When can we enhance a triangulated category? Fernando Muro - PowerPoint PPT Presentation

Minimal A ∞ -categories A minimal A ∞ -structure on a graded category A is a Hochschild cochain of total degree 2 = m 3 + m 4 + · · · + m n + · · · m concentrated in horizontal degrees ≥ 3 which is a solution of the Maurer–Cartan equation, ∂ ( m ) + 1 0 . 2 [ m , m ] = This equation can be decomposed as ∂ ( m n ) + 1 � 0 , n ≥ 3 , [ m p , m q ] = 2 p + q = n + 2 in particular m 3 is a cocycle, { m 3 } ∈ HH 3 , − 1 ( A ) . Fernando Muro When can we enhance a triangulated category?

Minimal A n -categories Definition A minimal A n -structure on a graded category A is given by Hochschild cochains m 3 , . . . , m i , . . . , m n of bidegree ( i , 2 − i ) such that � ∂ ( m i ) + 1 0 , n ≥ i ≥ 3 . [ m p , m q ] = 2 p + q = i + 2 An A 3 -structure on a graded category is just a 3-cocycle m 3 , { m 3 } ∈ HH 3 , − 1 ( A ) . An A ∞ -structure is a sequence of cochains m 3 , . . . , m n , . . . such that m 3 , . . . , m n is an A n -structure for all n ≥ 3. Fernando Muro When can we enhance a triangulated category?

Minimal A n -categories Definition A minimal A n -structure on a graded category A is given by Hochschild cochains m 3 , . . . , m i , . . . , m n of bidegree ( i , 2 − i ) such that � ∂ ( m i ) + 1 0 , n ≥ i ≥ 3 . [ m p , m q ] = 2 p + q = i + 2 An A 3 -structure on a graded category is just a 3-cocycle m 3 , { m 3 } ∈ HH 3 , − 1 ( A ) . An A ∞ -structure is a sequence of cochains m 3 , . . . , m n , . . . such that m 3 , . . . , m n is an A n -structure for all n ≥ 3. Fernando Muro When can we enhance a triangulated category?

Minimal A n -categories Definition A minimal A n -structure on a graded category A is given by Hochschild cochains m 3 , . . . , m i , . . . , m n of bidegree ( i , 2 − i ) such that � ∂ ( m i ) + 1 0 , n ≥ i ≥ 3 . [ m p , m q ] = 2 p + q = i + 2 An A 3 -structure on a graded category is just a 3-cocycle m 3 , { m 3 } ∈ HH 3 , − 1 ( A ) . An A ∞ -structure is a sequence of cochains m 3 , . . . , m n , . . . such that m 3 , . . . , m n is an A n -structure for all n ≥ 3. Fernando Muro When can we enhance a triangulated category?

Pretriangulated A ∞ -categories The derived category D ( A ) of an A ∞ -category A is the homotopy category of right A -modules, which is triangulated in a natural way. The inclusion of free modules induces a functor H 0 A − → D ( A ) �→ A ( · , X ) . X Definition An A ∞ -category A is pretriangulated if H 0 A is a triangulated subcategory of D ( A ) . If A is pretriangulated and T = H 0 ( A ) then ∼ H n A ( X , Y ) T ( X , Σ n Y ) , n ∈ Z , = where Σ is the suspension in T . Fernando Muro When can we enhance a triangulated category?

Pretriangulated A ∞ -categories The derived category D ( A ) of an A ∞ -category A is the homotopy category of right A -modules, which is triangulated in a natural way. The inclusion of free modules induces a functor H 0 A − → D ( A ) �→ A ( · , X ) . X Definition An A ∞ -category A is pretriangulated if H 0 A is a triangulated subcategory of D ( A ) . If A is pretriangulated and T = H 0 ( A ) then ∼ H n A ( X , Y ) T ( X , Σ n Y ) , n ∈ Z , = where Σ is the suspension in T . Fernando Muro When can we enhance a triangulated category?

Pretriangulated A ∞ -categories The derived category D ( A ) of an A ∞ -category A is the homotopy category of right A -modules, which is triangulated in a natural way. The inclusion of free modules induces a functor H 0 A − → D ( A ) �→ A ( · , X ) . X Definition An A ∞ -category A is pretriangulated if H 0 A is a triangulated subcategory of D ( A ) . If A is pretriangulated and T = H 0 ( A ) then ∼ H n A ( X , Y ) T ( X , Σ n Y ) , n ∈ Z , = where Σ is the suspension in T . Fernando Muro When can we enhance a triangulated category?

The problem Let T be a triangulated category over k with suspension Σ . When is T algebraic? Translation: For k a field, we wonder about the existence of a minimal pretriangulated A ∞ -category A = ( T Σ , m ) on the graded category T Σ with the same objects as T and morphisms � T ( X , Σ n Y ) , T Σ ( X , Y ) = n ∈ Z such that T embeds as a triangulated subcategory of D ( A ) . For this we have to find m 3 , m 4 , . . . adequately. Fernando Muro When can we enhance a triangulated category?

The problem Let T be a triangulated category over k with suspension Σ . When is T algebraic? Translation: For k a field, we wonder about the existence of a minimal pretriangulated A ∞ -category A = ( T Σ , m ) on the graded category T Σ with the same objects as T and morphisms � T ( X , Σ n Y ) , T Σ ( X , Y ) = n ∈ Z such that T embeds as a triangulated subcategory of D ( A ) . For this we have to find m 3 , m 4 , . . . adequately. Fernando Muro When can we enhance a triangulated category?

The problem Let T be a triangulated category over k with suspension Σ . When is T algebraic? Translation: For k a field, we wonder about the existence of a minimal pretriangulated A ∞ -category A = ( T Σ , m ) on the graded category T Σ with the same objects as T and morphisms � T ( X , Σ n Y ) , T Σ ( X , Y ) = n ∈ Z such that T embeds as a triangulated subcategory of D ( A ) . For this we have to find m 3 , m 4 , . . . adequately. Fernando Muro When can we enhance a triangulated category?

The problem Let T be a triangulated category over k with suspension Σ . When is T algebraic? Translation: For k a field, we wonder about the existence of a minimal pretriangulated A ∞ -category A = ( T Σ , m ) on the graded category T Σ with the same objects as T and morphisms � T ( X , Σ n Y ) , T Σ ( X , Y ) = n ∈ Z such that T embeds as a triangulated subcategory of D ( A ) . For this we have to find m 3 , m 4 , . . . adequately. Fernando Muro When can we enhance a triangulated category?

Secondary compositions A secondary composition or Massey product or Toda bracket in an additive graded category C is an operation which sends composable homogeneous morphisms g h f − → Y − → X − → W , Z with fg = 0 and gh = 0, to C ( Z , W ) � f , g , h � ∈ f · C ( Z , X ) + C ( Y , W ) · h , such that deg ( � f , g , h � ) deg ( f ) + deg ( g ) + deg ( h ) − 1 , = � f , g , h � · i ⊂ � f , g , h · i � ⊂ � f , g · h , i � ⊃ � f · g , h , i � ⊃ ( − 1 ) deg ( f ) f · � g , h , i � . Fernando Muro When can we enhance a triangulated category?

Secondary compositions A secondary composition or Massey product or Toda bracket in an additive graded category C is an operation which sends composable homogeneous morphisms g h f − → Y − → X − → W , Z with fg = 0 and gh = 0, to C ( Z , W ) � f , g , h � ∈ f · C ( Z , X ) + C ( Y , W ) · h , such that deg ( � f , g , h � ) deg ( f ) + deg ( g ) + deg ( h ) − 1 , = � f , g , h � · i ⊂ � f , g , h · i � ⊂ � f , g · h , i � ⊃ � f · g , h , i � ⊃ ( − 1 ) deg ( f ) f · � g , h , i � . Fernando Muro When can we enhance a triangulated category?

Secondary compositions in T Σ The graded category T Σ carries a secondary composition induced by the triangulated structure on T . Given g h f � Y � X � W Z exact This extends canonically to a secondary composition in T Σ . Fernando Muro When can we enhance a triangulated category?

Secondary compositions in T Σ The graded category T Σ carries a secondary composition induced by the triangulated structure on T . Given g h f � Y � X � W in T Z exact This extends canonically to a secondary composition in T Σ . Fernando Muro When can we enhance a triangulated category?

Secondary compositions in T Σ � � � � The graded category T Σ carries a secondary composition induced by the triangulated structure on T . Given 0 � � � � � � � g h f � W in T Z Y X � � � � � � � 0 exact This extends canonically to a secondary composition in T Σ . Fernando Muro When can we enhance a triangulated category?

Secondary compositions in T Σ The graded category T Σ carries a secondary composition induced by the triangulated structure on T . Given g h f � Y � X � W in T Z � V � X � W Σ − 1 W exact f This extends canonically to a secondary composition in T Σ . Fernando Muro When can we enhance a triangulated category?

Secondary compositions in T Σ � � The graded category T Σ carries a secondary composition induced by the triangulated structure on T . Given 0 � � � � � � � g h f � Y � W in T Z X � V � X � W Σ − 1 W exact f This extends canonically to a secondary composition in T Σ . Fernando Muro When can we enhance a triangulated category?

Secondary compositions in T Σ � � � � The graded category T Σ carries a secondary composition induced by the triangulated structure on T . Given 0 � � � � � � � g h f � Y � W in T Z X � V � W Σ − 1 W exact X f This extends canonically to a secondary composition in T Σ . Fernando Muro When can we enhance a triangulated category?

Secondary compositions in T Σ � � � � The graded category T Σ carries a secondary composition induced by the triangulated structure on T . Given 0 � � � � � � � g h f � X � W in T Z Y � V � W Σ − 1 W exact X f This extends canonically to a secondary composition in T Σ . Fernando Muro When can we enhance a triangulated category?

Secondary compositions in T Σ � � � � � � The graded category T Σ carries a secondary composition induced by the triangulated structure on T . Given 0 � � � � � � � g h f � X � W in T Z Y � W Σ − 1 W exact V X f This extends canonically to a secondary composition in T Σ . Fernando Muro When can we enhance a triangulated category?

Secondary compositions in T Σ � � � � The graded category T Σ carries a secondary composition induced by the triangulated structure on T . Given g h f � Y � X � W in T Z � f , g , h � ∋ � W Σ − 1 W exact V X f This extends canonically to a secondary composition in T Σ . Fernando Muro When can we enhance a triangulated category?

Secondary compositions in T Σ � � � � The graded category T Σ carries a secondary composition induced by the triangulated structure on T . Given g h f � Y � X � W in T Z � f , g , h � ∋ � W Σ − 1 W exact V X f This extends canonically to a secondary composition in T Σ . Fernando Muro When can we enhance a triangulated category?

Secondary compositions in T Σ Conversely, this secondary composition determines the exact triangles. Proposition f i q A triangle X → Y → C → Σ X is exact in T if and only if T ( U , X ) → T ( U , Y ) → T ( U , C ) → T ( U , Σ X ) → T ( U , Σ Y ) is exact for any object U in T and 1 X ∈ � q , i , f � ⊂ T ( X , X ) . Using [Heller’68] one can actually determine the subset { Puppe triangulated structures in T } ⊆ { Secondary compositions in T Σ } which is the intersection of an ‘open’ and a ‘closed’ subset. Fernando Muro When can we enhance a triangulated category?

Secondary compositions in T Σ Conversely, this secondary composition determines the exact triangles. Proposition f i q A triangle X → Y → C → Σ X is exact in T if and only if T ( U , X ) → T ( U , Y ) → T ( U , C ) → T ( U , Σ X ) → T ( U , Σ Y ) is exact for any object U in T and 1 X ∈ � q , i , f � ⊂ T ( X , X ) . Using [Heller’68] one can actually determine the subset { Puppe triangulated structures in T } ⊆ { Secondary compositions in T Σ } which is the intersection of an ‘open’ and a ‘closed’ subset. Fernando Muro When can we enhance a triangulated category?

Secondary compositions in T Σ Definition A finitely presented right T -module is a functor M : T op → k - Mod which fits into an exact sequence T ( · , X ) → T ( · , Y ) → M → 0 Theorem (Freyd’66) The category mod - T of finitely presented right T -modules is a Frobenius abelian category. Fernando Muro When can we enhance a triangulated category?

Secondary compositions in T Σ Definition A finitely presented right T -module is a functor M : T op → k - Mod which fits into an exact sequence T ( · , X ) → T ( · , Y ) → M → 0 Theorem (Freyd’66) The category mod - T of finitely presented right T -modules is a Frobenius abelian category. Fernando Muro When can we enhance a triangulated category?

Secondary compositions in T Σ The suspension functor in T extends uniquely to an exact equivalence Σ: mod - T − → mod - T . We can therefore define a graded category mod - T Σ with the same objects as mod - T and graded morphisms � Hom ∗ Hom T ( M , Σ n N ) , T ( M , N ) = n ∈ Z and also bigraded ext’s Ext ∗ , ∗ T . Proposition = HH 0 , − 1 ( mod - T Σ , Ext 3 , ∗ { Secondary compositions in T Σ } ∼ T ) . skip proof Fernando Muro When can we enhance a triangulated category?

Secondary compositions in T Σ The suspension functor in T extends uniquely to an exact equivalence Σ: mod - T − → mod - T . We can therefore define a graded category mod - T Σ with the same objects as mod - T and graded morphisms � Hom ∗ Hom T ( M , Σ n N ) , T ( M , N ) = n ∈ Z and also bigraded ext’s Ext ∗ , ∗ T . Proposition = HH 0 , − 1 ( mod - T Σ , Ext 3 , ∗ { Secondary compositions in T Σ } ∼ T ) . skip proof Fernando Muro When can we enhance a triangulated category?

Secondary compositions in T Σ The suspension functor in T extends uniquely to an exact equivalence Σ: mod - T − → mod - T . We can therefore define a graded category mod - T Σ with the same objects as mod - T and graded morphisms � Hom ∗ Hom T ( M , Σ n N ) , T ( M , N ) = n ∈ Z and also bigraded ext’s Ext ∗ , ∗ T . Proposition = HH 0 , − 1 ( mod - T Σ , Ext 3 , ∗ { Secondary compositions in T Σ } ∼ T ) . skip proof Fernando Muro When can we enhance a triangulated category?

Secondary compositions in T Σ Idea of the proof. Suppose we have a secondary composition �· , · , ·� . We now define an element in κ ∈ HH 0 , − 1 ( mod - T Σ , Ext 3 , ∗ T ) . Let M be in mod - T , T ( · , f ) � T ( · , Y ) p � � M , · T ( · , X ) . . i q f Σ − 1 Y exact. − → C − → X − → Y , Fernando Muro When can we enhance a triangulated category?

Secondary compositions in T Σ Idea of the proof. Suppose we have a secondary composition �· , · , ·� . We now define an element in κ ∈ HH 0 , − 1 ( mod - T Σ , Ext 3 , ∗ T ) . Let M be in mod - T , T ( · , f ) � T ( · , Y ) p � � M , · T ( · , X ) . . i q f Σ − 1 Y exact. − → C − → X − → Y , Fernando Muro When can we enhance a triangulated category?

Secondary compositions in T Σ Idea of the proof. Suppose we have a secondary composition �· , · , ·� . We now define an element in κ ∈ HH 0 , − 1 ( mod - T Σ , Ext 3 , ∗ T ) . Let M be in mod - T , T ( · , f ) � T ( · , Y ) p � � M , · T ( · , X ) . . i q f Σ − 1 Y exact. − → C − → X − → Y , Fernando Muro When can we enhance a triangulated category?

Secondary compositions in T Σ Idea of the proof. Suppose we have a secondary composition �· , · , ·� . We now define an element in κ ∈ HH 0 , − 1 ( mod - T Σ , Ext 3 , ∗ T ) . Let M be in mod - T , T ( · , i ) T ( · , q ) � T ( · , X ) T ( · , f ) � T ( · , Y ) p � T ( · , C ) · T ( · , Σ − 1 Y ) � � M , . . q Σ − 1 Y i f exact. − → C − → X − → Y , Fernando Muro When can we enhance a triangulated category?

Secondary compositions in T Σ � � Idea of the proof. Suppose we have a secondary composition �· , · , ·� . We now define an element in κ ∈ HH 0 , − 1 ( mod - T Σ , Ext 3 , ∗ T ) . Let M be in mod - T , T ( · , i ) T ( · , q ) � T ( · , X ) T ( · , f ) � T ( · , Y ) p · T ( · , Σ − 1 Y ) � � M , T ( · , C ) T ( · , � f , q , i � ) ∋ . T ( · , Σ − 1 Y ) . q Σ − 1 Y i f exact. − → C − → X − → Y , Fernando Muro When can we enhance a triangulated category?

Secondary compositions in T Σ � � Idea of the proof. Suppose we have a secondary composition �· , · , ·� . We now define an element in κ ∈ HH 0 , − 1 ( mod - T Σ , Ext 3 , ∗ T ) . Let M be in mod - T , T ( · , i ) T ( · , q ) � T ( · , X ) T ( · , f ) � T ( · , Y ) p · T ( · , Σ − 1 Y ) � � M , T ( · , C ) T ( · , � f , q , i � ) ∋ Σ − 1 p � � Σ − 1 M . T ( · , Σ − 1 Y ) . q Σ − 1 Y i f exact. − → C − → X − → Y , Fernando Muro When can we enhance a triangulated category?

Secondary compositions in T Σ � � � Idea of the proof. Suppose we have a secondary composition �· , · , ·� . We now define an element in κ ∈ HH 0 , − 1 ( mod - T Σ , Ext 3 , ∗ T ) . Let M be in mod - T , T ( · , i ) T ( · , q ) � T ( · , X ) T ( · , f ) � T ( · , Y ) p · T ( · , Σ − 1 Y ) � � M , T ( · , C ) � ∈ κ ( M ) ∈ Ext 3 , − 1 � ( M , M ) � � � T T ( · , � f , q , i � ) ∋ � � � � � Σ − 1 p � � Σ − 1 M � . T ( · , Σ − 1 Y ) . q Σ − 1 Y i f exact. − → C − → X − → Y , Fernando Muro When can we enhance a triangulated category?

The first obstructions Theorem For k a field and any r ∈ Z there is a spectral sequence E p , q = HH p , r ( mod - T Σ , Ext q , ∗ ⇒ HH p + q , r ( T Σ ) . T ) = 2 If T = H 0 A for some pretriangulated minimal A ∞ -category A then the edge homomorphism for r = − 1 satisfies HH 0 , − 1 ( mod - T Σ , Ext 3 , ∗ T ) = E 0 , 3 HH 3 , − 1 ( T Σ ) − → 2 { m 3 } �→ �· , · , ·� . Corollary If T is algebraic over a field k then the secondary composition �· , · , ·� in T Σ is a permanent cycle in the previous spectral sequence. Fernando Muro When can we enhance a triangulated category?

The first obstructions Theorem For k a field and any r ∈ Z there is a spectral sequence E p , q = HH p , r ( mod - T Σ , Ext q , ∗ ⇒ HH p + q , r ( T Σ ) . T ) = 2 If T = H 0 A for some pretriangulated minimal A ∞ -category A then the edge homomorphism for r = − 1 satisfies HH 0 , − 1 ( mod - T Σ , Ext 3 , ∗ T ) = E 0 , 3 HH 3 , − 1 ( T Σ ) − → 2 { m 3 } �→ �· , · , ·� . Corollary If T is algebraic over a field k then the secondary composition �· , · , ·� in T Σ is a permanent cycle in the previous spectral sequence. Fernando Muro When can we enhance a triangulated category?

The first obstructions Theorem For k a field and any r ∈ Z there is a spectral sequence E p , q = HH p , r ( mod - T Σ , Ext q , ∗ ⇒ HH p + q , r ( T Σ ) . T ) = 2 If T = H 0 A for some pretriangulated minimal A ∞ -category A then the edge homomorphism for r = − 1 satisfies HH 0 , − 1 ( mod - T Σ , Ext 3 , ∗ T ) = E 0 , 3 HH 3 , − 1 ( T Σ ) − → 2 { m 3 } �→ �· , · , ·� . Corollary If T is algebraic over a field k then the secondary composition �· , · , ·� in T Σ is a permanent cycle in the previous spectral sequence. Fernando Muro When can we enhance a triangulated category?

The first obstructions Conversely, if �· , · , ·� is a permanent cycle we can choose a ( 3 , − 1 ) -cocycle m 3 such that { m 3 } �→ �· , · , ·� through the edge homomorphism. Such a choice yields an A 3 -structure on T Σ that we can try to extend to an A ∞ -structure. Fernando Muro When can we enhance a triangulated category?

The first obstructions Conversely, if �· , · , ·� is a permanent cycle we can choose a ( 3 , − 1 ) -cocycle m 3 such that { m 3 } �→ �· , · , ·� through the edge homomorphism. Such a choice yields an A 3 -structure on T Σ that we can try to extend to an A ∞ -structure. Fernando Muro When can we enhance a triangulated category?

The first obstructions Therefore the first obstructions for the existence of an A ∞ -enhancement are E 2 , 2 = HH 2 , − 1 ( mod - T Σ , Ext 2 , ∗ T ) , if = 0 d 2 ( �· , · , ·� ) ∈ 2 E 3 , 1 then d 3 ( �· , · , ·� ) 3 , if = 0 ∈ E 4 , 0 և HH 4 , − 1 ( mod - T Σ , Hom ∗ then d 4 ( �· , · , ·� ) T ) , if = 0 ∈ 4 then there is an A 3 -enhancement of T Σ . Fernando Muro When can we enhance a triangulated category?

The first obstructions Therefore the first obstructions for the existence of an A ∞ -enhancement are E 2 , 2 = HH 2 , − 1 ( mod - T Σ , Ext 2 , ∗ T ) , if = 0 d 2 ( �· , · , ·� ) ∈ 2 E 3 , 1 then d 3 ( �· , · , ·� ) 3 , if = 0 ∈ E 4 , 0 և HH 4 , − 1 ( mod - T Σ , Hom ∗ then d 4 ( �· , · , ·� ) T ) , if = 0 ∈ 4 then there is an A 3 -enhancement of T Σ . Fernando Muro When can we enhance a triangulated category?

The first obstructions Therefore the first obstructions for the existence of an A ∞ -enhancement are E 2 , 2 = HH 2 , − 1 ( mod - T Σ , Ext 2 , ∗ T ) , if = 0 d 2 ( �· , · , ·� ) ∈ 2 E 3 , 1 then d 3 ( �· , · , ·� ) 3 , if = 0 ∈ E 4 , 0 և HH 4 , − 1 ( mod - T Σ , Hom ∗ then d 4 ( �· , · , ·� ) T ) , if = 0 ∈ 4 then there is an A 3 -enhancement of T Σ . Fernando Muro When can we enhance a triangulated category?

The first obstructions Therefore the first obstructions for the existence of an A ∞ -enhancement are E 2 , 2 = HH 2 , − 1 ( mod - T Σ , Ext 2 , ∗ T ) , if = 0 d 2 ( �· , · , ·� ) ∈ 2 E 3 , 1 then d 3 ( �· , · , ·� ) 3 , if = 0 ∈ E 4 , 0 և HH 4 , − 1 ( mod - T Σ , Hom ∗ then d 4 ( �· , · , ·� ) T ) , if = 0 ∈ 4 then there is an A 3 -enhancement of T Σ . Fernando Muro When can we enhance a triangulated category?

Higher obstructions Suppose that we have enhanced T Σ to an A n − 1 -category, n > 3, in a compatible way with the triangulated structure of T . Then     1 �  ∈ HH n + 1 , 2 − n ( T Σ ) . [ m p , m q ] 2  p + q = n + 2 If this cohomology class vanishes then any trivialising cochain m n yields an extension to an A n -category since ∂ ( m n ) + 1 � 0 , [ m p , m q ] = 2 p + q = n + 2 and conversely. Fernando Muro When can we enhance a triangulated category?

Higher obstructions Suppose that we have enhanced T Σ to an A n − 1 -category, n > 3, in a compatible way with the triangulated structure of T . Then     1 �  ∈ HH n + 1 , 2 − n ( T Σ ) . [ m p , m q ] 2  p + q = n + 2 If this cohomology class vanishes then any trivialising cochain m n yields an extension to an A n -category since ∂ ( m n ) + 1 � 0 , [ m p , m q ] = 2 p + q = n + 2 and conversely. Fernando Muro When can we enhance a triangulated category?

Higher obstructions Suppose that we have enhanced T Σ to an A n − 1 -category, n > 3, in a compatible way with the triangulated structure of T . Then     1 �  ∈ HH n + 1 , 2 − n ( T Σ ) . [ m p , m q ] 2  p + q = n + 2 If this cohomology class vanishes then any trivialising cochain m n yields an extension to an A n -category since ∂ ( m n ) + 1 � 0 , [ m p , m q ] = 2 p + q = n + 2 and conversely. Fernando Muro When can we enhance a triangulated category?

Higher obstructions The first of these higher obstructions is as follows. Example For n = 4 , if ( T Σ , m 3 ) is an A 3 -category, the obstruction for the existence of an A 4 -enhancement is obtained from { m 3 } ∈ HH 3 , − 1 ( T Σ ) , 1 HH 5 , − 2 ( T Σ ) . 2 [ { m 3 } , { m 3 } ] ∈ dual numbers Tate Amiot skip Fernando Muro When can we enhance a triangulated category?

The example of dual numbers Let T = finitely generated free modules over k [ ε ] / ( ε 2 ) and Σ is the identity on objects and such that Σ( ε ) = − ε . HH 0 , − 1 ( mod - T Σ , Ext 3 , ∗ ∼ ∼ { Secondary compositions in T Σ } = T ) = k . Each x ∈ k × corresponds to the secondary composition of an algebraic triangulated structure on T with exact triangle k [ ε ] / ( ε 2 ) → k [ ε ] / ( ε 2 ) → k [ ε ] / ( ε 2 ) x · ε → k [ ε ] / ( ε 2 ) . ε ε − − − And 0 ∈ k does not correspond to any triangulated structure. Fernando Muro When can we enhance a triangulated category?

The example of dual numbers Let T = finitely generated free modules over k [ ε ] / ( ε 2 ) and Σ is the identity on objects and such that Σ( ε ) = − ε . HH 0 , − 1 ( mod - T Σ , Ext 3 , ∗ ∼ ∼ { Secondary compositions in T Σ } = T ) = k . Each x ∈ k × corresponds to the secondary composition of an algebraic triangulated structure on T with exact triangle k [ ε ] / ( ε 2 ) → k [ ε ] / ( ε 2 ) → k [ ε ] / ( ε 2 ) x · ε → k [ ε ] / ( ε 2 ) . ε ε − − − And 0 ∈ k does not correspond to any triangulated structure. Fernando Muro When can we enhance a triangulated category?

The example of dual numbers Let T = finitely generated free modules over k [ ε ] / ( ε 2 ) and Σ is the identity on objects and such that Σ( ε ) = − ε . HH 0 , − 1 ( mod - T Σ , Ext 3 , ∗ ∼ ∼ { Secondary compositions in T Σ } = T ) = k . Each x ∈ k × corresponds to the secondary composition of an algebraic triangulated structure on T with exact triangle k [ ε ] / ( ε 2 ) → k [ ε ] / ( ε 2 ) → k [ ε ] / ( ε 2 ) x · ε → k [ ε ] / ( ε 2 ) . ε ε − − − And 0 ∈ k does not correspond to any triangulated structure. Fernando Muro When can we enhance a triangulated category?

The example of dual numbers The edge homomorphism is HH 0 , − 1 ( mod - T Σ , Ext 3 , ∗ HH 3 , − 1 ( T Σ ) ∼ T ) ∼ = k · α ⊕ k · β − → = k , 1 , α �→ 0 , β �→ x � = 0 . y ∈ k , x · α + y · β �→ Let { m 3 } = x · α + y · β . The obstruction to enhance ( T Σ , m 3 ) to an A 4 -category is 1 xy [ α, β ] + 1 2 y 2 [ β, β ] , 2 [ x · α + y · β, x · α + y · β ] = ∈ HH 5 , − 2 ( T Σ ) ∼ [ α, α ] = 0 , k · [ α, β ] ⊕ k · [ β, β ] , = so the obstruction vanishes if and only if y = 0. Tate Amiot skip Fernando Muro When can we enhance a triangulated category?

The example of dual numbers The edge homomorphism is HH 0 , − 1 ( mod - T Σ , Ext 3 , ∗ HH 3 , − 1 ( T Σ ) ∼ T ) ∼ = k · α ⊕ k · β − → = k , 1 , α �→ 0 , β �→ x � = 0 . y ∈ k , x · α + y · β �→ Let { m 3 } = x · α + y · β . The obstruction to enhance ( T Σ , m 3 ) to an A 4 -category is 1 xy [ α, β ] + 1 2 y 2 [ β, β ] , 2 [ x · α + y · β, x · α + y · β ] = ∈ HH 5 , − 2 ( T Σ ) ∼ [ α, α ] = 0 , k · [ α, β ] ⊕ k · [ β, β ] , = so the obstruction vanishes if and only if y = 0. Tate Amiot skip Fernando Muro When can we enhance a triangulated category?

The example of dual numbers The edge homomorphism is HH 0 , − 1 ( mod - T Σ , Ext 3 , ∗ HH 3 , − 1 ( T Σ ) ∼ T ) ∼ = k · α ⊕ k · β − → = k , 1 , α �→ 0 , β �→ x � = 0 . y ∈ k , x · α + y · β �→ Let { m 3 } = x · α + y · β . The obstruction to enhance ( T Σ , m 3 ) to an A 4 -category is 1 xy [ α, β ] + 1 2 y 2 [ β, β ] , 2 [ x · α + y · β, x · α + y · β ] = ∈ HH 5 , − 2 ( T Σ ) ∼ [ α, α ] = 0 , k · [ α, β ] ⊕ k · [ β, β ] , = so the obstruction vanishes if and only if y = 0. Tate Amiot skip Fernando Muro When can we enhance a triangulated category?

The example of dual numbers The edge homomorphism is HH 0 , − 1 ( mod - T Σ , Ext 3 , ∗ HH 3 , − 1 ( T Σ ) ∼ T ) ∼ = k · α ⊕ k · β − → = k , 1 , α �→ 0 , β �→ x � = 0 . y ∈ k , x · α + y · β �→ Let { m 3 } = x · α + y · β . The obstruction to enhance ( T Σ , m 3 ) to an A 4 -category is 1 xy [ α, β ] + 1 2 y 2 [ β, β ] , 2 [ x · α + y · β, x · α + y · β ] = ∈ HH 5 , − 2 ( T Σ ) ∼ [ α, α ] = 0 , k · [ α, β ] ⊕ k · [ β, β ] , = so the obstruction vanishes if and only if y = 0. Tate Amiot skip Fernando Muro When can we enhance a triangulated category?

The example of dual numbers The edge homomorphism is HH 0 , − 1 ( mod - T Σ , Ext 3 , ∗ HH 3 , − 1 ( T Σ ) ∼ T ) ∼ = k · α ⊕ k · β − → = k , 1 , α �→ 0 , β �→ x � = 0 . y ∈ k , x · α + y · β �→ Let { m 3 } = x · α + y · β . The obstruction to enhance ( T Σ , m 3 ) to an A 4 -category is 1 xy [ α, β ] + 1 2 y 2 [ β, β ] , 2 [ x · α + y · β, x · α + y · β ] = ∈ HH 5 , − 2 ( T Σ ) ∼ [ α, α ] = 0 , k · [ α, β ] ⊕ k · [ β, β ] , = so the obstruction vanishes if and only if y = 0. Tate Amiot skip Fernando Muro When can we enhance a triangulated category?

The example of dual numbers The edge homomorphism is HH 0 , − 1 ( mod - T Σ , Ext 3 , ∗ HH 3 , − 1 ( T Σ ) ∼ T ) ∼ = k · α ⊕ k · β − → = k , 1 , α �→ 0 , β �→ x � = 0 . y ∈ k , x · α + y · β �→ Let { m 3 } = x · α + y · β . The obstruction to enhance ( T Σ , m 3 ) to an A 4 -category is 1 xy [ α, β ] + 1 2 y 2 [ β, β ] , 2 [ x · α + y · β, x · α + y · β ] = ∈ HH 5 , − 2 ( T Σ ) ∼ [ α, α ] = 0 , k · [ α, β ] ⊕ k · [ β, β ] , = so the obstruction vanishes if and only if y = 0. Tate Amiot skip Fernando Muro When can we enhance a triangulated category?

Another application of the spectral sequence E p , q = HH p , r ( mod - T Σ , Ext q , ∗ HH p + q , r ( T Σ ) , T ) = ⇒ 2 E p , q H ∗ ( G , k ) , Ext q , ∗ = HH p , r ( Mod - � HH p + q , r ( � H ∗ ( G , k )) , H ∗ ( G , k ) ) = ⇒ 2 b Here G is a finite group and � H ∗ ( G , k ) is Tate cohomology. edge H ∗ ( G , k ) , Ext 3 , ∗ HH 3 , − 1 ( � HH 0 , − 1 ( Mod - � H ∗ ( G , k )) − → H ∗ ( G , k ) ) , b γ G �→ κ, Theorem (Benson–Krause–Schwede’03) Given a right � H ∗ ( G , k ) -module X, κ ( X ) = 0 ⇔ X is a direct summand of � H ∗ ( G , M ) for some kG-module M. Moreover, there is a class γ G such that the edge homomorphism maps γ G to κ . dual numbers Amiot skip Fernando Muro When can we enhance a triangulated category?

Another application of the spectral sequence E p , q = HH p , r ( mod - T Σ , Ext q , ∗ HH p + q , r ( T Σ ) , T ) = ⇒ 2 E p , q H ∗ ( G , k ) , Ext q , ∗ = HH p , r ( Mod - � HH p + q , r ( � H ∗ ( G , k )) , H ∗ ( G , k ) ) = ⇒ 2 b Here G is a finite group and � H ∗ ( G , k ) is Tate cohomology. edge H ∗ ( G , k ) , Ext 3 , ∗ HH 3 , − 1 ( � HH 0 , − 1 ( Mod - � H ∗ ( G , k )) − → H ∗ ( G , k ) ) , b γ G �→ κ, Theorem (Benson–Krause–Schwede’03) Given a right � H ∗ ( G , k ) -module X, κ ( X ) = 0 ⇔ X is a direct summand of � H ∗ ( G , M ) for some kG-module M. Moreover, there is a class γ G such that the edge homomorphism maps γ G to κ . dual numbers Amiot skip Fernando Muro When can we enhance a triangulated category?

Another application of the spectral sequence E p , q = HH p , r ( mod - T Σ , Ext q , ∗ HH p + q , r ( T Σ ) , T ) = ⇒ 2 E p , q H ∗ ( G , k ) , Ext q , ∗ = HH p , r ( Mod - � HH p + q , r ( � H ∗ ( G , k )) , H ∗ ( G , k ) ) = ⇒ 2 b Here G is a finite group and � H ∗ ( G , k ) is Tate cohomology. edge H ∗ ( G , k ) , Ext 3 , ∗ HH 3 , − 1 ( � HH 0 , − 1 ( Mod - � H ∗ ( G , k )) − → H ∗ ( G , k ) ) , b γ G �→ κ, Theorem (Benson–Krause–Schwede’03) Given a right � H ∗ ( G , k ) -module X, κ ( X ) = 0 ⇔ X is a direct summand of � H ∗ ( G , M ) for some kG-module M. Moreover, there is a class γ G such that the edge homomorphism maps γ G to κ . dual numbers Amiot skip Fernando Muro When can we enhance a triangulated category?

Another application of the spectral sequence E p , q = HH p , r ( mod - T Σ , Ext q , ∗ HH p + q , r ( T Σ ) , T ) = ⇒ 2 E p , q H ∗ ( G , k ) , Ext q , ∗ = HH p , r ( Mod - � HH p + q , r ( � H ∗ ( G , k )) , H ∗ ( G , k ) ) = ⇒ 2 b Here G is a finite group and � H ∗ ( G , k ) is Tate cohomology. edge H ∗ ( G , k ) , Ext 3 , ∗ HH 3 , − 1 ( � HH 0 , − 1 ( Mod - � H ∗ ( G , k )) − → H ∗ ( G , k ) ) , b γ G �→ κ, Theorem (Benson–Krause–Schwede’03) Given a right � H ∗ ( G , k ) -module X, κ ( X ) = 0 ⇔ X is a direct summand of � H ∗ ( G , M ) for some kG-module M. Moreover, there is a class γ G such that the edge homomorphism maps γ G to κ . dual numbers Amiot skip Fernando Muro When can we enhance a triangulated category?

Another application of the spectral sequence E p , q = HH p , r ( mod - T Σ , Ext q , ∗ HH p + q , r ( T Σ ) , T ) = ⇒ 2 E p , q H ∗ ( G , k ) , Ext q , ∗ = HH p , r ( Mod - � HH p + q , r ( � H ∗ ( G , k )) , H ∗ ( G , k ) ) = ⇒ 2 b Here G is a finite group and � H ∗ ( G , k ) is Tate cohomology. edge H ∗ ( G , k ) , Ext 3 , ∗ HH 3 , − 1 ( � HH 0 , − 1 ( Mod - � H ∗ ( G , k )) − → H ∗ ( G , k ) ) , b γ G �→ κ, Theorem (Benson–Krause–Schwede’03) Given a right � H ∗ ( G , k ) -module X, κ ( X ) = 0 ⇔ X is a direct summand of � H ∗ ( G , M ) for some kG-module M. Moreover, there is a class γ G such that the edge homomorphism maps γ G to κ . dual numbers Amiot skip Fernando Muro When can we enhance a triangulated category?

Another application of the spectral sequence E p , q = HH p , r ( mod - T Σ , Ext q , ∗ HH p + q , r ( T Σ ) , T ) = ⇒ 2 E p , q H ∗ ( G , k ) , Ext q , ∗ = HH p , r ( Mod - � HH p + q , r ( � H ∗ ( G , k )) , H ∗ ( G , k ) ) = ⇒ 2 b Here G is a finite group and � H ∗ ( G , k ) is Tate cohomology. edge H ∗ ( G , k ) , Ext 3 , ∗ HH 3 , − 1 ( � HH 0 , − 1 ( Mod - � H ∗ ( G , k )) − → H ∗ ( G , k ) ) , b γ G �→ κ, Theorem (Benson–Krause–Schwede’03) Given a right � H ∗ ( G , k ) -module X, κ ( X ) = 0 ⇔ X is a direct summand of � H ∗ ( G , M ) for some kG-module M. Moreover, there is a class γ G such that the edge homomorphism maps γ G to κ . dual numbers Amiot skip Fernando Muro When can we enhance a triangulated category?

An open problem There are relevant finiteness conditions on triangulated categories which may allow cohomological computations: Krull–Remak–Schmidt. Finitely many indecomposables. Finite-dimensional hom’s. Over an algebraically closed field k , [Amiot’06] has classified the underlying category of a wide class of triangulated categories satisfying these conditions. This class includes maximal d -Calabi–Yau’s, d ≥ 2. It could be interesting to determine how many of them are algebraic for k = Q . This could eventually yield examples of exotic triangulated categories where 2 and all primes are invertible. dual numbers Tate skip Fernando Muro When can we enhance a triangulated category?

An open problem There are relevant finiteness conditions on triangulated categories which may allow cohomological computations: Krull–Remak–Schmidt. Finitely many indecomposables. Finite-dimensional hom’s. Over an algebraically closed field k , [Amiot’06] has classified the underlying category of a wide class of triangulated categories satisfying these conditions. This class includes maximal d -Calabi–Yau’s, d ≥ 2. It could be interesting to determine how many of them are algebraic for k = Q . This could eventually yield examples of exotic triangulated categories where 2 and all primes are invertible. dual numbers Tate skip Fernando Muro When can we enhance a triangulated category?

An open problem There are relevant finiteness conditions on triangulated categories which may allow cohomological computations: Krull–Remak–Schmidt. Finitely many indecomposables. Finite-dimensional hom’s. Over an algebraically closed field k , [Amiot’06] has classified the underlying category of a wide class of triangulated categories satisfying these conditions. This class includes maximal d -Calabi–Yau’s, d ≥ 2. It could be interesting to determine how many of them are algebraic for k = Q . This could eventually yield examples of exotic triangulated categories where 2 and all primes are invertible. dual numbers Tate skip Fernando Muro When can we enhance a triangulated category?

Main ingredients The special features of fields which allow the definition of an obstruction theory for the existence of an A ∞ -enhancement of a triangulated category over a field k are: Kadeishvili’s theorem: any A ∞ -category is quasi-isomorphic to a minimal one. The spectral sequence: HH p , r ( mod - T Σ , Ext q , ∗ T ) ⇒ HH p + q , r ( T Σ ) . What happens when k is just a commutative ring? What about topological triangulated categories? Fernando Muro When can we enhance a triangulated category?

Main ingredients The special features of fields which allow the definition of an obstruction theory for the existence of an A ∞ -enhancement of a triangulated category over a field k are: Kadeishvili’s theorem: any A ∞ -category is quasi-isomorphic to a minimal one. The spectral sequence: HH p , r ( mod - T Σ , Ext q , ∗ T ) ⇒ HH p + q , r ( T Σ ) . What happens when k is just a commutative ring? What about topological triangulated categories? Fernando Muro When can we enhance a triangulated category?

Main ingredients The special features of fields which allow the definition of an obstruction theory for the existence of an A ∞ -enhancement of a triangulated category over a field k are: Kadeishvili’s theorem: any A ∞ -category is quasi-isomorphic to a minimal one. The spectral sequence: HH p , r ( mod - T Σ , Ext q , ∗ T ) ⇒ HH p + q , r ( T Σ ) . What happens when k is just a commutative ring? What about topological triangulated categories? Fernando Muro When can we enhance a triangulated category?

Main ingredients The special features of fields which allow the definition of an obstruction theory for the existence of an A ∞ -enhancement of a triangulated category over a field k are: Kadeishvili’s theorem: any A ∞ -category is quasi-isomorphic to a minimal one. The spectral sequence: HH p , r ( mod - T Σ , Ext q , ∗ T ) ⇒ HH p + q , r ( T Σ ) . What happens when k is just a commutative ring? What about topological triangulated categories? Fernando Muro When can we enhance a triangulated category?

Main ingredients The special features of fields which allow the definition of an obstruction theory for the existence of an A ∞ -enhancement of a triangulated category over a field k are: Kadeishvili’s theorem: any A ∞ -category is quasi-isomorphic to a minimal one. The spectral sequence: HH p , r ( mod - T Σ , Ext q , ∗ T ) ⇒ HH p + q , r ( T Σ ) . What happens when k is just a commutative ring? What about topological triangulated categories? Fernando Muro When can we enhance a triangulated category?

Arbitrary commutative ground ring k There is a version of Kadeishvili’s theorem over an arbitrary commutative ring. Theorem (Sagave’08) Any A ∞ -algebra is quasi-isomorphic to a minimal derived A ∞ -algebra. This theorem may be extended to A ∞ -categories. Derived A ∞ -algebras are related to Shukla cohomology (a.k.a. derived Hochschild cohomology) as A ∞ -algebras are related to Hochschild cohomology. In particular any derived A ∞ -algebra A yields a characteristic cohomology class γ A ∈ SH 3 , − 1 ( H ∗ A ) . Fernando Muro When can we enhance a triangulated category?