Massey products � � � � Idea of the proof. 0 � � � � � � � g f h � Y � U X Z a � Y � C exact X Σ X q f i q f i A triangle X → Σ X is exact if and only if − → Y − → C − (Σ f ) ∗ f ∗ i ∗ q ∗ T ( − , X ) − → T ( − , Y ) − → T ( − , C ) − → T ( − , Σ X ) − → T ( − , Σ Y ) is an exact sequence of T -modules and 1 Σ X ∈ � q , i , f � . Fernando Muro On Massey products and triangulated categories
Massey products � � � � � Idea of the proof. 0 � � � � � � � g f h � Y � U X Z a b � Y � C exact X Σ X q f i q f i A triangle X → Σ X is exact if and only if − → Y − → C − (Σ f ) ∗ f ∗ i ∗ q ∗ T ( − , X ) − → T ( − , Y ) − → T ( − , C ) − → T ( − , Σ X ) − → T ( − , Σ Y ) is an exact sequence of T -modules and 1 Σ X ∈ � q , i , f � . Fernando Muro On Massey products and triangulated categories
Massey products � � � Idea of the proof. g f h � Y � Z � U X a b ∈ � h , g , f � � Y � C exact X Σ X q f i q f i A triangle X → Σ X is exact if and only if − → Y − → C − (Σ f ) ∗ f ∗ i ∗ q ∗ T ( − , X ) − → T ( − , Y ) − → T ( − , C ) − → T ( − , Σ X ) − → T ( − , Σ Y ) is an exact sequence of T -modules and 1 Σ X ∈ � q , i , f � . Fernando Muro On Massey products and triangulated categories
Massey products � � � Idea of the proof. g f h � Y � Z � U X a b ∈ � h , g , f � � Y � C exact X Σ X q f i q f i A triangle X → Σ X is exact if and only if − → Y − → C − (Σ f ) ∗ f ∗ i ∗ q ∗ T ( − , X ) − → T ( − , Y ) − → T ( − , C ) − → T ( − , Σ X ) − → T ( − , Σ Y ) is an exact sequence of T -modules and 1 Σ X ∈ � q , i , f � . Fernando Muro On Massey products and triangulated categories
Heller’s theory When is a Massey product induced by a triangulated structure? Let mod - T be the stable category of coherent T -modules, Hom T ( M , N ) Hom T ( M , N ) = { M → T ( − , X ) → N } . The stable category is triangulated. The translation functor S : mod - T − → mod - T is determined by the choice of short exact sequences in mod - T , 0 → M − → SM → 0 . → T ( − , CM ) − Fernando Muro On Massey products and triangulated categories
Heller’s theory When is a Massey product induced by a triangulated structure? Let mod - T be the stable category of coherent T -modules, Hom T ( M , N ) Hom T ( M , N ) = { M → T ( − , X ) → N } . The stable category is triangulated. The translation functor S : mod - T − → mod - T is determined by the choice of short exact sequences in mod - T , 0 → M − → SM → 0 . → T ( − , CM ) − Fernando Muro On Massey products and triangulated categories
Heller’s theory When is a Massey product induced by a triangulated structure? Let mod - T be the stable category of coherent T -modules, Hom T ( M , N ) Hom T ( M , N ) = { M → T ( − , X ) → N } . The stable category is triangulated. The translation functor S : mod - T − → mod - T is determined by the choice of short exact sequences in mod - T , 0 → M − → SM → 0 . → T ( − , CM ) − Fernando Muro On Massey products and triangulated categories
Heller’s theory � �� � � �� � The functor Σ extends in an essentially unique way, Σ T T ∼ Yoneda Yoneda Σ mod - T mod - T exact ∼ � mod - T Σ triangle mod - T ∼ Fernando Muro On Massey products and triangulated categories
Heller’s theory � � � Theorem (Heller’68) There is a bijective correspondence between Puppe triangulated structures on ( T , Σ) and natural isomorphisms δ : Σ ∼ = S 3 such that for any coherent T -module M, δ SM S 4 M Σ SM � � � � � − 1 � � � � � � ∼ � � = S δ M � � � � � S Σ M Theorem There is an isomorphism which sends the Massey product of a triangulation on ( T , Σ) to Heller’s natural isomorphism, Hom (Σ , S 3 ) . ∼ MP ( T , Σ) = skip proof Fernando Muro On Massey products and triangulated categories
Heller’s theory � � � Theorem (Heller’68) There is a bijective correspondence between Puppe triangulated structures on ( T , Σ) and natural isomorphisms δ : Σ ∼ = S 3 such that for any coherent T -module M, δ SM S 4 M Σ SM � � � � � − 1 � � � � � � ∼ � � = S δ M � � � � � S Σ M Theorem There is an isomorphism which sends the Massey product of a triangulation on ( T , Σ) to Heller’s natural isomorphism, Hom (Σ , S 3 ) . ∼ MP ( T , Σ) = skip proof Fernando Muro On Massey products and triangulated categories
Heller’s theory �� � �� � �� � �� � � Idea of the proof. Let �− , − , −� be a Massey product. We need to define a morphism δ M : Σ M → S 3 M for any coherent T -module M . g � f h � T ( − , CS 2 M ) T ( − , CS 3 M ) T ( − , CM ) T ( − , CSM ) � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � S 2 M S 3 M M SM Fernando Muro On Massey products and triangulated categories
Heller’s theory �� � �� � �� � �� � � Idea of the proof. Let �− , − , −� be a Massey product. We need to define a morphism δ M : Σ M → S 3 M for any coherent T -module M . g � f h � T ( − , CS 2 M ) T ( − , CS 3 M ) T ( − , CM ) T ( − , CSM ) � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � S 2 M S 3 M M SM Fernando Muro On Massey products and triangulated categories
Heller’s theory �� � �� � �� � �� � � Idea of the proof. Let �− , − , −� be a Massey product. We need to define a morphism δ M : Σ M → S 3 M for any coherent T -module M . g � f h � T ( − , CS 2 M ) T ( − , CS 3 M ) T ( − , CM ) T ( − , CSM ) � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � S 2 M S 3 M M SM Fernando Muro On Massey products and triangulated categories
Heller’s theory � � �� �� � � �� �� � Idea of the proof. Let �− , − , −� be a Massey product. We need to define a morphism δ M : Σ M → S 3 M for any coherent T -module M . g � f h � T ( − , CS 2 M ) T ( − , CS 3 M ) T ( − , CM ) T ( − , CSM ) � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � h , g , f � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � S 2 M S 3 M T ( − , Σ CM ) M SM Fernando Muro On Massey products and triangulated categories
Heller’s theory �� � �� � �� � �� � � �� � Idea of the proof. Let �− , − , −� be a Massey product. We need to define a morphism δ M : Σ M → S 3 M for any coherent T -module M . g � f h � T ( − , CS 2 M ) T ( − , CS 3 M ) T ( − , CM ) T ( − , CSM ) � � � � � � � � � � � � � � � � � � � � � � � � � � � h , g , f � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � S 2 M S 3 M T ( − , Σ CM ) M SM � � � � � � � Σ M Fernando Muro On Massey products and triangulated categories
Heller’s theory � � �� �� � �� �� � �� � � Idea of the proof. Let �− , − , −� be a Massey product. We need to define a morphism δ M : Σ M → S 3 M for any coherent T -module M . g � f h � T ( − , CS 2 M ) T ( − , CS 3 M ) T ( − , CM ) T ( − , CSM ) � � � � � � � � � � � � � � � � � � � � � � � � � � � h , g , f � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � S 2 M S 3 M T ( − , Σ CM ) M SM � � � � � � � � � � � � δ M � � � Σ M Fernando Muro On Massey products and triangulated categories
An example of Heller’s theory Let T = F ( Z / 4 ) be the category of finitely generated free Z / 4-modules and Σ = 1 F ( Z / 4 ) the identity functor. In this case mod - T = mod - Z / 4, mod - T = F ( Z / 2 ) and S = 1 F ( Z / 2 ) . ∼ ∼ MP ( F ( Z / 4 ) , 1 F ( Z / 4 ) ) Hom ( 1 F ( Z / 2 ) , 1 F ( Z / 2 ) ) Z / 2 . = = Theorem (M.-Schwede-Strickland’07) The non-trivial Massey product in ( F ( Z / 4 ) , 1 F ( Z / 4 ) ) is induced by a Verdier triangulated structure where the triangle 2 2 2 Z / 4 → Z / 4 → Z / 4 → Z / 4 − − − is exact. Fernando Muro On Massey products and triangulated categories
An example of Heller’s theory Let T = F ( Z / 4 ) be the category of finitely generated free Z / 4-modules and Σ = 1 F ( Z / 4 ) the identity functor. In this case mod - T = mod - Z / 4, mod - T = F ( Z / 2 ) and S = 1 F ( Z / 2 ) . ∼ ∼ MP ( F ( Z / 4 ) , 1 F ( Z / 4 ) ) Hom ( 1 F ( Z / 2 ) , 1 F ( Z / 2 ) ) Z / 2 . = = Theorem (M.-Schwede-Strickland’07) The non-trivial Massey product in ( F ( Z / 4 ) , 1 F ( Z / 4 ) ) is induced by a Verdier triangulated structure where the triangle 2 2 2 Z / 4 → Z / 4 → Z / 4 → Z / 4 − − − is exact. Fernando Muro On Massey products and triangulated categories
An example of Heller’s theory Let T = F ( Z / 4 ) be the category of finitely generated free Z / 4-modules and Σ = 1 F ( Z / 4 ) the identity functor. In this case mod - T = mod - Z / 4, mod - T = F ( Z / 2 ) and S = 1 F ( Z / 2 ) . ∼ ∼ MP ( F ( Z / 4 ) , 1 F ( Z / 4 ) ) Hom ( 1 F ( Z / 2 ) , 1 F ( Z / 2 ) ) Z / 2 . = = Theorem (M.-Schwede-Strickland’07) The non-trivial Massey product in ( F ( Z / 4 ) , 1 F ( Z / 4 ) ) is induced by a Verdier triangulated structure where the triangle 2 2 2 Z / 4 → Z / 4 → Z / 4 → Z / 4 − − − is exact. Fernando Muro On Massey products and triangulated categories
An example of Heller’s theory Let T = F ( Z / 4 ) be the category of finitely generated free Z / 4-modules and Σ = 1 F ( Z / 4 ) the identity functor. In this case mod - T = mod - Z / 4, mod - T = F ( Z / 2 ) and S = 1 F ( Z / 2 ) . ∼ ∼ MP ( F ( Z / 4 ) , 1 F ( Z / 4 ) ) Hom ( 1 F ( Z / 2 ) , 1 F ( Z / 2 ) ) Z / 2 . = = Theorem (M.-Schwede-Strickland’07) The non-trivial Massey product in ( F ( Z / 4 ) , 1 F ( Z / 4 ) ) is induced by a Verdier triangulated structure where the triangle 2 2 2 Z / 4 → Z / 4 → Z / 4 → Z / 4 − − − is exact. Fernando Muro On Massey products and triangulated categories
Hochschild-Mitchell cohomology A T -bimodule is a T ⊗ T op -module. The bar complex C ∗ ( T ) is the complex of T -bimodules � C ∗ ( T ) = T ( X 0 , − ) ⊗ · · · ⊗ T ( X i , X i − 1 ) ⊗ · · · ⊗ T ( − , X n ) , X 0 ,..., X n with differential n � ( − 1 ) i α 0 ⊗ · · · ⊗ ( α i α i + 1 ) ⊗ · · · ⊗ α n + 1 . ∂ ( α 0 ⊗ · · · ⊗ α n + 1 ) = i = 0 The Hochschild-Mitchell cohomology of T with coefficients in M , HH ∗ ( T , M ) , is the cohomology of C ∗ ( T , M ) Hom T - bimod ( C ∗ ( T ) , M ) . = Fernando Muro On Massey products and triangulated categories
Hochschild-Mitchell cohomology A T -bimodule is a T ⊗ T op -module. The bar complex C ∗ ( T ) is the complex of T -bimodules � C ∗ ( T ) = T ( X 0 , − ) ⊗ · · · ⊗ T ( X i , X i − 1 ) ⊗ · · · ⊗ T ( − , X n ) , X 0 ,..., X n with differential n � ( − 1 ) i α 0 ⊗ · · · ⊗ ( α i α i + 1 ) ⊗ · · · ⊗ α n + 1 . ∂ ( α 0 ⊗ · · · ⊗ α n + 1 ) = i = 0 The Hochschild-Mitchell cohomology of T with coefficients in M , HH ∗ ( T , M ) , is the cohomology of C ∗ ( T , M ) Hom T - bimod ( C ∗ ( T ) , M ) . = Fernando Muro On Massey products and triangulated categories
Hochschild-Mitchell cohomology A T -bimodule is a T ⊗ T op -module. The bar complex C ∗ ( T ) is the complex of T -bimodules � C ∗ ( T ) = T ( X 0 , − ) ⊗ · · · ⊗ T ( X i , X i − 1 ) ⊗ · · · ⊗ T ( − , X n ) , X 0 ,..., X n with differential n � ( − 1 ) i α 0 ⊗ · · · ⊗ ( α i α i + 1 ) ⊗ · · · ⊗ α n + 1 . ∂ ( α 0 ⊗ · · · ⊗ α n + 1 ) = i = 0 The Hochschild-Mitchell cohomology of T with coefficients in M , HH ∗ ( T , M ) , is the cohomology of C ∗ ( T , M ) Hom T - bimod ( C ∗ ( T ) , M ) . = Fernando Muro On Massey products and triangulated categories
Hochschild-Mitchell cohomology Example T = T ( − , − ) is a T -bimodule, and we denote HH ∗ ( T ) HH ∗ ( T , T ) . = More generally, for any q ∈ Z we consider HH p , q ( T ) HH p ( T , T ( − , Σ q )) HH p ( T , T (Σ − q , − )) . = = We also consider the ( mod - T ) -bimodules Ext q , r Ext q ∼ Ext q T ( − , Σ r ) T (Σ − r , − ) , q ≥ 0 , r ∈ Z , = = T and the cohomology HH p ( mod - T , Ext q , r T ) . Fernando Muro On Massey products and triangulated categories
Hochschild-Mitchell cohomology Example T = T ( − , − ) is a T -bimodule, and we denote HH ∗ ( T ) HH ∗ ( T , T ) . = More generally, for any q ∈ Z we consider HH p , q ( T ) HH p ( T , T ( − , Σ q )) HH p ( T , T (Σ − q , − )) . = = We also consider the ( mod - T ) -bimodules Ext q , r Ext q ∼ Ext q T ( − , Σ r ) T (Σ − r , − ) , q ≥ 0 , r ∈ Z , = = T and the cohomology HH p ( mod - T , Ext q , r T ) . Fernando Muro On Massey products and triangulated categories
Hochschild-Mitchell cohomology Example T = T ( − , − ) is a T -bimodule, and we denote HH ∗ ( T ) HH ∗ ( T , T ) . = More generally, for any q ∈ Z we consider HH p , q ( T ) HH p ( T , T ( − , Σ q )) HH p ( T , T (Σ − q , − )) . = = We also consider the ( mod - T ) -bimodules Ext q , r Ext q ∼ Ext q T ( − , Σ r ) T (Σ − r , − ) , q ≥ 0 , r ∈ Z , = = T and the cohomology HH p ( mod - T , Ext q , r T ) . Fernando Muro On Massey products and triangulated categories
Baues-Wirsching cohomology The Baues-Wirsching cohomology of T with coefficients in M , H ∗ ( T , M ) , is the cohomology of the ‘group ring’ k -category k [ T ] obtained by taking free k -modules on morphism pointed sets, free k -module on T ( X , Y ) . k [ T ]( X , Y ) = The natural k -linear functor k [ T ] → T induces a homomorphism HH ∗ ( T , M ) − → H ∗ ( T , M ) . Example We consider H p , q ( T ) = H p ( T , T (Σ − q , − )) and H p ( mod - T , Ext q , r T ) . Fernando Muro On Massey products and triangulated categories
Baues-Wirsching cohomology The Baues-Wirsching cohomology of T with coefficients in M , H ∗ ( T , M ) , is the cohomology of the ‘group ring’ k -category k [ T ] obtained by taking free k -modules on morphism pointed sets, free k -module on T ( X , Y ) . k [ T ]( X , Y ) = The natural k -linear functor k [ T ] → T induces a homomorphism HH ∗ ( T , M ) − → H ∗ ( T , M ) . Example We consider H p , q ( T ) = H p ( T , T (Σ − q , − )) and H p ( mod - T , Ext q , r T ) . Fernando Muro On Massey products and triangulated categories
Baues-Wirsching cohomology The Baues-Wirsching cohomology of T with coefficients in M , H ∗ ( T , M ) , is the cohomology of the ‘group ring’ k -category k [ T ] obtained by taking free k -modules on morphism pointed sets, free k -module on T ( X , Y ) . k [ T ]( X , Y ) = The natural k -linear functor k [ T ] → T induces a homomorphism HH ∗ ( T , M ) − → H ∗ ( T , M ) . Example We consider H p , q ( T ) = H p ( T , T (Σ − q , − )) and H p ( mod - T , Ext q , r T ) . Fernando Muro On Massey products and triangulated categories
Massey products and H 3 A Baues-Wirsching ( 3 , − 1 ) -cocycle z 3 , − 1 of T sends any three composable morphisms f g h X − → Y − → Z − → U to an element z 3 , − 1 ( h , g , f ) ∈ T (Σ X , U ) , in such a way that i · z 3 , − 1 ( h , g , f ) − z 3 , − 1 ( i · h , g , f ) + z 3 , − 1 ( i , h · g , f ) 0 . − z 3 , − 1 ( i , h , g · f ) + z 3 , − 1 ( i , h , g ) · (Σ f ) = It is a Hochschild-Mitchell cocycle if z 3 , − 1 is k -multilinear. Fernando Muro On Massey products and triangulated categories
Massey products and H 3 A Baues-Wirsching ( 3 , − 1 ) -cocycle z 3 , − 1 of T sends any three composable morphisms f g h X − → Y − → Z − → U to an element z 3 , − 1 ( h , g , f ) ∈ T (Σ X , U ) , in such a way that i · z 3 , − 1 ( h , g , f ) − z 3 , − 1 ( i · h , g , f ) + z 3 , − 1 ( i , h · g , f ) 0 . − z 3 , − 1 ( i , h , g · f ) + z 3 , − 1 ( i , h , g ) · (Σ f ) = It is a Hochschild-Mitchell cocycle if z 3 , − 1 is k -multilinear. Fernando Muro On Massey products and triangulated categories
Massey products and H 3 Lemma Given a Baues-Wirsching ( 3 , − 1 ) -cocycle z 3 , − 1 there is defined a unique Massey product in ( T , Σ) such that z 3 , − 1 ( h , g , f ) ∈ � h , g , f � ⊂ T (Σ X , U ) . This defines a homomorphism HH 3 , − 1 ( T ) − → H 3 , − 1 ( T ) − → MP ( T , Σ) . Fernando Muro On Massey products and triangulated categories
Massey products and H 3 Lemma Given a Baues-Wirsching ( 3 , − 1 ) -cocycle z 3 , − 1 there is defined a unique Massey product in ( T , Σ) such that z 3 , − 1 ( h , g , f ) ∈ � h , g , f � ⊂ T (Σ X , U ) . This defines a homomorphism HH 3 , − 1 ( T ) − → H 3 , − 1 ( T ) − → MP ( T , Σ) . Fernando Muro On Massey products and triangulated categories
Massey products and H 3 Theorem (Pirashvili’88, Baues-Dreckmann’89) The Massey product of a topological triangulated category is in the image of H 3 , − 1 ( T ) − → MP ( T , Σ) . The Massey product of a locally projective algebraic triangulated category is in the image of HH 3 , − 1 ( T ) − → MP ( T , Σ) . skip proof Is there any triangulated category whose Massey product does not come from HH 3 , − 1 or H 3 , − 1 ? Fernando Muro On Massey products and triangulated categories
Massey products and H 3 Theorem (Pirashvili’88, Baues-Dreckmann’89) The Massey product of a topological triangulated category is in the image of H 3 , − 1 ( T ) − → MP ( T , Σ) . The Massey product of a locally projective algebraic triangulated category is in the image of HH 3 , − 1 ( T ) − → MP ( T , Σ) . skip proof Is there any triangulated category whose Massey product does not come from HH 3 , − 1 or H 3 , − 1 ? Fernando Muro On Massey products and triangulated categories
Massey products and H 3 Theorem (Pirashvili’88, Baues-Dreckmann’89) The Massey product of a topological triangulated category is in the image of H 3 , − 1 ( T ) − → MP ( T , Σ) . The Massey product of a locally projective algebraic triangulated category is in the image of HH 3 , − 1 ( T ) − → MP ( T , Σ) . skip proof Is there any triangulated category whose Massey product does not come from HH 3 , − 1 or H 3 , − 1 ? Fernando Muro On Massey products and triangulated categories
Massey products and H 3 � Idea of the proof. Let M be a topological or algebraic model of T such that T ⊂ D ( M ) as a full triangulated subcategory. There is defined a derived 2-category D 2 ( M ) , and a projection � � D ( M ) ⊃ T . D 2 ( M ) � � � � � � � The obstruction to the existence of a splitting pseudofunctor is � D 2 ( M ) � | T ∈ H 3 , − 1 ( T ) universal Massey product and maps to the Massey product of T by H 3 , − 1 ( T ) − → MP ( T , Σ) . Fernando Muro On Massey products and triangulated categories
Massey products and H 3 � Idea of the proof. Let M be a topological or algebraic model of T such that T ⊂ D ( M ) as a full triangulated subcategory. There is defined a derived 2-category D 2 ( M ) , and a projection � � D ( M ) ⊃ T . D 2 ( M ) � � � � � � � The obstruction to the existence of a splitting pseudofunctor is � D 2 ( M ) � | T ∈ H 3 , − 1 ( T ) universal Massey product and maps to the Massey product of T by H 3 , − 1 ( T ) − → MP ( T , Σ) . Fernando Muro On Massey products and triangulated categories
Massey products and H 3 � Idea of the proof. Let M be a topological or algebraic model of T such that T ⊂ D ( M ) as a full triangulated subcategory. There is defined a derived 2-category D 2 ( M ) , and a projection � � D ( M ) ⊃ T . D 2 ( M ) � � � � � � � The obstruction to the existence of a splitting pseudofunctor is � D 2 ( M ) � | T ∈ H 3 , − 1 ( T ) universal Massey product and maps to the Massey product of T by H 3 , − 1 ( T ) − → MP ( T , Σ) . Fernando Muro On Massey products and triangulated categories
Massey products and H 3 � Idea of the proof. Let M be a topological or algebraic model of T such that T ⊂ D ( M ) as a full triangulated subcategory. There is defined a derived 2-category D 2 ( M ) , and a projection � � D ( M ) ⊃ T . D 2 ( M ) � � � � � � � The obstruction to the existence of a splitting pseudofunctor is � D 2 ( M ) � | T ∈ H 3 , − 1 ( T ) universal Massey product and maps to the Massey product of T by H 3 , − 1 ( T ) − → MP ( T , Σ) . Fernando Muro On Massey products and triangulated categories
Massey products and H 3 � Idea of the proof. Let M be a topological or algebraic model of T such that T ⊂ D ( M ) as a full triangulated subcategory. There is defined a derived 2-category D 2 ( M ) , and a projection � � D ( M ) ⊃ T . D 2 ( M ) � � � � � � � The obstruction to the existence of a splitting pseudofunctor is � D 2 ( M ) � | T ∈ H 3 , − 1 ( T ) universal Massey product and maps to the Massey product of T by H 3 , − 1 ( T ) − → MP ( T , Σ) . Fernando Muro On Massey products and triangulated categories
Massey products and H 0 Proposition There is an isomorphism H 0 ( mod - T , Ext 3 , − 1 ∼ MP ( T , Σ) = ) . T skip proof Proof. ∼ Hom (Σ , S 3 ) MP ( T , Σ) = H 0 ( mod - T , Hom T (Σ , S 3 )) ∼ = H 0 ( mod - T , Ext 3 , − 1 ∼ = ) , T since Hom T (Σ M , S 3 N ) ∼ = Ext 3 T (Σ M , N ) and mod - T ։ mod - T is full and the identity on objects. Fernando Muro On Massey products and triangulated categories
Massey products and H 0 Proposition There is an isomorphism H 0 ( mod - T , Ext 3 , − 1 ∼ MP ( T , Σ) = ) . T skip proof Proof. ∼ Hom (Σ , S 3 ) MP ( T , Σ) = H 0 ( mod - T , Hom T (Σ , S 3 )) ∼ = H 0 ( mod - T , Ext 3 , − 1 ∼ = ) , T since Hom T (Σ M , S 3 N ) ∼ = Ext 3 T (Σ M , N ) and mod - T ։ mod - T is full and the identity on objects. Fernando Muro On Massey products and triangulated categories
Massey products and H 0 Proposition There is an isomorphism H 0 ( mod - T , Ext 3 , − 1 ∼ MP ( T , Σ) = ) . T skip proof Proof. ∼ Hom (Σ , S 3 ) MP ( T , Σ) = H 0 ( mod - T , Hom T (Σ , S 3 )) ∼ = H 0 ( mod - T , Ext 3 , − 1 ∼ = ) , T since Hom T (Σ M , S 3 N ) ∼ = Ext 3 T (Σ M , N ) and mod - T ։ mod - T is full and the identity on objects. Fernando Muro On Massey products and triangulated categories
Massey products and H 0 Proposition There is an isomorphism H 0 ( mod - T , Ext 3 , − 1 ∼ MP ( T , Σ) = ) . T skip proof Proof. ∼ Hom (Σ , S 3 ) MP ( T , Σ) = H 0 ( mod - T , Hom T (Σ , S 3 )) ∼ = H 0 ( mod - T , Ext 3 , − 1 ∼ = ) , T since Hom T (Σ M , S 3 N ) ∼ = Ext 3 T (Σ M , N ) and mod - T ։ mod - T is full and the identity on objects. Fernando Muro On Massey products and triangulated categories
Massey products and H 0 � � Theorem ( Ulmer’69 + Jibladze-Pirashvili’91, Lowen-van den Bergh’05) There is a spectral sequence for any r ∈ Z , H p ( mod - T , Ext q , r ⇒ H p + q , r ( T ) , T ) = and also for HH ∗ if for instance k is a field. Proposition The following diagram commutes (also for HH ∗ if k is a field). H 0 ( mod - T , Ext 3 , − 1 ) T � � edge � � � � ∼ � = � � � � � � � � MP ( T , Σ) H 3 , − 1 ( T ) the previous one Fernando Muro On Massey products and triangulated categories
Massey products and H 0 � � Theorem ( Ulmer’69 + Jibladze-Pirashvili’91, Lowen-van den Bergh’05) There is a spectral sequence for any r ∈ Z , H p ( mod - T , Ext q , r ⇒ H p + q , r ( T ) , T ) = and also for HH ∗ if for instance k is a field. Proposition The following diagram commutes (also for HH ∗ if k is a field). H 0 ( mod - T , Ext 3 , − 1 ) T � � edge � � � � ∼ � = � � � � � � � � MP ( T , Σ) H 3 , − 1 ( T ) the previous one Fernando Muro On Massey products and triangulated categories
Massey products and H 0 � � Theorem ( Ulmer’69 + Jibladze-Pirashvili’91, Lowen-van den Bergh’05) There is a spectral sequence for any r ∈ Z , H p ( mod - T , Ext q , r ⇒ H p + q , r ( T ) , T ) = and also for HH ∗ if for instance k is a field. Proposition The following diagram commutes (also for HH ∗ if k is a field). H 0 ( mod - T , Ext 3 , − 1 ) T � � edge � � � � ∼ � = � � � � � � � � MP ( T , Σ) H 3 , − 1 ( T ) the previous one Fernando Muro On Massey products and triangulated categories
The example F ( Z / 4 ) Theorem For ( F ( Z / 4 ) , 1 F ( Z / 4 ) ) the edge homomorphism is trivial. 0 = HML 3 ( Z / 4 ) ∼ = H 3 , − 1 ( F ( Z / 4 )) → H 0 ( mod - Z / 4 , Ext 3 , − 1 Z / 2 ∼ Z / 4 ) ∼ = Z / 2 . − Corollary (M.-Schwede-Strickland’07) The triangulated category F ( Z / 4 ) does not have any algebraic or topological model. When does a triangulated category have a model? Is there an obstruction theory for the existence of models of any kind? Fernando Muro On Massey products and triangulated categories
The example F ( Z / 4 ) Theorem For ( F ( Z / 4 ) , 1 F ( Z / 4 ) ) the edge homomorphism is trivial. 0 = HML 3 ( Z / 4 ) ∼ = H 3 , − 1 ( F ( Z / 4 )) → H 0 ( mod - Z / 4 , Ext 3 , − 1 Z / 2 ∼ Z / 4 ) ∼ = Z / 2 . − Corollary (M.-Schwede-Strickland’07) The triangulated category F ( Z / 4 ) does not have any algebraic or topological model. When does a triangulated category have a model? Is there an obstruction theory for the existence of models of any kind? Fernando Muro On Massey products and triangulated categories
The example F ( Z / 4 ) Theorem For ( F ( Z / 4 ) , 1 F ( Z / 4 ) ) the edge homomorphism is trivial. 0 = HML 3 ( Z / 4 ) ∼ = H 3 , − 1 ( F ( Z / 4 )) → H 0 ( mod - Z / 4 , Ext 3 , − 1 Z / 2 ∼ Z / 4 ) ∼ = Z / 2 . − Corollary (M.-Schwede-Strickland’07) The triangulated category F ( Z / 4 ) does not have any algebraic or topological model. When does a triangulated category have a model? Is there an obstruction theory for the existence of models of any kind? Fernando Muro On Massey products and triangulated categories
The example F ( Z / 4 ) Theorem For ( F ( Z / 4 ) , 1 F ( Z / 4 ) ) the edge homomorphism is trivial. 0 = HML 3 ( Z / 4 ) ∼ = H 3 , − 1 ( F ( Z / 4 )) → H 0 ( mod - Z / 4 , Ext 3 , − 1 Z / 2 ∼ Z / 4 ) ∼ = Z / 2 . − Corollary (M.-Schwede-Strickland’07) The triangulated category F ( Z / 4 ) does not have any algebraic or topological model. When does a triangulated category have a model? Is there an obstruction theory for the existence of models of any kind? Fernando Muro On Massey products and triangulated categories
The example F ( Z / 4 ) Theorem For ( F ( Z / 4 ) , 1 F ( Z / 4 ) ) the edge homomorphism is trivial. 0 = HML 3 ( Z / 4 ) ∼ = H 3 , − 1 ( F ( Z / 4 )) → H 0 ( mod - Z / 4 , Ext 3 , − 1 Z / 2 ∼ Z / 4 ) ∼ = Z / 2 . − Corollary (M.-Schwede-Strickland’07) The triangulated category F ( Z / 4 ) does not have any algebraic or topological model. When does a triangulated category have a model? Is there an obstruction theory for the existence of models of any kind? Fernando Muro On Massey products and triangulated categories
Stable Massey products A Massey product on ( T , Σ) is stable if � Σ h , Σ g , Σ f � = − Σ � h , g , f � . Therefore the submodule of stable Massey products MP s ( T , Σ) is the kernel of Σ − 1 ∗ Σ ∗ + 1 = HH 0 ( mod - T , Ext 3 , − 1 HH 0 ( mod - T , Ext 3 , − 1 MP ( T , Σ) ∼ ) − → ) . T T Moreover, { triangulated structures on ( T , Σ) } ⊂ MP s ( T , Σ) ⊂ MP ( T , Σ) . Fernando Muro On Massey products and triangulated categories
Stable Massey products A Massey product on ( T , Σ) is stable if Σ − 1 � Σ h , Σ g , Σ f � = −� h , g , f � . Therefore the submodule of stable Massey products MP s ( T , Σ) is the kernel of Σ − 1 ∗ Σ ∗ + 1 = HH 0 ( mod - T , Ext 3 , − 1 HH 0 ( mod - T , Ext 3 , − 1 MP ( T , Σ) ∼ ) − → ) . T T Moreover, { triangulated structures on ( T , Σ) } ⊂ MP s ( T , Σ) ⊂ MP ( T , Σ) . Fernando Muro On Massey products and triangulated categories
Stable Massey products A Massey product on ( T , Σ) is stable if Σ − 1 � Σ h , Σ g , Σ f � = −� h , g , f � . Therefore the submodule of stable Massey products MP s ( T , Σ) is the kernel of Σ − 1 ∗ Σ ∗ + 1 = HH 0 ( mod - T , Ext 3 , − 1 HH 0 ( mod - T , Ext 3 , − 1 MP ( T , Σ) ∼ ) − → ) . T T Moreover, { triangulated structures on ( T , Σ) } ⊂ MP s ( T , Σ) ⊂ MP ( T , Σ) . Fernando Muro On Massey products and triangulated categories
Stable Massey products A Massey product on ( T , Σ) is stable if Σ − 1 � Σ h , Σ g , Σ f � = −� h , g , f � . Therefore the submodule of stable Massey products MP s ( T , Σ) is the kernel of Σ − 1 ∗ Σ ∗ + 1 = HH 0 ( mod - T , Ext 3 , − 1 HH 0 ( mod - T , Ext 3 , − 1 MP ( T , Σ) ∼ ) − → ) . T T Moreover, { triangulated structures on ( T , Σ) } ⊂ MP s ( T , Σ) ⊂ MP ( T , Σ) . Fernando Muro On Massey products and triangulated categories
Cohomology of graded categories Let k be a field and T Σ the Z -graded k -category with T Σ ( X , Y ) n T ( X , Σ n Y ) , = n ∈ Z . A T Σ -bimodule is a degree 0 functor T op Σ ⊗ T Σ → Mod Z - k to Z -graded k -modules. The bar complex C ∗ ( T Σ ) is now a complex of T Σ -bimodules. Given a T Σ -bimodule M the Hochschild-Mitchell cohomology HH p , q ( T Σ , M ) , is the p th cohomology of C ∗ ( T Σ , M [ q ]) Hom T Σ - bimod ( C ∗ ( T Σ ) , M [ q ]) . = Example T Σ = T Σ ( − , − ) is a T Σ -bimodule and HH p , q ( T Σ ) = HH p , q ( T Σ , T Σ ) . Fernando Muro On Massey products and triangulated categories
Cohomology of graded categories Let k be a field and T Σ the Z -graded k -category with T Σ ( X , Y ) n T ( X , Σ n Y ) , = n ∈ Z . A T Σ -bimodule is a degree 0 functor T op Σ ⊗ T Σ → Mod Z - k to Z -graded k -modules. The bar complex C ∗ ( T Σ ) is now a complex of T Σ -bimodules. Given a T Σ -bimodule M the Hochschild-Mitchell cohomology HH p , q ( T Σ , M ) , is the p th cohomology of C ∗ ( T Σ , M [ q ]) Hom T Σ - bimod ( C ∗ ( T Σ ) , M [ q ]) . = Example T Σ = T Σ ( − , − ) is a T Σ -bimodule and HH p , q ( T Σ ) = HH p , q ( T Σ , T Σ ) . Fernando Muro On Massey products and triangulated categories
Cohomology of graded categories Let k be a field and T Σ the Z -graded k -category with T Σ ( X , Y ) n T ( X , Σ n Y ) , = n ∈ Z . A T Σ -bimodule is a degree 0 functor T op Σ ⊗ T Σ → Mod Z - k to Z -graded k -modules. The bar complex C ∗ ( T Σ ) is now a complex of T Σ -bimodules. Given a T Σ -bimodule M the Hochschild-Mitchell cohomology HH p , q ( T Σ , M ) , is the p th cohomology of C ∗ ( T Σ , M [ q ]) Hom T Σ - bimod ( C ∗ ( T Σ ) , M [ q ]) . = Example T Σ = T Σ ( − , − ) is a T Σ -bimodule and HH p , q ( T Σ ) = HH p , q ( T Σ , T Σ ) . Fernando Muro On Massey products and triangulated categories
Cohomology of graded categories Proposition For any q ∈ Z , the complex C ∗ ( T Σ , T Σ [ q ]) is the homotopy fiber of ∗ Σ ∗ + 1 : C ∗ ( T , T ( − , Σ q )) − Σ − 1 → C ∗ ( T , T ( − , Σ q )) . This homotopy fiber is strongly related to the stability equation for Massey products, Σ − 1 � Σ h , Σ g , Σ f � = −� h , g , f � . Fernando Muro On Massey products and triangulated categories
Cohomology of graded categories Proposition For any q ∈ Z , the complex C ∗ ( T Σ , T Σ [ q ]) is the homotopy fiber of ∗ Σ ∗ + 1 : C ∗ ( T , T ( − , Σ q )) − Σ − 1 → C ∗ ( T , T ( − , Σ q )) . This homotopy fiber is strongly related to the stability equation for Massey products, Σ − 1 � Σ h , Σ g , Σ f � = −� h , g , f � . Fernando Muro On Massey products and triangulated categories
Cohomology of graded categories � � � � Corollary There is a long exact sequence for any q ∈ Z , Σ − 1 ∗ Σ ∗ + 1 HH p , q ( T ) → HH p + 1 , q ( T Σ ) → · · · · · · → HH p , q ( T Σ ) → HH p , q ( T ) − → Moreover, there is a commutative diagram edge � HH 0 ( mod - T , Ext 3 , − 1 HH 3 , − 1 ( T ) ) ∼ = MP ( T , Σ) T � MP s ( T , Σ) HH 3 , − 1 ( T Σ ) Fernando Muro On Massey products and triangulated categories
A ∞ -categories An element { m 3 } ∈ HH 3 , − 1 ( T Σ ) is the same as an A 4 -category structure ( m 1 = 0 , m 2 , m 3 ) in T Σ , with m 2 the composition in T Σ . An A ∞ -category A consists of Objects X , Y , . . . Morphism Z -graded k -modules A ( X , Y ) , Identity morphisms id X ∈ A ( X , X ) 0 , n -Fold composition law, n ≥ 1, m n : A ( X 1 , X 0 ) ⊗ · · · ⊗ A ( X n , X n − 1 ) − → A ( X n , X 0 ) , deg ( m n ) 2 − n . = Fernando Muro On Massey products and triangulated categories
A ∞ -categories An element { m 3 } ∈ HH 3 , − 1 ( T Σ ) is the same as an A 4 -category structure ( m 1 = 0 , m 2 , m 3 ) in T Σ , with m 2 the composition in T Σ . An A ∞ -category A consists of Objects X , Y , . . . Morphism Z -graded k -modules A ( X , Y ) , Identity morphisms id X ∈ A ( X , X ) 0 , n -Fold composition law, n ≥ 1, m n : A ( X 1 , X 0 ) ⊗ · · · ⊗ A ( X n , X n − 1 ) − → A ( X n , X 0 ) , deg ( m n ) 2 − n . = Fernando Muro On Massey products and triangulated categories
A ∞ -categories An element { m 3 } ∈ HH 3 , − 1 ( T Σ ) is the same as an A 4 -category structure ( m 1 = 0 , m 2 , m 3 ) in T Σ , with m 2 the composition in T Σ . An A ∞ -category A consists of Objects X , Y , . . . Morphism Z -graded k -modules A ( X , Y ) , Identity morphisms id X ∈ A ( X , X ) 0 , n -Fold composition law, n ≥ 1, m n : A ( X 1 , X 0 ) ⊗ · · · ⊗ A ( X n , X n − 1 ) − → A ( X n , X 0 ) , deg ( m n ) 2 − n . = Fernando Muro On Massey products and triangulated categories
A ∞ -categories An element { m 3 } ∈ HH 3 , − 1 ( T Σ ) is the same as an A 4 -category structure ( m 1 = 0 , m 2 , m 3 ) in T Σ , with m 2 the composition in T Σ . An A ∞ -category A consists of Objects X , Y , . . . Morphism Z -graded k -modules A ( X , Y ) , Identity morphisms id X ∈ A ( X , X ) 0 , n -Fold composition law, n ≥ 1, m n : A ( X 1 , X 0 ) ⊗ · · · ⊗ A ( X n , X n − 1 ) − → A ( X n , X 0 ) , deg ( m n ) 2 − n . = Fernando Muro On Massey products and triangulated categories
A ∞ -categories An element { m 3 } ∈ HH 3 , − 1 ( T Σ ) is the same as an A 4 -category structure ( m 1 = 0 , m 2 , m 3 ) in T Σ , with m 2 the composition in T Σ . An A ∞ -category A consists of Objects X , Y , . . . Morphism Z -graded k -modules A ( X , Y ) , Identity morphisms id X ∈ A ( X , X ) 0 , n -Fold composition law, n ≥ 1, m n : A ( X 1 , X 0 ) ⊗ · · · ⊗ A ( X n , X n − 1 ) − → A ( X n , X 0 ) , deg ( m n ) 2 − n . = Fernando Muro On Massey products and triangulated categories
A ∞ -categories An element { m 3 } ∈ HH 3 , − 1 ( T Σ ) is the same as an A 4 -category structure ( m 1 = 0 , m 2 , m 3 ) in T Σ , with m 2 the composition in T Σ . An A ∞ -category A consists of Objects X , Y , . . . Morphism Z -graded k -modules A ( X , Y ) , Identity morphisms id X ∈ A ( X , X ) 0 , n -Fold composition law, n ≥ 1, m n : A ( X 1 , X 0 ) ⊗ · · · ⊗ A ( X n , X n − 1 ) − → A ( X n , X 0 ) , deg ( m n ) 2 − n . = Fernando Muro On Massey products and triangulated categories
A ∞ -categories The composition laws must satisfy the following equations, ( − 1 ) jp + q m i ( 1 ⊗ j ⊗ m p ⊗ 1 ⊗ q ) , 0 � n ≥ 1 . = j + p + q = n i = j + 1 + q n = 1, m 2 1 = 0, i.e. A ( X , Y ) are complexes. n = 2, m 1 m 2 = m 2 ( 1 ⊗ m 1 + m 1 ⊗ 1 ) , i.e. m 1 is a derivation for the product m 2 . n = 3, m 2 ( m 2 ⊗ 1 − 1 ⊗ m 2 ) = m 1 m 3 + m 3 ( 1 ⊗ 1 ⊗ m 1 + 1 ⊗ m 1 ⊗ 1 + m 1 ⊗ 1 ⊗ 1 ) , i.e. m 2 is associative up to homotopy. Fernando Muro On Massey products and triangulated categories
A ∞ -categories The composition laws must satisfy the following equations, ( − 1 ) jp + q m i ( 1 ⊗ j ⊗ m p ⊗ 1 ⊗ q ) , 0 � n ≥ 1 . = j + p + q = n i = j + 1 + q n = 1, m 2 1 = 0, i.e. A ( X , Y ) are complexes. n = 2, m 1 m 2 = m 2 ( 1 ⊗ m 1 + m 1 ⊗ 1 ) , i.e. m 1 is a derivation for the product m 2 . n = 3, m 2 ( m 2 ⊗ 1 − 1 ⊗ m 2 ) = m 1 m 3 + m 3 ( 1 ⊗ 1 ⊗ m 1 + 1 ⊗ m 1 ⊗ 1 + m 1 ⊗ 1 ⊗ 1 ) , i.e. m 2 is associative up to homotopy. Fernando Muro On Massey products and triangulated categories
A ∞ -categories The composition laws must satisfy the following equations, ( − 1 ) jp + q m i ( 1 ⊗ j ⊗ m p ⊗ 1 ⊗ q ) , 0 � n ≥ 1 . = j + p + q = n i = j + 1 + q n = 1, m 2 1 = 0, i.e. A ( X , Y ) are complexes. n = 2, m 1 m 2 = m 2 ( 1 ⊗ m 1 + m 1 ⊗ 1 ) , i.e. m 1 is a derivation for the product m 2 . n = 3, m 2 ( m 2 ⊗ 1 − 1 ⊗ m 2 ) = m 1 m 3 + m 3 ( 1 ⊗ 1 ⊗ m 1 + 1 ⊗ m 1 ⊗ 1 + m 1 ⊗ 1 ⊗ 1 ) , i.e. m 2 is associative up to homotopy. Fernando Muro On Massey products and triangulated categories
A ∞ -categories The composition laws must satisfy the following equations, ( − 1 ) jp + q m i ( 1 ⊗ j ⊗ m p ⊗ 1 ⊗ q ) , 0 � n ≥ 1 . = j + p + q = n i = j + 1 + q n = 1, m 2 1 = 0, i.e. A ( X , Y ) are complexes. n = 2, m 1 m 2 = m 2 ( 1 ⊗ m 1 + m 1 ⊗ 1 ) , i.e. m 1 is a derivation for the product m 2 . n = 3, m 2 ( m 2 ⊗ 1 − 1 ⊗ m 2 ) = m 1 m 3 + m 3 ( 1 ⊗ 1 ⊗ m 1 + 1 ⊗ m 1 ⊗ 1 + m 1 ⊗ 1 ⊗ 1 ) , i.e. m 2 is associative up to homotopy. Fernando Muro On Massey products and triangulated categories
A ∞ -categories An A ∞ -category is pretriangulated if the full subcategory of the derived category H 0 A ⊂ D ( A ) is a triangulated subcategory. An A ∞ -category is minimal if m 1 = 0. Proposition (Lefèvre-Hasegawa’03) A compactly generated algebraic triangulated k-category T is H 0 A of a minimal pretringulated A ∞ -category A . The underlying Z -graded k -category of A is actually T Σ , so in order to reconstruct A one just has to find m 3 , m 4 , . . . Fernando Muro On Massey products and triangulated categories
A ∞ -categories An A ∞ -category is pretriangulated if the full subcategory of the derived category H 0 A ⊂ D ( A ) is a triangulated subcategory. An A ∞ -category is minimal if m 1 = 0. Proposition (Lefèvre-Hasegawa’03) A compactly generated algebraic triangulated k-category T is H 0 A of a minimal pretringulated A ∞ -category A . The underlying Z -graded k -category of A is actually T Σ , so in order to reconstruct A one just has to find m 3 , m 4 , . . . Fernando Muro On Massey products and triangulated categories
A ∞ -categories An A ∞ -category is pretriangulated if the full subcategory of the derived category H 0 A ⊂ D ( A ) is a triangulated subcategory. An A ∞ -category is minimal if m 1 = 0. Proposition (Lefèvre-Hasegawa’03) A compactly generated algebraic triangulated k-category T is H 0 A of a minimal pretringulated A ∞ -category A . The underlying Z -graded k -category of A is actually T Σ , so in order to reconstruct A one just has to find m 3 , m 4 , . . . Fernando Muro On Massey products and triangulated categories
A ∞ -categories An A ∞ -category is pretriangulated if the full subcategory of the derived category H 0 A ⊂ D ( A ) is a triangulated subcategory. An A ∞ -category is minimal if m 1 = 0. Proposition (Lefèvre-Hasegawa’03) A compactly generated algebraic triangulated k-category T is H 0 A of a minimal pretringulated A ∞ -category A . The underlying Z -graded k -category of A is actually T Σ , so in order to reconstruct A one just has to find m 3 , m 4 , . . . Fernando Muro On Massey products and triangulated categories
A ∞ -obstructions for triangulated categories The existence of m 3 is equivalent to say that the Massey product of T is in the image of the composite edge HH 3 , − 1 ( T Σ ) − → HH 3 , − 1 ( T ) → HH 0 ( mod - T , Ext 3 , − 1 ) ∼ = MP ( T , Σ) . − T In order to check this fact, one can use the spectral sequence HH p ( mod - T , Ext q , r ⇒ HH p + q , r ( T ) T ) = and the long exact sequence Σ − 1 ∗ Σ ∗ + 1 HH p , q ( T ) → HH p + 1 , q ( T Σ ) → · · · · · · → HH p , q ( T Σ ) → HH p , q ( T ) − → Fernando Muro On Massey products and triangulated categories
A ∞ -obstructions for triangulated categories The existence of m 3 is equivalent to say that the Massey product of T is in the image of the composite edge HH 3 , − 1 ( T Σ ) − → HH 3 , − 1 ( T ) → HH 0 ( mod - T , Ext 3 , − 1 ) ∼ = MP ( T , Σ) . − T In order to check this fact, one can use the spectral sequence HH p ( mod - T , Ext q , r ⇒ HH p + q , r ( T ) T ) = and the long exact sequence Σ − 1 ∗ Σ ∗ + 1 HH p , q ( T ) → HH p + 1 , q ( T Σ ) → · · · · · · → HH p , q ( T Σ ) → HH p , q ( T ) − → Fernando Muro On Massey products and triangulated categories
A ∞ -obstructions for triangulated categories Lemma (Lefèvre-Hasegawa’03) Let n ≥ 5 . Given a minimal A n − 1 -category structure on T Σ , defined by ( m 1 = 0 , m 2 , m 3 , . . . , m n − 2 ) , there is a well-defined θ ( m 3 ,..., m n − 2 ) ∈ HH n , 3 − n ( T Σ ) , which vanishes if and only if there exists m n − 1 such that ( m 1 = 0 , m 2 , m 3 , . . . , m n − 2 , m n − 1 ) is an A n -category structure on T Σ . Fernando Muro On Massey products and triangulated categories
Summing up the A ∞ -obstruction theory Let T be triangulated � δ ∈ H 0 ( mod - T , Ext 3 , − 1 ) . T δ must be a perm. cycle of HH p ( mod - T , Ext q , − 1 ) ⇒ HH p + q , − 1 ( T ) , T edge H 0 ( mod - T , Ext 3 , − 1 HH 3 , − 1 ( T ) − → ) , T ∆ �→ δ. Σ − 1 ∗ Σ ∗ + 1 ∆ must be in the kernel of HH 3 , − 1 ( T ) HH 3 , − 1 ( T ) , so − → HH 3 , − 1 ( T Σ ) HH 3 , − 1 ( T ) , − → { m 3 } �→ ∆ . The higher obstructions must vanish, θ ( m 3 ,..., m n − 2 ) ∈ H n , 3 − n ( T Σ ) , n ≥ 5 . Then T can be enhanced to an A ∞ -category defined over T Σ . Fernando Muro On Massey products and triangulated categories
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