Triangulated categories with universal Toda bracket Fernando Muro - PowerPoint PPT Presentation

5 An example Z /p 2 ) and N = mod ( Z Z /p [ t ] /t 2 ) , for p ∈ Z Consider M = mod ( Z Z prime. In both cases Ho M ≃ Ho N ≃ mod ( Z Z /p ) , and Σ = 1 is the identity functor. The first k -invariants are Z /p ) , Hom) ∼ k M 1 , k N 1 ∈ H 3 ( mod ( Z = H 3 ML ( Z Z /p, Z Z /p ) = 0 , therefore P 1 L M ≃ P 1 L N . However L M ≃ / L N , [Schlichting, To¨ en-Vezzosi] . • Is there any n for which P n L M ≃ / P n L N ? Notice that for any pair of objects A , B , L M ( A, B ) ≃ L N ( A, B ) !!!

6 Cohomology of categories Let C be a category. A C -bimodule M is a functor M : C op × C − → Ab .

6 Cohomology of categories Let C be a category. A C -bimodule M is a functor M : C op × C − → Ab . Example . Z Z [Hom C ( − , − )] .

6 Cohomology of categories Let C be a category. A C -bimodule M is a functor M : C op × C − → Ab . Example . Z Z [Hom C ( − , − )] . The Hochschild-Mitchell cohomology of C with coefficients in M is given by H ∗ ( C , M ) Ext ∗ = C -bimod ( Z Z [Hom C ] , M ) .

6 Cohomology of categories Let C be a category. A C -bimodule M is a functor M : C op × C − → Ab . Example . Z Z [Hom C ( − , − )] . The Hochschild-Mitchell cohomology of C with coefficients in M is given by H ∗ ( C , M ) Ext ∗ = C -bimod ( Z Z [Hom C ] , M ) . A functor ϕ : D → C induces a homomorphism ϕ ∗ : H ∗ ( C , M ) − → H ∗ ( D , ϕ ∗ M ) , where ϕ ∗ M = M ( ϕ, ϕ ) .

7 Cohomology of categories This can be computed as the cohomology of a cobar-like complex F ∗ ( C , M ) where an n -cochain c is a function sending a chain of n composable morphisms in C σ 1 σn A 0 ← A 1 ← · · · ← A n − 1 ← A n to an element c ( σ 1 , · · · , σ n ) ∈ M ( A n , A 0 ) .

7 Cohomology of categories This can be computed as the cohomology of a cobar-like complex F ∗ ( C , M ) where an n -cochain c is a function sending a chain of n composable morphisms in C σ 1 σn A 0 ← A 1 ← · · · ← A n − 1 ← A n to an element c ( σ 1 , · · · , σ n ) ∈ M ( A n , A 0 ) . If S is an S -category and M is a π 0 S bimodule then H ∗ diag F ∗ ( S , M ) . H ∗ DK ( S , M ) =

7 Cohomology of categories This can be computed as the cohomology of a cobar-like complex F ∗ ( C , M ) where an n -cochain c is a function sending a chain of n composable morphisms in C σ 1 σn A 0 ← A 1 ← · · · ← A n − 1 ← A n to an element c ( σ 1 , · · · , σ n ) ∈ M ( A n , A 0 ) . If S is an S -category and M is a π 0 S bimodule then H ∗ diag F ∗ ( S , M ) . H ∗ DK ( S , M ) = The universal Toda bracket of a stable model category M is the first k -invariant of L M k 1 ∈ H 3 (Ho M , Hom Ho M (Σ , − )) .

8 Toda brackets Suppose that ( A , Σ , E ) is a triangulated category. f g h � B � C � D gf = 0 , hg = 0 . A

8 Toda brackets Suppose that ( A , Σ , E ) is a triangulated category. f q i � B � C f � Σ A A f g h � B � C � D gf = 0 , hg = 0 . A

� 8 Toda brackets Suppose that ( A , Σ , E ) is a triangulated category. f q i � B � C f � Σ A A a f g h � B � C � D gf = 0 , hg = 0 . A

� � 8 Toda brackets Suppose that ( A , Σ , E ) is a triangulated category. f q i � B � C f � Σ A A a b f g h � B � C � D gf = 0 , hg = 0 . A

� � 8 Toda brackets Suppose that ( A , Σ , E ) is a triangulated category. f q i � B � C f � Σ A A a b f g h � B � C � D gf = 0 , hg = 0 . A A (Σ A, D ) b ∈ � h, g, f � ∈ h A (Σ A, C ) + A (Σ B, D )(Σ f ) .

� � 8 Toda brackets Suppose that ( A , Σ , E ) is a triangulated category. f q i � B � C f � Σ A A a b f g h � B � C � D gf = 0 , hg = 0 . A A (Σ A, D ) b ∈ � h, g, f � ∈ h A (Σ A, C ) + A (Σ B, D )(Σ f ) . Example . 1 Σ A ∈ � q, i, f � .

� � 8 Toda brackets Suppose that ( A , Σ , E ) is a triangulated category. f q i � B � C f � Σ A A a b f g h � B � C � D gf = 0 , hg = 0 . A A (Σ A, D ) b ∈ � h, g, f � ∈ h A (Σ A, C ) + A (Σ B, D )(Σ f ) . Example . 1 Σ A ∈ � q, i, f � . Actually it is immediate to see for D = Σ A that the lower triangle is an exact triangle if and only if it is coexact and 1 Σ A ∈ � h, g, f � .

� 9 Toda brackets Example . − 1 Σ B ∈ � Σ f, q, i � , q − Σ f i � C f � Σ A � Σ B B − 1 q Σ f i � C f � Σ A � Σ B B

� 9 Toda brackets Example . − 1 Σ B ∈ � Σ f, q, i � , q − Σ f i � C f � Σ A � Σ B B − 1 q Σ f i � C f � Σ A � Σ B B 1 Σ Cf ∈ � Σ i, Σ f, q � .

� � � 10 Toda brackets f g h Suppose that our triangulated category is A = Ho M . Then A → B → C → D with gf = 0 , hg = 0 , is the same as a functor 0 ϕ � • � • Toda = • • − → A . 0

� � � 10 Toda brackets f g h Suppose that our triangulated category is A = Ho M . Then A → B → C → D with gf = 0 , hg = 0 , is the same as a functor 0 ϕ � • � • Toda = • • − → A . 0 H 3 ( Toda , ϕ ∗ Hom A (Σ , − ))

� � � 10 Toda brackets f g h Suppose that our triangulated category is A = Ho M . Then A → B → C → D with gf = 0 , hg = 0 , is the same as a functor 0 ϕ � • � • Toda = • • − → A . 0 H 3 ( Toda , ϕ ∗ Hom A (Σ , − )) A (Σ A,D ) h A (Σ A,C )+ A (Σ B,D )(Σ f )

� � � 10 Toda brackets f g h Suppose that our triangulated category is A = Ho M . Then A → B → C → D with gf = 0 , hg = 0 , is the same as a functor 0 ϕ � • � • Toda = • • − → A . 0 H 3 ( Toda , ϕ ∗ Hom A (Σ , − )) A (Σ A,D ) � h, g, f � ∈ h A (Σ A,C )+ A (Σ B,D )(Σ f )

� � � 10 Toda brackets f g h Suppose that our triangulated category is A = Ho M . Then A → B → C → D with gf = 0 , hg = 0 , is the same as a functor 0 ϕ � • � • Toda = • • − → A . 0 H 3 ( Toda , ϕ ∗ Hom A (Σ , − )) ϕ ∗ k 1 ∈ [Baues-Dreckmann] A (Σ A,D ) � h, g, f � ∈ h A (Σ A,C )+ A (Σ B,D )(Σ f )

11 Toda brackets Example . Let free ( S ) ⊂ L Spectra be the full S -category of the simplicial localization of spectra given by n S ∨ · · · ∨ S, n ≥ 0 . where S is the sphere spectrum.

11 Toda brackets Example . Let free ( S ) ⊂ L Spectra be the full S -category of the simplicial localization of spectra given by n S ∨ · · · ∨ S, n ≥ 0 . where S is the sphere spectrum. All triple Toda brackets vanish in free ( S ) .

11 Toda brackets Example . Let free ( S ) ⊂ L Spectra be the full S -category of the simplicial localization of spectra given by n S ∨ · · · ∨ S, n ≥ 0 . where S is the sphere spectrum. All triple Toda brackets vanish in free ( S ) . However, the universal Toda bracket of free ( S ) is the generator of Z / 2)) ∼ Z / 2) ∼ H 3 ( free ( Z = H 3 Z ) , Hom( − , − ⊗ Z ML ( Z Z , Z = Z Z / 2 .

12 Detecting the exact triangles ∼ → A a self-equivalence, and θ ∈ H 3 ( A , Hom A (Σ , − )) Let A be any additive category, Σ: A any cohomology class (which needs not be the universal Toda bracket of any stable model category).

12 Detecting the exact triangles ∼ → A a self-equivalence, and θ ∈ H 3 ( A , Hom A (Σ , − )) Let A be any additive category, Σ: A any cohomology class (which needs not be the universal Toda bracket of any stable model category). Let I = (0 → 1) and let [ I , A ] be the category of functors, called pairs. Objects are regarded as cochain complexes d A : A 0 → A 1 concentrated in dimensions 0 and 1 .

12 Detecting the exact triangles ∼ → A a self-equivalence, and θ ∈ H 3 ( A , Hom A (Σ , − )) Let A be any additive category, Σ: A any cohomology class (which needs not be the universal Toda bracket of any stable model category). Let I = (0 → 1) and let [ I , A ] be the category of functors, called pairs. Objects are regarded as cochain complexes d A : A 0 → A 1 concentrated in dimensions 0 and 1 . Consider the evaluation functor ev : [ I , A ] × I − → A .

12 Detecting the exact triangles ∼ → A a self-equivalence, and θ ∈ H 3 ( A , Hom A (Σ , − )) Let A be any additive category, Σ: A any cohomology class (which needs not be the universal Toda bracket of any stable model category). Let I = (0 → 1) and let [ I , A ] be the category of functors, called pairs. Objects are regarded as cochain complexes d A : A 0 → A 1 concentrated in dimensions 0 and 1 . Consider the evaluation functor ev : [ I , A ] × I − → A . ev ∗ � H 3 ([ I , A ] × I , ev ∗ Hom A (Σ , − )) k 1 ∈ H 3 ( A , Hom A (Σ , − ))

� 12 Detecting the exact triangles ∼ → A a self-equivalence, and θ ∈ H 3 ( A , Hom A (Σ , − )) Let A be any additive category, Σ: A any cohomology class (which needs not be the universal Toda bracket of any stable model category). Let I = (0 → 1) and let [ I , A ] be the category of functors, called pairs. Objects are regarded as cochain complexes d A : A 0 → A 1 concentrated in dimensions 0 and 1 . Consider the evaluation functor ev : [ I , A ] × I − → A . ev ∗ � H 3 ([ I , A ] × I , ev ∗ Hom A (Σ , − )) k 1 ∈ H 3 ( A , Hom A (Σ , − )) K¨ unneth SS k 1 ∈ H 2 ([ I , A ] , H 1 Hom A (Σ , − )) ¯

13 Detecting the exact triangles The image ¯ k 1 of k 1 determines a linear extension called the category of homotopy pairs, H 1 Hom A (Σ , − ) ֒ → [ I , B ] ։ [ I , A ] .

13 Detecting the exact triangles The image ¯ k 1 of k 1 determines a linear extension called the category of homotopy pairs, H 1 Hom A (Σ , − ) ֒ → [ I , B ] ։ [ I , A ] . For any two pairs d A , d B there is a short exact sequence H 1 Hom A (Σ d A , d B ) ֒ → [ I , B ]( d A , d B ) ։ [ I , A ]( d A , d B )

13 Detecting the exact triangles The image ¯ k 1 of k 1 determines a linear extension called the category of homotopy pairs, H 1 Hom A (Σ , − ) ֒ → [ I , B ] ։ [ I , A ] . For any two pairs d A , d B there is a short exact sequence H 1 Hom A (Σ d A , d B ) ֒ → [ I , B ]( d A , d B ) ։ [ I , A ]( d A , d B ) = H 0 Hom A ( d A , d B ) .

14 Detecting the exact triangles In particular given a morphism f : A → B and an object X in A there is a long exact sequence (Σ f ) ∗ f ∗ (S) A (Σ B, U ) → A (Σ A, X ) → [ I , B ]( f, 0 → X ) → A ( B, X ) → A ( A, X ) .

14 Detecting the exact triangles In particular given a morphism f : A → B and an object X in A there is a long exact sequence (Σ f ) ∗ f ∗ (S) A (Σ B, U ) → A (Σ A, X ) → [ I , B ]( f, 0 → X ) → A ( B, X ) → A ( A, X ) . Suppose that [ I , B ]( f, 0 → X ) is representable as a functor in X , [ I , B ]( f, 0 → X ) ∼ = Hom A ( C f , X ) .

14 Detecting the exact triangles In particular given a morphism f : A → B and an object X in A there is a long exact sequence (Σ f ) ∗ f ∗ (S) A (Σ B, U ) → A (Σ A, X ) → [ I , B ]( f, 0 → X ) → A ( B, X ) → A ( A, X ) . Suppose that [ I , B ]( f, 0 → X ) is representable as a functor in X , [ I , B ]( f, 0 → X ) ∼ = Hom A ( C f , X ) . Then (S) and Yoneda’s lemma yield a triangle f (T) A − → B − → C f − → Σ A.

14 Detecting the exact triangles In particular given a morphism f : A → B and an object X in A there is a long exact sequence (Σ f ) ∗ f ∗ (S) A (Σ B, U ) → A (Σ A, X ) → [ I , B ]( f, 0 → X ) → A ( B, X ) → A ( A, X ) . Suppose that [ I , B ]( f, 0 → X ) is representable as a functor in X , [ I , B ]( f, 0 → X ) ∼ = Hom A ( C f , X ) . Then (S) and Yoneda’s lemma yield a triangle f (T) A − → B − → C f − → Σ A. Theorem . For A = Ho M the triangles (T) are the exact triangles.

15 Detecting the exact triangles • Are there conditions on θ which imply that the triangles (T) induce a triangulated structure in A with translation functor Σ ?

15 Detecting the exact triangles • Are there conditions on θ which imply that the triangles (T) induce a triangulated structure in A with translation functor Σ ? The situation when the triangles (T) define a triangulated structure is very convenient since, for example, cofibers are automatically functorial in the category [ I , B ] . One can also construct the differential d 2 of Adams spectral sequence [Baues-Jibladze] . . .

� 16 Cohomology of diagrams Consider the diagram �� �� Σ �� A

� 16 Cohomology of diagrams Consider the diagram �� �� Σ �� A and the bimodule morphism → Σ ∗ Hom A (Σ , − ) = Hom A (Σ 2 , Σ) . ¯ Σ = − Σ: Hom A (Σ , − ) −

� 16 Cohomology of diagrams Consider the diagram �� �� Σ �� A and the bimodule morphism → Σ ∗ Hom A (Σ , − ) = Hom A (Σ 2 , Σ) . ¯ Σ = − Σ: Hom A (Σ , − ) − The diagram cohomology of Σ with coefficients in ¯ Σ can be obtained as Σ ∗ − ¯ “ ” H ∗ (Σ , ¯ H ∗ Fib Σ ∗ : F ∗ ( A , Hom A (Σ , − )) − → F ∗ ( A , Hom A (Σ 2 , Σ)) Σ) = .

� 16 Cohomology of diagrams Consider the diagram �� �� Σ �� A and the bimodule morphism → Σ ∗ Hom A (Σ , − ) = Hom A (Σ 2 , Σ) . ¯ Σ = − Σ: Hom A (Σ , − ) − The diagram cohomology of Σ with coefficients in ¯ Σ can be obtained as Σ ∗ − ¯ “ ” H ∗ (Σ , ¯ H ∗ Fib Σ ∗ : F ∗ ( A , Hom A (Σ , − )) − → F ∗ ( A , Hom A (Σ 2 , Σ)) Σ) = . In particular there is a long exact sequence Σ ∗− ¯ j Σ ∗ · · · → H n (Σ , ¯ → H n ( A , Hom A (Σ , − )) → H n ( A , Hom A (Σ 2 , Σ)) → H n +1 (Σ , ¯ Σ) − Σ) → · · ·

17 Cohomology of diagrams In a triangulated category A the following formula for Toda brackets holds � Σ h, Σ g, Σ f � = − Σ � h, g, f � .

17 Cohomology of diagrams In a triangulated category A the following formula for Toda brackets holds � Σ h, Σ g, Σ f � = − Σ � h, g, f � . This indicates that if A has a universal Toda bracket θ it is reasonable to think that Σ ∗ θ ¯ = Σ ∗ θ.

17 Cohomology of diagrams In a triangulated category A the following formula for Toda brackets holds � Σ h, Σ g, Σ f � = − Σ � h, g, f � . This indicates that if A has a universal Toda bracket θ it is reasonable to think that Σ ∗ θ ¯ = Σ ∗ θ. In particular for some ∇ ∈ H 3 (Σ , ¯ θ = j ∇ , Σ) .

17 Cohomology of diagrams In a triangulated category A the following formula for Toda brackets holds � Σ h, Σ g, Σ f � = − Σ � h, g, f � . This indicates that if A has a universal Toda bracket θ it is reasonable to think that Σ ∗ θ ¯ = Σ ∗ θ. In particular for some ∇ ∈ H 3 (Σ , ¯ θ = j ∇ , Σ) . Remark . If A = Ho M the class ∇ is the first k -invariant of the simplicial endofunctor Σ: L M − → L M .

17 Cohomology of diagrams In a triangulated category A the following formula for Toda brackets holds � Σ h, Σ g, Σ f � = − Σ � h, g, f � . This indicates that if A has a universal Toda bracket θ it is reasonable to think that Σ ∗ θ ¯ = Σ ∗ θ. In particular for some ∇ ∈ H 3 (Σ , ¯ θ = j ∇ , Σ) . Remark . If A = Ho M the class ∇ is the first k -invariant of the simplicial endofunctor Σ: L M − → L M . This k -invariant for diagrams is completely determined by k 2 ∈ H 4 ( P 1 L M , π 2 L M ) .

18 The conditions There is a K¨ unneth spectral sequence for the computation of cohomology of products of diagrams of categories.

18 The conditions There is a K¨ unneth spectral sequence for the computation of cohomology of products of diagrams of categories. This spectral sequence induces a filtration D 3 , 0 ⊂ D 2 , 1 ⊂ D 1 , 2 ⊂ D 0 , 3 ⊂ H 3 (Σ , ¯ Σ) .

18 The conditions There is a K¨ unneth spectral sequence for the computation of cohomology of products of diagrams of categories. This spectral sequence induces a filtration D 3 , 0 ⊂ D 2 , 1 ⊂ D 1 , 2 ⊂ D 0 , 3 ⊂ H 3 (Σ , ¯ Σ) . In the conditions above if ∇ ∈ D 1 , 2 then A with the triangles (T) above Theorem . [Baues-M.] satisfies all axioms except from the octahedral axiom.

18 The conditions There is a K¨ unneth spectral sequence for the computation of cohomology of products of diagrams of categories. This spectral sequence induces a filtration D 3 , 0 ⊂ D 2 , 1 ⊂ D 1 , 2 ⊂ D 0 , 3 ⊂ H 3 (Σ , ¯ Σ) . In the conditions above if ∇ ∈ D 1 , 2 then A with the triangles (T) above Theorem . [Baues-M.] satisfies all axioms except from the octahedral axiom. Moreover, if ∇ ∈ D 2 , 1 then the octahedral axiom is also satisfied and hence the triangles (T) yield a triangulated structure on A .

18 The conditions There is a K¨ unneth spectral sequence for the computation of cohomology of products of diagrams of categories. This spectral sequence induces a filtration D 3 , 0 ⊂ D 2 , 1 ⊂ D 1 , 2 ⊂ D 0 , 3 ⊂ H 3 (Σ , ¯ Σ) . In the conditions above if ∇ ∈ D 1 , 2 then A with the triangles (T) above Theorem . [Baues-M.] satisfies all axioms except from the octahedral axiom. Moreover, if ∇ ∈ D 2 , 1 then the octahedral axiom is also satisfied and hence the triangles (T) yield a triangulated structure on A . Definition . A cohomologically triangulted category is a triple ( A , Σ , ∇ ) where A is an additive ∼ → A is a self-equivalence, and ∇ ∈ H 3 (Σ , ¯ category, Σ: A Σ) satisfying the second condition in the Theorem, so that the universal Toda bracket j ∇ ∈ H 3 ( A , Hom A (Σ , − )) induces a triangulated structure in A .

19 The last example Z /p 2 ) and N = mod ( Z Z /p [ t ] /t 2 ) . Recall that in both cases the homotopy Consider M = mod ( Z category is A = mod ( Z Z /p ) , the suspension functor is the identity Σ = 1 , and k 1 = 0 .

19 The last example Z /p 2 ) and N = mod ( Z Z /p [ t ] /t 2 ) . Recall that in both cases the homotopy Consider M = mod ( Z category is A = mod ( Z Z /p ) , the suspension functor is the identity Σ = 1 , and k 1 = 0 . What happens with ∇ ?

19 The last example Z /p 2 ) and N = mod ( Z Z /p [ t ] /t 2 ) . Recall that in both cases the homotopy Consider M = mod ( Z category is A = mod ( Z Z /p ) , the suspension functor is the identity Σ = 1 , and k 1 = 0 . What happens with ∇ ? Σ ∗− ¯ Σ ∗ � H 2 ( mod ( Z � H 3 (Σ , ¯ � 0 H 2 ( mod ( Z Z /p ) , Hom) Z /p ) , Hom) Σ)

� � � 19 The last example Z /p 2 ) and N = mod ( Z Z /p [ t ] /t 2 ) . Recall that in both cases the homotopy Consider M = mod ( Z category is A = mod ( Z Z /p ) , the suspension functor is the identity Σ = 1 , and k 1 = 0 . What happens with ∇ ? Σ ∗− ¯ Σ ∗ � H 2 ( mod ( Z � H 3 (Σ , ¯ � 0 H 2 ( mod ( Z Z /p ) , Hom) Z /p ) , Hom) Σ) ∼ ∼ = = 2 H 2 H 2 ML ( Z Z /p, Z Z /p ) ML ( Z Z /p, Z Z /p )

� � � � � � 19 The last example Z /p 2 ) and N = mod ( Z Z /p [ t ] /t 2 ) . Recall that in both cases the homotopy Consider M = mod ( Z category is A = mod ( Z Z /p ) , the suspension functor is the identity Σ = 1 , and k 1 = 0 . What happens with ∇ ? Σ ∗− ¯ Σ ∗ � H 2 ( mod ( Z � H 3 (Σ , ¯ � 0 H 2 ( mod ( Z Z /p ) , Hom) Z /p ) , Hom) Σ) ∼ ∼ = = 2 H 2 H 2 ML ( Z Z /p, Z Z /p ) ML ( Z Z /p, Z Z /p ) ∼ ∼ = = 2 Z Z /p Z Z /p

� � � � � � 19 The last example Z /p 2 ) and N = mod ( Z Z /p [ t ] /t 2 ) . Recall that in both cases the homotopy Consider M = mod ( Z category is A = mod ( Z Z /p ) , the suspension functor is the identity Σ = 1 , and k 1 = 0 . What happens with ∇ ? Σ ∗− ¯ Σ ∗ � H 2 ( mod ( Z � H 3 (Σ , ¯ � 0 H 2 ( mod ( Z Z /p ) , Hom) Z /p ) , Hom) Σ) ∼ ∼ = = 2 H 2 H 2 ML ( Z Z /p, Z Z /p ) ML ( Z Z /p, Z Z /p ) ∼ ∼ = = 2 Z Z /p Z Z /p Therefore  Z Z / 2 , p = 2 , H 3 (Σ , ¯ Σ) = 0 , p � = 2 .

20 The last example For p = 2 one can check that ∇ N = 0 and ∇ M � = 0 , hence the cohomologically triangulated Z /p 2 ) and N = mod ( Z Z /p [ t ] /t 2 ) are different structures associated to M = mod ( Z ( mod ( Z Z / 2) , Σ , 1) , ( mod ( Z Z / 2) , Σ , 0) , respectively, and k M 2 � = k N 2 .