Serre A -functors Oleksandr Manzyuk joint work with Volodymyr Lyubashenko math.ct/0701165 0. Notation 1. Preliminaries on A -categories 2. Serre functors 3. Serre A -functors
0. Notation k denotes a (ground) commutative ring k-linear = enriched in k -mod graded = Z-graded C k = the category of complexes of k-modules A differential graded (dg) category is a category enriched in C k. In particular, since C k is a closed monoidal category, it gives rise to a category C k enriched in C k, i.e., to a dg category. We use geometric notation for composition: f g = fg 2
1. Preliminaries on A -categories A -categories Definition. An A -category A consists of a set of objects Ob A for each X, Y Ob A, a graded k-module A(X, Y ) for each n 1 and X 0,...,X n ObA, a k-linear map m n : A(X 0, X 1 ) A(X n 1, X n ) A(X 0, X n ) of degree 2 n, satisfying the equations ( 1) pk+q (1 p m k 1 q )m p+1+q = 0, n 1. p+k+q=n (n = 1) m 2 1 = 0 (n = 2) m 2 m 1 = (m 1 1 + 1 m 1 )m 2 (n = 3) (m 2 1)m 2 (1 m 2 )m 2 = m 3 m 1 + (m 1 1 1 + 1 m 1 1 + 1 1 m 1 )m 3 Example. A dg category can be viewed as an A -category in which m n = 0 for n 3. 3
A -functors Definition. An A -functor f : A B consists of a function Ob f : ObA ObB, X Xf for each n 1 and X 0,...,X n ObA, a k-linear map f n : A(X 0, X 1 ) A(X n 1, X n ) B(X 0 f, X n f) of degree 1 n, satisfying the equations l>0 i 1 + +i l =n ( 1) σ (f i1 f il )m l = p+k+q=n ( 1) pk+q (1 p m k 1 q )f p+1+q, n 1, where σ = (i 2 1) + 2(i 3 1) + + (l 1)(i l 1). (n = 1) f 1 m 1 = m 1 f 1 (n = 2) m 2 f 1 (f 1 f 1 )m 2 = f 2 m 1 + (m 1 1 + 1 m 1 )f 2 Example. A dg functor can be viewed as an A -functor with f n = 0 for n 2. 4
Graded quivers Definition. A graded quiver A consists of a set of objects Ob A for each X, Y Ob A, a graded k-module A(X, Y ). A morphism f : A B of graded quivers consists of a function Ob f : ObA ObB, X Xf for each X, Y ObA, a k-linear map f = f X,Y : A(X, Y ) B(Xf, Y f) of degree 0. Let Q denote the category of graded quivers. It is symmetric monoidal with tensor product (A, B) A B given by ObA B = Ob A Ob B, (A B)((X, U), (Y, V )) = A(X, Y ) B(U, V ), (X, Y Ob A, U, V Ob B). The unit object is the graded quiver 1 with Ob 1 = { }, 1(, ) = k. 5
Graded quivers with a fixed set of objects For a set S, denote by Q/S the fibre of the functor Ob : Q Set over S, i.e., the subcategory of Q whose objects are graded quivers with the set of objects S and whose morphisms are morphisms of graded quivers acting as the identity on objects. Q/S is a monoidal category with tensor product (A, B) A B given by (A B)(X, Z) = Y S A(X, Y ) B(Y, Z), X, Z S. The unit object is the discrete quiver ks given by k if X = Y, Ob ks = S, ks(x, Y ) = 0 if X Y, X, Y S. 6
Cocategories Definition. An augmented graded cocategory is a graded quiver C equipped with the structure of an augmented counital coassociative coalgebra in the monoidal category Q/ ObC. Therefore, C comes with a comultiplication : C C C a counit ε : C k ObC an augmentation η : k ObC C which are morphisms in Q/ ObC satisfying the usual axioms. A morphism of augmented graded cocategories is a morphism of graded quivers that preserves the comultiplication, counit, and augmentation. The category of augmented graded cocategories inherits the tensor product from Q. 7
Main example: tensor cocategory of a quiver Let A be a graded quiver. Denote T n A = A n = A A }{{} n times The graded quiver TA = (tensor product in Q/ ObA). T n A n=0 equipped with the cut comultiplication the counit 0 : h 1 h n and the augmentation n h 1 h k hk+1 h n, k=0 ε = pr 0 : TA T 0 A = k ObA, η = in 0 : k ObA = T 0 A TA is an augmented graded cocategory. 8
Cocategory approach to A -categories For a graded quiver A, denote by sa its suspension: Ob sa = Ob A, (sa(x, Y )) d = A(X, Y ) d+1, X, Y ObA. Let s : A sa denote the identity map of degree 1. Proposition (folklore). There is a bijection between structures (m n ) n 1 of an A -category on a graded quiver A and differentials b : TsA TsA of degree 1 such that (TsA, 0, pr 0, in 0, b) is an augmented differential graded cocategory, i.e., b 2 = 0, b 0 = 0 (b 1 + 1 b), b pr 0 = 0, in 0 b = 0. The bijection is given by the formulas m n = [ A n s n (sa) n b n sa s 1 A ], b n = [ (sa) n inn TsA b TsA pr1 sa ], b nm = 1 p b k 1 q : T n sa T m sa. p+k+q=n p+1+q=m We may think of A -categories as of augmented dg cocategories of particular form. Then A -functors f : A B correspond precisely to morphisms of augmented dg cocategories. (TsA, b) (TsB, b) The advantage is that we can easily define A -functors of many arguments! 9
A short reminder about multicategories A multicategory is just like a category, the only difference being the shape of arrows. An arrow in a multicategory looks like X 1 X 2 X n. with a finite family of objects as the source and one object as the target. Composition turns a tree of arrows into a single arrow, e.g. Y X 1 X 2 X 3 X 4 X 5 f 1 f 2 Y 1 Y 2 g Y X 1 X 2 X 3 (f 1, f 2 )g X 4 X 5 Example. A one-object multicategory is an operad (multicategories are sometimes called many-object operads, or colored operads ). Y Example. A monoidal category C gives rise to a multicategory Ĉ with the same set of objects. An arrow X 1,..., X n Y in Ĉ is an arrow X1 Xn Y in C. Composition in Ĉ is derived from composition and tensor product in C. 10
A -categories constitute a symmetric multicategory Definition. Let A 1,..., A n, B be A -categories. An A -functor f : A 1,...,A n B is a morphism of augmented dg cocategories TsA 1 TsA n TsB. Explicitly, an A -functor f : A 1,..., A n B consists of a function Ob f : n Ob A i ObB, (X 1,...,X n ) (X 1,..., X n )f i=1 for each k = (k 1,..., k n ) N n {0} and X j i ObA i, i = 1,..., n, j = 1,..., k i, a k-linear map [A 1 (X 0 1, X1 1 ) A 1(X k 1 1 1, X k 1 1 )] [A n (Xn 0, X1 n ) A n(x k n 1 n, X k n n )] f k B((X1, 0..., Xn)f, 0 (X k 1 1,...,Xk n n )f) of degree 1 (k 1 + + k n ) subject to equations. Denote by A the symmetric multicategory of A -categories and A -functors. 11
The multicategory A is closed For each collection of A -categories A 1,..., A n, B, there exists a functor A -category A (A 1,..., A n ; B) and an evaluation A -functor ev A : A 1,..., A n, A (A 1,..., A n ; B) B such that the mapping A (B 1,..., B m ; A (A 1,..., A n ; C)) A (A 1,..., A n, B 1...,B m ; C), f (1 A1,...,1 An, f) ev A is a bijection. The objects of the A -category A (A 1,..., A n ; B) are A -functors A 1,...,A n B. For A -functors f, g : A 1,...,A n B, A (A 1,...,A n ; B)(f, g) = { A -transformations f g} = { (f, g)-coderivations TsA 1 TsA n TsB}. The evaluation A -functor acts on objects as expected: A 1,..., A n, A (A 1,...,A n ; B) B, (X 1,..., X n, f) (X 1,...,X n )f. In the case n = 1, the A -category A (A; B) has been considered by many authors (Keller, Kontsevich, Lefèvre-Hasegawa, Lyubashenko, Soibelman...). 12
Unital A -categories Definition. An A -category A is called unital if, for each X ObA, there is a cycle 1 X A(X, X) 0, called the identity of X, such that (1 X id)m 2, (id 1 Y )m 2 id : A(X, Y ) A(X, Y ), for each X, Y ObA. A unital A -category A gives rise to a k-linear category H 0 (A): Ob H 0 (A) = Ob A, H 0 (A)(X, Y ) = H 0 (A(X, Y ), m 1 ), X, Y ObA. Composition is induced by m 2, and the identity of an object X is the class [1 X ] H 0 (A)(X, X). The category H 0 (A) is called the homotopy category of A. An A -functor f : A B is unital if it preserves identities modulo boundaries: 1 X f 1 1 Xf Im m 1. A unital A -functor f : A B gives rise to a k-linear functor H 0 (f) : H 0 (A) H 0 (B) such that ObH 0 (f) = Obf, and for each X, Y ObA, the k-linear map H 0 (f) : H 0 (A)(X, Y ) H 0 (B)(Xf, Y f) is induced by f 1 : A(X, Y ) B(Xf, Y f). An A -functor of many argument is unital if it is unital in each argument. 13
The symmetric closed multicategory of unital A -categories Composition of unital A -functors is unital. Let A u A denote the submulticategory of unital A -categories and unital A -functors. It is also closed: A u (A 1,..., A n ; B) A (A 1,..., A n ; B) is the full A -subcategory whose objects are unital A -functors. It is a unital A -category. The evaluation A -functor ev Au is the restriction of ev A. It is a unital A -functor. Definition. Unital A -functors f, g : A 1,...,A n B are called isomorphic if they are isomorphic as objects of the category H 0 (A u (A 1,..., A n ; B)). 14
Opposite A -categories Definition. Let A be an A -category. The opposite A -category A op is given by ObA op = Ob A, A op (X, Y ) = A(Y, X), X, Y ObA, and operations m Aop n m Aop n are given by = ( 1) n(n+1)/2+1 signed permutation of arguments m A n. The correspondence A A op extends to A -functors and yields a symmetric multifunctor op : A A. The opposite of a unital A -category (resp. A -functor) is again unital, hence op restricts to a symmetric multifunctor op : A u Au. 15
2. Serre functors Hereafter, k is a field. Definition (Bondal Kapranov). Let C be a k-linear category. A k-linear functor S : C C is called a (right) Serre functor if there exists an isomorphism C(X, Y S) = C(Y, X) natural in X, Y Ob C, where denotes the dual vector space. A right Serre functor, if it exists, is unique up to isomorphism. Example. Let X be a smooth projective variety of dimension n over the field k. Let ω X denote the canonical sheaf on X. Let C = D b (Coh X ) be the bounded derived category of coherent sheaves on X. Then the functor S = ω X [n] is a right Serre functor. 16
3. Serre A -functors Definition For an A -category A, there is an A -functor Hom A : A op, A C k, (X, Y ) (A(X, Y ), m 1 ). It is unital if so is A. The A -functor A A (A op ; C k ) that corresponds to Hom A : A op, A C k by closedness of the multicategory A is precisely the Yoneda embedding. Definition (Kontsevich Soibelman). Let A be a unital A -category. A unital A -functor S : A A is called a (right) Serre A -functor if the diagram A op, A 1,S A op, A Hom op A op op C k D Hom A Ck commutes up to isomorphism (in H 0 (A u (Aop, A; C k ))). Here D : C k op C k, M M = C k (M, k), is the duality dg functor. Proposition. As in the case of ordinary Serre functors, if a right Serre A -functor exists, then it is unique up to isomorphism. 17
A -categories closed under shifts (see also V. Lyubashenko s talk) Let A be an A -category. It gives rise to an A -category A [ ] obtained from A by formally adding shifts of objects: ObA [ ] = ObA Z, A [ ] ((X, n), (Y, m)) = A(X, Y )[m n]. A embeds as a full A -subcategory into A [ ] via u : A A [ ], X (X, 0). Definition. A unital A -category A is called closed under shifts if u is an A -equivalence. Equivalently, each object (X, n) of A [ ] is isomorphic in H 0 (A [ ] ) to an object of the form (Y, 0). Example. Pretriangulated A -categories (to be defined by V. Lyubashenko) are closed under shifts. 18
Main theorem As above, assume that k is a field. Theorem. (1) If S : A A is a right Serre A -functor, then the induced functor H 0 (S) : H 0 (A) H 0 (A) is an ordinary right Serre functor. (2) Conversely, suppose that A is closed under shifts and that H 0 (A) admits a right Serre functor S : H 0 (A) H 0 (A). Then there exists a right Serre A -functor S : A A such that H 0 (S) = S. Example. By results of Drinfeld, we know that D b (Coh X ) is of the form H 0 (A), where A is the dg quotient of the dg category of complexes of coherent sheaves over the full dg subcategory of acyclic complexes. Therefore, the Serre functor S = ω X [n] lifts to a Serre A -functor A A. 19