada / Journal of lgebra 321 (2009) its intelligent truncation τ n (M ) : 0 M 0 M n 1 M n / Im(M n+1 ) 0 Notice that when n varies, we obtain the following natural tower of complexes τ 2 (M ) τ 1 (M ) M τ 0 (M ) Moreover each vertical map is a fibration and the induced map on the homology R-modules ( H i (M ) H i τ n (M ) ) is an isomorphism for i n Notice also that the homology R-modules H i (τ n (M )) are zero for i > n Now, let be a positively graded dg category Since for every n 0, the truncation functor τ n is lax monoidal (see 214), the above remarks imply the following: if we apply the intelligent truncation functors to each complex of morphisms of, we obtain a Postnikov tower τ 2 () τ 1 () τ 0 () for Moreover, by construction, all the dg functors in the diagram induce the identity map on the set of objects Notice also that since the morphisms of complexes τ n+1 (M ) τ n (M ) are fibrations, Remark 32 and Corollary 26 imply that the dg functors

11 3860 G Tabuada / Journal of lgebra 321 (2009) τ n+1 () τ n () are fibrations in dgcat Notation 38 We denote by P() the small Postnikov model obtained In particular denotes the dg category τ n () 32 Big model We start by recalling from [18, 13] some generating cofibrations for the Quillen model structure on dgcat Definition 39 For n Z, let S n be the complex R[n] (with R concentrated in degree n) and let D n+1 be the mapping cone on the identity of S n Wedenoteby1 n the element of degree n in S n, which corresponds to the unit of R LetC(n) be the dg category with two objects 1 and 2 such that C(n)(1, 1) = R, C(n)(2, 2) = R, C(n)(2, 1) = 0, C(n)(1, 2) = S n and composition given by multiplicationwedenotebyd(n + 1) the dg category with two objects 3 and 4 such that D(n + 1)(3, 3) = R, D(n + 1)(4, 4) = R, D(n + 1)(4, 3) = 0, D(n + 1)(3, 4) = D n+1 and with composition given by multiplication Finally, let S(n) be the dg functor from C(n) to D(n + 1) that sends 1 to 3, 2 to 4 and S n to D n+1 by the identity on R in degree n 321 Let B be a dg category and m an integer By inspiring ourselves in Drinfeld s construction (see Remark 310), we now describe the pushouts in dgcat along the dg functor S(m) Consider the following diagram S(m) C(m) D(m + 1) T B B Notice first that by construction, the dg functor T : C(m) B corresponds to specifying two objects T (1) and T (3) in B plus an element T (1 m ) Z m (B)(T (1), T (3)) By definition of S(m) (see 39), the dg category B is obtained from B by adding a new morphism h of degree m + 1fromT (1) to T (3) such that d(h) = T (1 m ) (we add neither new objects nor new relations between morphisms) Therefore for x, y B we have an isomorphism of graded R-modules (but not an isomorphism of complexes) where B l (x, y) is given by B l (x, y) B(x, y), l=0 B ( T (3), ) y R[m + 1] B ( T (3), T (1) ) R[m + 1] B ( x, T (1) ) }{{} l factors R[m+1] Intuitively B l (x, y) consists of length-l formal tensor products of elements in B with the new morphism h R[m + 1] However, since we impose the relation d(h) = T (1 m ), the above isomorphism is not an isomorphism of complexes Given an element g l+1 h g 2 h g 1 B l (x, y), }{{} l factors h

12 G Tabuada / Journal of lgebra 321 (2009) its differential equals d(g l+1 ) h g 2 h g 1 + ( 1) g l+1 g l+1 T (1 m ) g 2 h g 1 + }{{} (l 1) factors h and so it belongs to B (l 1) (x, y) This implies that, for every j 0, the sum j B l (x, y) B(x, y) l 0 is a subcomplex and so we obtain an exhaustive filtration of B(x, y) Remark 310 Suppose that in the above pushout we have n = 0, T (1) = T (3) and T (1 0 ) = 1 T (1) Inthis situation, B is the Drinfeld s DG quotient of B with respect to the object T (1), see [7, 31] Therefore, we recover the description of the Hom complexes in Drinfeld s DG quotient Lemma 311 Let be a small dg category and n 0 Suppose that the dg functor 0(where 0 denotes the terminal object in dgcat) has the right lifting property with respect to the set {S(m) m > n} Then for all objects x, y, the homology R-modules H i ((x, y)) are zero for i > n Proof This follows easily from the above definitions Lemma 312 Let π : M N be a fibration in Ch and n + 1 > 0 If the induced map on the homology R-modules H i (M ) H i (N ) is an isomorphism for i > n + 1, thenπ has the right lifting property with respect to the set {S m D m+1 m > n + 1}, see Definition 39 Proof Consider the short exact sequence of complexes 0 K i M π N 0, where K denotes the kernel of π Notice that in the induced long exact sequence on homology, the isomorphisms (m > n + 1) H m+1 (M ) H m+1 (N ) H m (K ) H m (M ) H m (N ) imply that H m (K ) = 0 Now a simple diagram chasing argument (see [12, 235]) allows us to conclude the proof Corollary 313 Let F : B be a dg functor such that for all objects x, y, the induced morphism F (x, y) : (x, y) B(Fx, Fy) in Ch satisfies the conditions of Lemma 312 Then F has the right lifting property with respect to the elements of the set {S(m) m > n + 1}

13 3862 G Tabuada / Journal of lgebra 321 (2009) Now, let be a positively graded dg category For each n 0, we apply the small object argument [11, 10514] to the dg functor 0, using the set {S(m) m > n} of generating cofibrations (see 39) We obtain the following factorization 0, P n where the dg functor P n is obtained by an infinite composition of pushouts along the elements of the set {S(m) m > n} Notice that the small object argument furnishes us natural dg functors P n+1 () making the following diagram P 2 () P 2 P 1 () P 1 P 0 P 0 () commutative Moreover, by construction, all the dg functors in the diagram induce the identity map on the set of objects Proposition 314 The above construction is a Postnikov tower for Proof We verify the conditions of Definition 31: 1) Since is obtained by an infinite composition of pushouts along the elements of the set {S(m) m > n} and the homology functors commute with infinite compositions, it is enough to prove the following: let B be a positively graded dg category and consider the following pushout (m > n) S(m) C(m) D(m + 1) T B B in dgcat We need to show that B is also positively graded and that for all objects x, y B, the induced map on the homology R-modules ( ) ( H i B(x, y) Hi B(x, y) )

14 G Tabuada / Journal of lgebra 321 (2009) is an isomorphism for i n By321wehave,forevery j 0, an exhaustive filtration j B l (x, y) B(x, y) l 0 Since m > n and B is positively graded the natural inclusion B(x, y) = B 0 (x, y) B(x, y) induces isomorphisms B(x, y) i B(x, y) i for i n + 1 and so an isomorphism τ n B(x, y) τ n B(x, y) between the truncated complexes We conclude that B is positively graded and that the induced map on the homology R-modules ( ) ( H i B(x, y) Hi B(x, y) ) is an isomorphism for i n 2) By condition 1), for all objects x, y, the induced map on the homology R-modules ( ) ( H 0 (x, y) H0 Pn ()(x, y) ) is an isomorphism Since the dg functor P n : n induces the identity map on the set of objects, we conclude that the induced functor H 0 () H 0 ( n ) is an equivalence of categories B) By construction, the dg functor 0 has the right lifting property with respect to the set {S(m) m > n} This implies, by Lemma 311, that for all objects x, y the homology R-modules H i ((x, y)) are zero for i > n Notation 315 We denote by P() the Big Postnikov model thus obtained 33 Uniqueness and homotopy type Proposition 316 Let ( n ) n 0 be a Postnikov tower for a homologically connective dg category, where all the dg functors n+1 n are fibrations Then there exists a morphism M : P() ( n ) n 0 between Postnikov towers Proof We will construct M recursively, starting with the case n = 0 and then going upwards (n = 0) Notice that the small object argument allows us to construct inductively a dg functor M 0 : P 0 () 0 as follows:

15 3864 G Tabuada / Journal of lgebra 321 (2009) step: suppose we have the following (solid) diagram (i 0, P 0 () 0 = ) T i m>0 C(m) P 0 () i C(m) P 0 () i M i 0 0 m>0 C(m) P 0 () i D(m + 1) P 0 () i+1 M i+1 0 Recall from that we denote by 1 m the cycle of degree m in S m (and so in C(m)(1, 2)) which corresponds to the unit of R Since 0 satisfies condition B) of Definition 31, we can choose a bounding chain b in 0 for each cycle T i (1 m ), m > 0 (ie d(b) = T i (1 m )) These choices give rise to a dg functor M i+1 0 which makes the above diagram commute By passing to the colimit on i, we obtain our desired dg functor M 0 = colim i M i 0 : P 0() = colim P 0 () i 0 i (n n + 1) Suppose we have a dg functor M n : n between the nth Postnikov sections We will construct a lift M n+1 which makes the square P n+1 () M n+1 n+1 M n n commutative Our argument is also an inductive one: step: suppose we have the following (solid) diagram (i 0, P n+1 () 0 = ) m>n+1 C(m) P n+1 () i C(m) P n+1 () i i Mn+1 n+1 m>n+1 C(m) P n+1 () i D(m + 1) P n+1 () M i+1 n+1 i+1 Mi+1 n n Notice that the left (solid) square appears in the construction of i+1 This implies that the dg functor Mn i+1 : i+1 i+1 n restricts to a dg functor M n, which makes the right square commutative Now, observe that the dg functor n+1 n satisfies the conditions of Corollary 313 and so it has the right lifting property with respect to the elements of the set {S(m) m > n} This implies that there exists an induced dg functor M i+1 n+1 which makes the above diagram commute By passing to the colimit on i, we obtain our desired morphism M n+1 = colim i M i n+1 : P n+1() = colim P n+1 () i n+1 i The proof is now finished

16 G Tabuada / Journal of lgebra 321 (2009) Remark 317 Since in the small Postnikov model P() for, the dg functors τ n+1 () τ n () are fibrations, Proposition 316 implies the existence of a morphism M : P() P() from the Big to the small Postnikov model Moreover, the bounding chains in P() used in the construction of M are all trivial and so this morphism is well defined Notice also that for n 0, the dg functor M n satisfies all the conditions of Corollary 26 and so it is a fibration in dgcat We now prove that Postnikov towers are essentially unique Theorem 318 Let be a homologically connective dg category Given two objects in Post() (see 34),there exists a zig-zag of weak equivalences (see 35) relating the two Proof Let ( n ) n 0 and ( n ) n 0 be two Postnikov towers for By Remark 36, we can construct morphisms in Post() ( n ) n 0 F ( n ) n 0, ( n ) n 0 F ( n ) n 0, such that the dg functors F ( n+1 ) F ( n ), F ( n+1 ) F ( n ) are fibrations in dgcat Moreover, by Proposition 316, we can also construct morphisms as follows P() F ( n ) 0, P() F ( n ) n 0 We obtain finally, the following zig-zag ( n ) n 0 F ( n ) n 0 P() F ( n ) n 0 ( n ) n 0 of weak equivalences in Post() Remark 319 Notice that by Theorem 318, the classifying space [11, 14] of Post() has a single connected component We now show how the full homotopy type of a homologically connective dg category can be recovered from any of its Postnikov towers Proposition 320 Let be a homologically connective dg category and ( n ) n 0 a Postnikov tower for Then the natural dg functor is a quasi-equivalence holim n n

17 3866 G Tabuada / Journal of lgebra 321 (2009) Proof Notice that Theorem 318 and Remark 35 imply that the homotopy limit of any Postnikov tower for is well defined up to quasi-equivalence We can then consider the small Postnikov model P() for Since every object in dgcat is fibrant (see 25) and the dg functors τ n+1 () τ n () are fibrations in dgcat, we have a natural quasi-equivalence lim n τ n () holim n τ n () By construction of limits in dgcat, we conclude that the natural dg functor lim n is an isomorphism 4 k-invariants In this section we construct k-invariants for homologically connective dg categories (see Definitions 412 and 414) We show that these invariants correspond to derived derivations with values in a certain bimodule (see 413) Then we prove our main theorem (416), which shows how the full homotopy type of the n + 1 Postnikov section of a homologically connective dg category can be recovered from the nth k-invariant of For constructions of k-invariants in the context of spectral algebra see [5,6,16] Let us start with some general constructions Definition 41 Let be a small dg category and M an -bimodule (see 21) The square zero extension Mof by M is the dg category defined as follows: its objects are those of and for objects x, y we have M(x, y) := (x, y) M(x, y) The composition in M is defined using the composition on, the above bimodule structure and by imposing that the composition between M-factors is zero Remark 42 Notice that is a (non-full) dg subcategory of M and that we have a natural projection dg functor M, which is clearly a fibration in dgcat, see Proposition 24 Definition 43 derivation of with values in an -bimodule M is a morphism in dgcat (see 210) from to M, or equivalently a section of the natural projection dg functor M derived derivation of with values in an -bimodule M is a morphism in the homotopy category Ho(dgcat ) (see 211) from to M Notation 44 We denote by Der(, M) (resp R Der(, M)) the set of derivations (resp derived derivations) of with values in M The (derived) derivation obtained by considering as a dg subcategory of M is called the trivial one

18 G Tabuada / Journal of lgebra 321 (2009) Remark 45 Notice that if is an R-algebra (ie has only one object and its endomorphisms R-algebra is ), the notion of derivation coincides with the classical one, ie an R-linear map D : M which satisfies the Leibniz relation D(ab) = a(db) + (Da)b, a, b Proposition 46 Let F : B be an object in dgcat B and M a B B-bimodule Then the set Ho(dgcat B)(, B M) is naturally isomorphic to the set of derived derivations R Der(, F (M)) of with values in the -bimodule F (M) obtained by restricting M along F Proof Recall from Remark 211, the (derived) Quillen adjunction Ho(dgcat ) F! RF! Ho(dgcat B) Notice that we have the following pull-back square F (M) F Id B M F B, which shows us that the image of B M under the functor RF! is isomorphic to F (M) Moreover the image of under the functor F! is isomorphic to the object F : B in Ho(dgcat B) and so by adjunction we obtain the desired isomorphism We now define the dg categories which play the same role as the Eilenberg Mac Lane spaces in the classical theory of k-invariants Definition 47 Let be a positively graded dg category and n 0 Consider the following bimodule: H n+1 ()[n + 2] : H 0 () op H 0 () Ch (x, y) H n+1 ( ()(x, y) ) [n + 2], where the complex H n+1 (()(x, y))[n + 2] is simply the R-module H n+1 ((x, y)) concentrated in degree n + 2 Notice that the natural projection dg functor P 0 () = H 0 () endow H n+1 ()[n + 2] with a structure of -bimodule Finally, we denote by H n+1 ()[n + 2] the square zero extension obtained (41) using this bimodule structure Remark 48 Notice that by Remark 42, is a dg subcategory of H n+1 ()[n + 2] and we have a natural projection dg functor H n+1 ()[n + 2]

19 3868 G Tabuada / Journal of lgebra 321 (2009) Definition 49 Let γ n : H n+1 ()[n + 2] be the natural dg functor obtained by modifying the dg functor M n : follows: step: suppose we have the following (solid) diagram (i 0, 0 = ) (see 317) as m>n C(m) i C(m) T i i γ i n H n+1 ()[n + 2] i+1 γn m>n C(m) i D(m + 1) i+1 For every cycle T i (1 m ), m > n + 1, choose 0 as a bounding chain in (and so in H n+1 ()[n + 2]), as in the case of the dg functor M n Now,letT i (1 n+1 ) i (T i (1), T i (3)) be a cycle of degree n + 1 Since m > n, the description of the complexes of morphisms in i (see 321) implies that we have natural isomorphisms ( T i (1), T i (3) ) j i( T i (1), T i (3) ) j for j n + 1 We can then choose for bounding chain for T i (1 n+1 ) its homology class in H n+1 ((T i (1), T i (3))) These choices give rise to a dg functor γn i+1 which makes the above diagram commute By passing to the colimit on i, we obtain our desired dg functor γ n = colim γn i : P n() = colim i H n+1 ()[n + 2] i i Remark 410 Notice that by construction, the dg functor γ n satisfies all the conditions of Corollary 26 and so it is a fibration in dgcat Moreover for n 0, we have the following commutative diagram in dgcat γ n H n+1 ()[n + 2] M n Notation 411 We denote by dgcat the category of objects in dgcat over, seenotation 210 Definition 412 Let be a positively graded dg category and n 0 Its nth k-invariant α n () is by definition the image of the dg functor γ n in the homotopy category Ho(dgcat ), seeremark410 Pn () is a quasi-equivalence, we have an isomor- Remark 413 Since the dg functor M n : phism between Ho ( dgcat )(, H n+1 ()[n + 2] )

20 G Tabuada / Journal of lgebra 321 (2009) and Ho ( dgcat )(, H n+1 ()[n + 2] ) which implies that α n () corresponds to a derived derivation of with values in the -bimodule H n+1 ()[n + 2], see Definition 43 Definition 414 Let be a homologically connective dg category Its nth k-invariant α n () is by definition the nth k-invariant of τ 0 (), see Remark 29 Remark 415 Notice that although the category dgcat is not pointed (the initial and terminal objects are not isomorphic), there is a natural morphism (in dgcat ) from its terminal object to H n+1 ()[n + 2] (see Remark 48) We now show how the full homotopy type of P n+1 () in dgcat can be entirely recovered from the nth k-invariant α n () Theorem 416 We have a homotopy fiber sequence P n+1 () γ n H n+1 ()[n + 2] in Ho(dgcat ) Proof We need to show that P n+1 () is quasi-equivalent in dgcat to the homotopy pullback of the diagram γ n H n+1 ()[n + 2] Since γ n is a fibration (see 410) and every dg category is fibrant (see 25), the homotopy pullback and the pullback are quasi-equivalent Notice that we have the following commutative diagram P n+1 P n P n+1 () W γ n H n+1 ()[n + 2]

21 3870 G Tabuada / Journal of lgebra 321 (2009) This diagram gives rise to the following factorization P n+1 P n+1 () φ θ W, where θ and φ are the induced dg functors to the pullback W We need to show that θ is a quasiequivalence By construction of limits in dgcat, all the dg functors in the previous diagrams induce the identity map on the set of objects and so it is enough to prove that for all objects x, y P n+1 (), the morphism of complexes θ(x, y) : P n+1 ()(x, y) W(x, y) is a quasi-isomorphism Let us denote by 0 M 0 M 1 M n M n+1 M n+2 M n+3 the complex (x, y) Notice that by construction of (see 321), the complex (x, y) is of the following shape 0 M 0 M 1 M n M n+1 Mn+1 Mn+3 The complex W(x, y) identifies then with the pullback of the following diagram M n M n+2 H n+1 ((x, y)) 0 M n M n M n / Im(M n+1 ) M n / Im(M n+1 ) M 1 M 1 M 1 M 0 M 0 M

22 G Tabuada / Journal of lgebra 321 (2009) The above diagram allows us to conclude that H j W(x, y) = 0for j n + 2 and that the induced map is an isomorphism for j n + 1 We now prove that the induced map ( H j Pn+1 ()(x, y) ) H j W(x, y) H n+1 (x, y) H n+1 W(x, y) is an isomorphism Notice that this implies (by the 2 out of 3 property) that θ(x, y) is a quasiisomorphism In order to prove this, we start by observing that in dgcat, pullbacks commute with filtered colimits Since is constructed as a filtered colimit and the homology functor H n+1 ( ) preserves filtered colimits it is then enough to prove the following: start: consider the following pullback square H n+1 ()[n + 2] step: consider the commutative diagram (i 0, 0 = ) C(n + 1) T i H n+1 ()[n + 2] S(n+1) D(n + 2) Pn () i γ i (T ) used in the construction of the natural dg functor γ n functor from to the pullback W i (T ) (see 49), and suppose that the induced dg i H n+1 ()[n + 2] induces an isomorphism ( H n+1 (x, y) H n+1 Wi (T )(x, y) ) We need to show that the induced dg functor from W i (T ) to the pullback W i (T ) i γ i (T ) H n+1 ()[n + 2]

23 3872 G Tabuada / Journal of lgebra 321 (2009) induces an isomorphism ( H n+1 Wi (T )(x, y) ) ( Hn+1 W i (T )(x, y) ) Recall from 321, that for all objects x, y i, we have an isomorphism of graded R-modules l=0 il (x, y) i (x, y), where il (x, y) is the graded R-module i( T (3), ) y R[n + 2] i( T (3), T (1) ) R[n + 2] i( x, T (1) ) }{{} l factors R[n+2] The differential of an element is equal to g l+1 h g 2 h g 1 }{{} il (x, y) l factors h d(g l+1 ) h g 2 h g 1 + ( 1) g l+1 g l+1 d(h) g 2 h g 1 +, }{{} (l 1) factors h where d(h) i (T (1), T (3)) corresponds to the image of 1 n+1 S n+1 (see 39) under the dg functor T This description show us that the unique elements in W i (T )(x, y), which eventually destroy the (n + 1)-homology of the complex W i (T )(x, y) belong to the graded R-module i 1 (x, y) = i( T (3), y ) R[n + 2] i( x, T (1) ) We now show that if g 2 h g 1 is a (homogeneous) element of degree n + 2in differential i 1 (x, y), whose g 2 d(h) g 1 ( i (x, y) ) n+1 ( W i (T )(x, y) ) n+1 is non-trivial in the homology R-module H n+1 (W i (T )(x, y)), then the element g 2 h g 1 does not belong to W i (T )(x, y) By hypothesis we have an induced isomorphism ( H n+1 (x, y) H n+1 Wi (T )(x, y) ) and so by Definition 47, the image of g 2 h g 1 under the dg functor γ i (T ) corresponds precisely to this non-trivial element in the homology R-module H n+1 (x, y) This implies that g 2 h g 1 does not belong to the pullback complex W i (T )(x, y) and so we conclude that we have an induced isomorphism ( H n+1 Wi (T )(x, y) ) ( Hn+1 W i (T )(x, y) )

24 G Tabuada / Journal of lgebra 321 (2009) Finally, by and infinite composition procedure, we obtain the pullback W Since the homology functor H n+1 ( ) commutes with filtered colimits, the induced map is an isomorphism and so we conclude that H n+1 (x, y) H n+1 W(x, y) θ(x, y) : P n+1 ()(x, y) W(x, y) is a quasi-isomorphism This proves the theorem 5 Obstruction theory By inspiring ourselves in Example 11, we formulate the following general rigidification problem The rigidification problem Let be a positively graded dg category and F 0 : B H 0 () adg functor with values in its homotopy category, with B a cofibrant dg category Is there a lift F : B making the diagram B F F 0 τ 0 H 0 () commute? Intuitively the dg functor F 0 represents the up-to-homotopy information that one would like to rigidify, ie lift to the dg category Remark 51 Notice that if is a homologically connective dg category, we have a zig-zag of dg functors τ 0 τ 0 () H 0 () In this situation we search for a lift B which factors through τ 0 () In order to solve this problem we consider the following notion: let be a positively graded dg category and recall from Section 32 its Big Postnikov model P 2 () P 2 P 1 () P 1 P 0 P 0 ()

25 3874 G Tabuada / Journal of lgebra 321 (2009) Definition 52 Let F : B be a dg functor Its obstruction class ω n (F ) is the image of the composed dg functor (see 49) F B γ n H n+1 ()[n + 2] in the homotopy category Ho(dgcat ), seeremark410 We say that the obstruction class ω n (F ) vanishes if it factors through the canonical morphism H n+1 ()[n + 2] in dgcat, seeremark415 Remark 53 Consider the composed dg functor B F By Proposition 46, the set M n P n() as an object in dgcat Ho ( dgcat )( B, H n+1 ()[n + 2] ) is naturally isomorphic to the set R Der ( B,(M n F ) ( H n+1 ()[n + 2] )) of derived derivations of B with values in (M n F ) (H n+1 ()[n + 2]) This implies that the obstruction class ω n (F ) of F corresponds to a derived derivation of B with values in the B B-bimodule (M n F ) (H n+1 ()[n + 2]) Moreover by the above isomorphism, the obstruction class ω n (F ) of F vanishes if and only if the associated derived derivation of B is the trivial one, see Notation 44 Proposition 54 Let B be a cofibrant dg category If two dg functors F 1, F 2 : B become equal in the homotopy category Ho(dgcat)(B, ), they give rise to isomorphic obstruction classes In particular ω n (F 1 ) vanishes if and only if ω n (F 2 ) vanishes Proof Notice that since every object in dgcat is fibrant (see 25) and B is cofibrant, two dg functors F 1 and F 2 become equal in Ho(dgcat)(B, ) if and only if they are left homotopic We can then construct the following diagram B F 1 i 0 I(B) H, i 1 B F 2 where I(B) is a cylinder object for B and i 0 and i 1 are quasi-equivalences Observe that the previous diagram gives rise to a zig-zag of weak equivalences in dgcat between ω n (F 1 ) and ω n (F 2 ), which implies that the obstruction classes are isomorphic In particular ω n (F 1 ) vanishes if and only if so does ω n (F 2 )

26 G Tabuada / Journal of lgebra 321 (2009) Let us return to our rigidification problem: let be a positively graded dg category and F : B H 0 () a dg functor, with B a cofibrant dg category Consider the diagram P 2 () M 2 P 2 () P 1 () M 1 P 1 () F 2 F 1 P 0 () M 0 P 0 () = H 0 () B, F 0 where the left (resp right) column is the Big (resp small) Postnikov model for and the morphism between the two is the one of Remark 317 Our strategy will be to try to lift F 0 : B H 0 () to dg functors F n : B for n = 1, 2, in succession If we are able to find all these lifts, there will be no difficulty in constructing the desired lift F = lim n F n : B lim n For the inductive step, we have a commutative (solid) diagram as follows (n 0) P n+1 () M n+1 P n+1 () M n F n+1 F n F n B Since B is cofibrant and M n is a trivial fibration, there exits a lift Fn of F n such that M n Fn = F n Moreover since M n is a quasi-equivalence, any two such lifts become equal in Ho(dgcat)(B, ) and so by Proposition 54 they give rise to isomorphic obstruction classes In what follows, we denote by ω n (F n ) the obstruction class of Fn Proposition 55 liftf n+1 of F n, making the diagram P n+1 () F n+1 F n B commute, exists if and only if the obstruction class ω n (F n ) vanishes (see 52)

27 3876 G Tabuada / Journal of lgebra 321 (2009) Proof Let us suppose first that ω n (F n ) vanishes Recall from Theorem 416, that we have a homotopy fiber sequence P n+1 () γ n H n+1 ()[n + 2] in Ho(dgcat ) By hypothesis, the obstruction class ω n (F n ) vanishes and so the choice of a homotopy in dgcat between γ n Fn and B H n+1 ()[n + 2] (see 415) induces a morphism in Ho(dgcat )(B, P n+1 ()) SinceB is cofibrant (and P n+1 () is fibrant) in dgcat (see 211), we can represent this morphism by a dg functor ψ : B P n+1 () Moreover, by Lemma 212, any two such representatives become equal in Ho(dgcat)(B, P n+1 ()) This implies that F n and the composition B M n+1 ψ P n+1 () becomes equal in Ho(dgcat)(B, ) Finally, by Lemma 213, we conclude that there exists a desired lift F n+1 as in the proposition Let us now prove the converse Suppose we have a lift F n+1 of F n as in the proposition Since B is cofibrant and M n+1 is a trivial fibration there exists a lift Fn+1 of F n+1 such that M n+1 Fn+1 = F n+1 Observe that Fn and the composition B F n+1 P n+1 () becomes equal in Ho(dgcat)(B, ) This implies, by Theorem 416 and Proposition 54, that the obstruction class ω n (F n ) vanishes Remark 56 By Proposition 55, if the obstruction class ω n (F n ) vanishes, then there exists a lift F n+1 of F n However this lift depends on the choice of a certain homotopy (see the proof of Proposition 55) Therefore there are many choices for F n+1, and different choices could lead to different obstruction classes ω n+1 (F n+1 ), some who vanish others who do not The conclusion of this section is that if at each stage of the inductive process of constructing lifts F n : B, the obstruction class ω n (F n ) vanishes, then the rigidification problem has a solution, ie we have the following result: Theorem 57 Let beapositivelygradeddgcategoryandf 0 : B H 0 () a dg functor, with B a cofibrant dg category If the inductive family {ω n ( Fn )} n 0 of obstruction classes vanishes, then there exists a lift F : B of F 0, making the diagram F τ 0 B F 0 H 0 () commute

28 G Tabuada / Journal of lgebra 321 (2009) cknowledgments It is a great pleasure to thank Carlos Simpson for motivating conversations and Gustavo Granja and Bertrand Toën for important comments on an older version of this article I would like to thank the Laboratoire J- Dieudonné at Nice, France for his hospitality, where some of this work was carried out Finally, I also would like to thank the anonymous referee for several important comments and corrections This work was supported by the grant SFRH/BPD/34575/2007 FCT-Portugal References [1] J Bergner, Quillen model structure on the category of simplicial categories, Trans mer Math Soc 359 (2007) [2] Bondal, M Van den Bergh, Generators and representability of functors in commutative and noncommutative geometry, Mosc Math J 3 (1) (2003) 1 37 [3] Bondal, M Kapranov, Framed triangulated categories, Mat Sb 181 (5) (1990) (in Russian); Math USSR Sb 70 (1) (1990) [4] F Borceux, Handbook of Categorical lgebra 2, Encyclopedia Math ppl, vol 51, Cambridge Univ Press, 1994 [5] D Dugger, B Shipley, Postnikov extensions of ring spectra, lgebr Geom Topol 6 (2006) (electronic) [6] D Dugger, B Shipley, Topological equivalences for differential graded algebras, dv Math 212 (1) (2007) [7] V Drinfeld, DG quotients of DG categories, J lgebra 272 (2004) [8] V Drinfeld, DG categories, University of Chicago Geometric Langlands Seminar, notes available at: edu/users/benzvi/grsp/lectures/langlandshtml [9] W Dwyer, D Kan, Simplicial localizations of categories, J Pure ppl lgebra 17 (3) (1980) [10] W Dwyer, D Kan, Equivalences between homotopy theories of diagrams, in: lgebraic Topology and lgebraic K-Theory, in: nn of Math Stud, vol 113, Princeton Univ Press, Princeton, 1987, pp [11] P Hirschhorn, Model Categories and Their Localizations, Math Surveys Monogr, vol 99, mer Math Soc, 2003 [12] M Hovey, Model Categories, Math Surveys Monogr, vol 63, mer Math Soc, 1999 [13] B Keller, On differential graded categories, in: International Congress of Mathematicians, vol II, Eur Math Soc, Zürich, 2006, pp [14] M Kontsevich, Categorification, NC Motives, Geometric Langlands and Lattice Models, University of Chicago Geometric Langlands Seminar, notes available at: [15] M Kontsevich, Notes on motives in finite characteristic, Manin Festschrift, in press, arxiv:math/ [16] Lazarev, Homotopy theory of ring spectra and applications to MU-modules, K-Theory 24 (3) (2001) [17] D Quillen, Higher algebraic K -theory, I: Higher K -theories, in: Proc Conf, Battelle Memorial Inst, Seattle, W, 1972, in: Lecture Notes in Math, vol 341, 1972, pp [18] G Tabuada, Théorie homotopique des DG-catégories, PhD thesis, available at: arxiv: [19] G Tabuada, Higher K -theory via universal invariants, Duke Math J 145 (1) (2008) [20] G Tabuada, Homotopy theory of spectral categories, dv Math, in press, arxiv: [21] G Tabuada, Differential graded versus simplicial categories, arxiv: [22] B Toën, The homotopy theory of dg-categories and derived Morita theory, Invent Math 167 (3) (2007) [23] F Waldhausen, lgebraic K-theory of spaces, in: lgebraic and Geometric Topology, New Brunswick, NJ, 1983, in: Lecture Notes in Math, vol 1126, Springer, Berlin, 1985, pp