Invertible braided module categories and graded braided extensions - PowerPoint PPT Presentation

Explanation of terms Let M be a right B -module category and N be a left B -module category. Dmitri Nikshych (University of New Hampshire) Braided extensions October 15, 2018 6 / 28

Explanation of terms Let M be a right B -module category and N be a left B -module category. The B -module tensor product M ⊠ B N consists of pairs ( V ∈ M ⊠ N , γ = { γ X } ), where the middle balancing γ X : V ⊗ ( X ⊠ 1) → (1 ⊠ X ) ⊗ V , X ∈ B is associative. Dmitri Nikshych (University of New Hampshire) Braided extensions October 15, 2018 6 / 28

Explanation of terms Let M be a right B -module category and N be a left B -module category. The B -module tensor product M ⊠ B N consists of pairs ( V ∈ M ⊠ N , γ = { γ X } ), where the middle balancing γ X : V ⊗ ( X ⊠ 1) → (1 ⊠ X ) ⊗ V , X ∈ B is associative. This is similar to tensor product of modules over a ring. Dmitri Nikshych (University of New Hampshire) Braided extensions October 15, 2018 6 / 28

Explanation of terms Let M be a right B -module category and N be a left B -module category. The B -module tensor product M ⊠ B N consists of pairs ( V ∈ M ⊠ N , γ = { γ X } ), where the middle balancing γ X : V ⊗ ( X ⊠ 1) → (1 ⊠ X ) ⊗ V , X ∈ B is associative. This is similar to tensor product of modules over a ring. If M , N are bimodule categories then so is M ⊠ B N . B - Bimod is a monoidal 2 -category via ⊠ B Objects = B -bimodule categories, 1-cells = B -bimodule functors, 2-cells = B -bimodule natural transformations. Dmitri Nikshych (University of New Hampshire) Braided extensions October 15, 2018 6 / 28

Explanation of terms Let M be a right B -module category and N be a left B -module category. The B -module tensor product M ⊠ B N consists of pairs ( V ∈ M ⊠ N , γ = { γ X } ), where the middle balancing γ X : V ⊗ ( X ⊠ 1) → (1 ⊠ X ) ⊗ V , X ∈ B is associative. This is similar to tensor product of modules over a ring. If M , N are bimodule categories then so is M ⊠ B N . B - Bimod is a monoidal 2 -category via ⊠ B Objects = B -bimodule categories, 1-cells = B -bimodule functors, 2-cells = B -bimodule natural transformations. We will suppress the assocativity 2-cells for ⊠ B . Dmitri Nikshych (University of New Hampshire) Braided extensions October 15, 2018 6 / 28

Explanation of terms Let M be a right B -module category and N be a left B -module category. The B -module tensor product M ⊠ B N consists of pairs ( V ∈ M ⊠ N , γ = { γ X } ), where the middle balancing γ X : V ⊗ ( X ⊠ 1) → (1 ⊠ X ) ⊗ V , X ∈ B is associative. This is similar to tensor product of modules over a ring. If M , N are bimodule categories then so is M ⊠ B N . B - Bimod is a monoidal 2 -category via ⊠ B Objects = B -bimodule categories, 1-cells = B -bimodule functors, 2-cells = B -bimodule natural transformations. We will suppress the assocativity 2-cells for ⊠ B . The Brauer-Picard categorical 2-group BrPic( B ) is the “pointed part” of B - Bimod Objects are invertible w.r.t ⊠ B , all cells are isomorphisms. Dmitri Nikshych (University of New Hampshire) Braided extensions October 15, 2018 6 / 28

Outline Graded extensions of fusion categories 1 Braided module categories over braided fusion categories 2 Braided extensions 3 Dmitri Nikshych (University of New Hampshire) Braided extensions October 15, 2018 7 / 28

Dmitri Nikshych (University of New Hampshire) Braided extensions October 15, 2018 8 / 28

Let B be a braided fusion category with braiding c X , Y : X ⊗ Y → Y ⊗ X . Dmitri Nikshych (University of New Hampshire) Braided extensions October 15, 2018 8 / 28

Let B be a braided fusion category with braiding c X , Y : X ⊗ Y → Y ⊗ X . Let M be a B -module category, i.e., there is ⊗ : B × M → M Dmitri Nikshych (University of New Hampshire) Braided extensions October 15, 2018 8 / 28

Let B be a braided fusion category with braiding c X , Y : X ⊗ Y → Y ⊗ X . Let M be a B -module category, i.e., there is ⊗ : B × M → M B - Mod := the 2-category of B -module categories Two tensor products on B - Mod Dmitri Nikshych (University of New Hampshire) Braided extensions October 15, 2018 8 / 28

Let B be a braided fusion category with braiding c X , Y : X ⊗ Y → Y ⊗ X . Let M be a B -module category, i.e., there is ⊗ : B × M → M B - Mod := the 2-category of B -module categories Two tensor products on B - Mod Dmitri Nikshych (University of New Hampshire) Braided extensions October 15, 2018 8 / 28

Let B be a braided fusion category with braiding c X , Y : X ⊗ Y → Y ⊗ X . Let M be a B -module category, i.e., there is ⊗ : B × M → M B - Mod := the 2-category of B -module categories Two tensor products on B - Mod There are two tensor functors α M ± : B → End B ( M ) ( α -inductions of B¨ ockenhauer-Evans-Kawahigashi): Dmitri Nikshych (University of New Hampshire) Braided extensions October 15, 2018 8 / 28

Let B be a braided fusion category with braiding c X , Y : X ⊗ Y → Y ⊗ X . Let M be a B -module category, i.e., there is ⊗ : B × M → M B - Mod := the 2-category of B -module categories Two tensor products on B - Mod There are two tensor functors α M ± : B → End B ( M ) ( α -inductions of ockenhauer-Evans-Kawahigashi): α M ± ( X ) = X ⊗ ? with the B¨ B -module structure given by c X , Y (resp. c − 1 Y , X ) Dmitri Nikshych (University of New Hampshire) Braided extensions October 15, 2018 8 / 28

Let B be a braided fusion category with braiding c X , Y : X ⊗ Y → Y ⊗ X . Let M be a B -module category, i.e., there is ⊗ : B × M → M B - Mod := the 2-category of B -module categories Two tensor products on B - Mod There are two tensor functors α M ± : B → End B ( M ) ( α -inductions of ockenhauer-Evans-Kawahigashi): α M ± ( X ) = X ⊗ ? with the B¨ B -module structure given by c X , Y (resp. c − 1 Y , X ) Since M is a B − End B ( M )-bimodule, one can turn M into a B -bimodule category in 2 different ways: M ± (using α M ± ). Dmitri Nikshych (University of New Hampshire) Braided extensions October 15, 2018 8 / 28

Let B be a braided fusion category with braiding c X , Y : X ⊗ Y → Y ⊗ X . Let M be a B -module category, i.e., there is ⊗ : B × M → M B - Mod := the 2-category of B -module categories Two tensor products on B - Mod There are two tensor functors α M ± : B → End B ( M ) ( α -inductions of ockenhauer-Evans-Kawahigashi): α M ± ( X ) = X ⊗ ? with the B¨ B -module structure given by c X , Y (resp. c − 1 Y , X ) Since M is a B − End B ( M )-bimodule, one can turn M into a B -bimodule category in 2 different ways: M ± (using α M ± ). Two monoidal 2-categories: B - Mod ± with products M ± ⊠ B N . Relation between ± products: Dmitri Nikshych (University of New Hampshire) Braided extensions October 15, 2018 8 / 28

Let B be a braided fusion category with braiding c X , Y : X ⊗ Y → Y ⊗ X . Let M be a B -module category, i.e., there is ⊗ : B × M → M B - Mod := the 2-category of B -module categories Two tensor products on B - Mod There are two tensor functors α M ± : B → End B ( M ) ( α -inductions of ockenhauer-Evans-Kawahigashi): α M ± ( X ) = X ⊗ ? with the B¨ B -module structure given by c X , Y (resp. c − 1 Y , X ) Since M is a B − End B ( M )-bimodule, one can turn M into a B -bimodule category in 2 different ways: M ± (using α M ± ). Two monoidal 2-categories: B - Mod ± with products M ± ⊠ B N . Relation between ± products: There is natural equivalence N − ⊠ B M ∼ − → M + ⊠ B N Dmitri Nikshych (University of New Hampshire) Braided extensions October 15, 2018 8 / 28

Let B be a braided fusion category with braiding c X , Y : X ⊗ Y → Y ⊗ X . Let M be a B -module category, i.e., there is ⊗ : B × M → M B - Mod := the 2-category of B -module categories Two tensor products on B - Mod There are two tensor functors α M ± : B → End B ( M ) ( α -inductions of ockenhauer-Evans-Kawahigashi): α M ± ( X ) = X ⊗ ? with the B¨ B -module structure given by c X , Y (resp. c − 1 Y , X ) Since M is a B − End B ( M )-bimodule, one can turn M into a B -bimodule category in 2 different ways: M ± (using α M ± ). Two monoidal 2-categories: B - Mod ± with products M ± ⊠ B N . Relation between ± products: There is natural equivalence N − ⊠ B M ∼ − → M + ⊠ B N given by the transposition of factors N ⊠ M → M ⊠ N , Dmitri Nikshych (University of New Hampshire) Braided extensions October 15, 2018 8 / 28

Let B be a braided fusion category with braiding c X , Y : X ⊗ Y → Y ⊗ X . Let M be a B -module category, i.e., there is ⊗ : B × M → M B - Mod := the 2-category of B -module categories Two tensor products on B - Mod There are two tensor functors α M ± : B → End B ( M ) ( α -inductions of ockenhauer-Evans-Kawahigashi): α M ± ( X ) = X ⊗ ? with the B¨ B -module structure given by c X , Y (resp. c − 1 Y , X ) Since M is a B − End B ( M )-bimodule, one can turn M into a B -bimodule category in 2 different ways: M ± (using α M ± ). Two monoidal 2-categories: B - Mod ± with products M ± ⊠ B N . Relation between ± products: There is natural equivalence N − ⊠ B M ∼ − → M + ⊠ B N given by the transposition of factors N ⊠ M → M ⊠ N , so that B - Mod − ≃ B - Mod op + Dmitri Nikshych (University of New Hampshire) Braided extensions October 15, 2018 8 / 28

Let B be a braided fusion category with braiding c X , Y : X ⊗ Y → Y ⊗ X . Let M be a B -module category, i.e., there is ⊗ : B × M → M B - Mod := the 2-category of B -module categories Two tensor products on B - Mod There are two tensor functors α M ± : B → End B ( M ) ( α -inductions of ockenhauer-Evans-Kawahigashi): α M ± ( X ) = X ⊗ ? with the B¨ B -module structure given by c X , Y (resp. c − 1 Y , X ) Since M is a B − End B ( M )-bimodule, one can turn M into a B -bimodule category in 2 different ways: M ± (using α M ± ). Two monoidal 2-categories: B - Mod ± with products M ± ⊠ B N . Relation between ± products: There is natural equivalence N − ⊠ B M ∼ − → M + ⊠ B N given by the transposition of factors N ⊠ M → M ⊠ N , so that B - Mod − ≃ B - Mod op + Dmitri Nikshych (University of New Hampshire) Braided extensions October 15, 2018 8 / 28

� � � � � � � � � � � A braided B -module category [Brochier, Ben-Zvi - Brochier - Jordan] is a B -module category M equipped with a collection of isomorphisms σ M X , M : X ∗ M → X ∗ M ( module braiding ) natural in X ∈ B , M ∈ M with σ 1 , M = 1 M and such that the diagrams σ M σ M X , Y ∗ M X ⊗ Y , M X ∗ ( Y ∗ M ) X ∗ ( Y ∗ M ) ( X ⊗ Y ) ∗ M ( X ⊗ Y ) ∗ M c − 1 m X , Y , M m X , Y , M c X , Y Y , X � ( X ⊗ Y ) ∗ M ( X ⊗ Y ) ∗ M ( Y ⊗ X ) ∗ M ( Y ⊗ X ) ∗ M c − 1 m − 1 m − 1 c X , Y Y , X Y , X , M � Y , X , M ( Y ⊗ X ) ∗ M ( Y ⊗ X ) ∗ M Y ∗ ( X ∗ M ) Y ∗ ( X ∗ M ) m − 1 m − 1 Y , X , M � Y , X , M σ M σ M σ M X , M Y , X ∗ M X , M � Y ∗ ( X ∗ M ) Y ∗ ( X ∗ M ) Y ∗ ( X ∗ M ) commute for all X , Y ∈ B and M ∈ M . Dmitri Nikshych (University of New Hampshire) Braided extensions October 15, 2018 9 / 28

� � � � � � � � � � � A braided B -module category [Brochier, Ben-Zvi - Brochier - Jordan] is a B -module category M equipped with a collection of isomorphisms σ M X , M : X ∗ M → X ∗ M ( module braiding ) natural in X ∈ B , M ∈ M with σ 1 , M = 1 M and such that the diagrams σ M σ M X , Y ∗ M X ⊗ Y , M X ∗ ( Y ∗ M ) X ∗ ( Y ∗ M ) ( X ⊗ Y ) ∗ M ( X ⊗ Y ) ∗ M c − 1 m X , Y , M m X , Y , M c X , Y Y , X � ( X ⊗ Y ) ∗ M ( X ⊗ Y ) ∗ M ( Y ⊗ X ) ∗ M ( Y ⊗ X ) ∗ M c − 1 m − 1 m − 1 c X , Y Y , X Y , X , M � Y , X , M ( Y ⊗ X ) ∗ M ( Y ⊗ X ) ∗ M Y ∗ ( X ∗ M ) Y ∗ ( X ∗ M ) m − 1 m − 1 Y , X , M � Y , X , M σ M σ M σ M X , M Y , X ∗ M X , M � Y ∗ ( X ∗ M ) Y ∗ ( X ∗ M ) Y ∗ ( X ∗ M ) commute for all X , Y ∈ B and M ∈ M . B -module braided functors are required to respect module braiding. Dmitri Nikshych (University of New Hampshire) Braided extensions October 15, 2018 9 / 28

Interpretation of module braidings Dmitri Nikshych (University of New Hampshire) Braided extensions October 15, 2018 10 / 28

Interpretation of module braidings Terminology justification Dmitri Nikshych (University of New Hampshire) Braided extensions October 15, 2018 10 / 28

Interpretation of module braidings Terminology justification A module braiding on M gives rise to the pure braid group representation on End M ( X 1 ⊗ · · · ⊗ X n ⊗ M ) for X 1 , . . . , X n ∈ B and M ∈ M . Dmitri Nikshych (University of New Hampshire) Braided extensions October 15, 2018 10 / 28

Interpretation of module braidings Terminology justification A module braiding on M gives rise to the pure braid group representation on End M ( X 1 ⊗ · · · ⊗ X n ⊗ M ) for X 1 , . . . , X n ∈ B and M ∈ M . A module braiding on M is precisely an isomorphism of tensor functors ∼ α M → α M − − . + Dmitri Nikshych (University of New Hampshire) Braided extensions October 15, 2018 10 / 28

Interpretation of module braidings Terminology justification A module braiding on M gives rise to the pure braid group representation on End M ( X 1 ⊗ · · · ⊗ X n ⊗ M ) for X 1 , . . . , X n ∈ B and M ∈ M . A module braiding on M is precisely an isomorphism of tensor functors ∼ ∼ α M → α M − − . It gives a B -bimodule equivalence M + − → M − . + Dmitri Nikshych (University of New Hampshire) Braided extensions October 15, 2018 10 / 28

Interpretation of module braidings Terminology justification A module braiding on M gives rise to the pure braid group representation on End M ( X 1 ⊗ · · · ⊗ X n ⊗ M ) for X 1 , . . . , X n ∈ B and M ∈ M . A module braiding on M is precisely an isomorphism of tensor functors ∼ ∼ α M → α M − − . It gives a B -bimodule equivalence M + − → M − . + Tensor product of braided module categories Dmitri Nikshych (University of New Hampshire) Braided extensions October 15, 2018 10 / 28

Interpretation of module braidings Terminology justification A module braiding on M gives rise to the pure braid group representation on End M ( X 1 ⊗ · · · ⊗ X n ⊗ M ) for X 1 , . . . , X n ∈ B and M ∈ M . A module braiding on M is precisely an isomorphism of tensor functors ∼ ∼ α M → α M − − . It gives a B -bimodule equivalence M + − → M − . + Tensor product of braided module categories ( M , σ M ) ⊠ B ( N , σ N ) := ( M + ⊠ B N , σ M ⊠ B σ N ). Dmitri Nikshych (University of New Hampshire) Braided extensions October 15, 2018 10 / 28

Interpretation of module braidings Terminology justification A module braiding on M gives rise to the pure braid group representation on End M ( X 1 ⊗ · · · ⊗ X n ⊗ M ) for X 1 , . . . , X n ∈ B and M ∈ M . A module braiding on M is precisely an isomorphism of tensor functors ∼ ∼ α M → α M − − . It gives a B -bimodule equivalence M + − → M − . + Tensor product of braided module categories ( M , σ M ) ⊠ B ( N , σ N ) := ( M + ⊠ B N , σ M ⊠ B σ N ). The unit object is the regular B with σ B X , Y = c Y , X ◦ c X , Y . Dmitri Nikshych (University of New Hampshire) Braided extensions October 15, 2018 10 / 28

Interpretation of module braidings Terminology justification A module braiding on M gives rise to the pure braid group representation on End M ( X 1 ⊗ · · · ⊗ X n ⊗ M ) for X 1 , . . . , X n ∈ B and M ∈ M . A module braiding on M is precisely an isomorphism of tensor functors ∼ ∼ α M → α M − − . It gives a B -bimodule equivalence M + − → M − . + Tensor product of braided module categories ( M , σ M ) ⊠ B ( N , σ N ) := ( M + ⊠ B N , σ M ⊠ B σ N ). The unit object is the regular B with σ B X , Y = c Y , X ◦ c X , Y . Denote B - Mod br the resulting monoidal 2-category. Dmitri Nikshych (University of New Hampshire) Braided extensions October 15, 2018 10 / 28

Interpretation of module braidings Terminology justification A module braiding on M gives rise to the pure braid group representation on End M ( X 1 ⊗ · · · ⊗ X n ⊗ M ) for X 1 , . . . , X n ∈ B and M ∈ M . A module braiding on M is precisely an isomorphism of tensor functors ∼ ∼ α M → α M − − . It gives a B -bimodule equivalence M + − → M − . + Tensor product of braided module categories ( M , σ M ) ⊠ B ( N , σ N ) := ( M + ⊠ B N , σ M ⊠ B σ N ). The unit object is the regular B with σ B X , Y = c Y , X ◦ c X , Y . Denote B - Mod br the resulting monoidal 2-category. Dmitri Nikshych (University of New Hampshire) Braided extensions October 15, 2018 10 / 28

Module braiding = central structure Dmitri Nikshych (University of New Hampshire) Braided extensions October 15, 2018 11 / 28

Module braiding = central structure Let N = ( N , σ N ) be a braided B -module category and M be any B -module category. Dmitri Nikshych (University of New Hampshire) Braided extensions October 15, 2018 11 / 28

Module braiding = central structure Let N = ( N , σ N ) be a braided B -module category and M be any B -module category. Let us combine previously mentioned equivalences: Dmitri Nikshych (University of New Hampshire) Braided extensions October 15, 2018 11 / 28

Module braiding = central structure Let N = ( N , σ N ) be a braided B -module category and M be any B -module category. Let us combine previously mentioned equivalences: transposition module braiding of N B M , N : M + ⊠ B N − − − − − − − → N − ⊠ B M − − − − − − − − − − − − − → N + ⊠ B M . Dmitri Nikshych (University of New Hampshire) Braided extensions October 15, 2018 11 / 28

Module braiding = central structure Let N = ( N , σ N ) be a braided B -module category and M be any B -module category. Let us combine previously mentioned equivalences: transposition module braiding of N B M , N : M + ⊠ B N − − − − − − − → N − ⊠ B M − − − − − − − − − − − − − → N + ⊠ B M . Let us denote B− Mod + simply B− Mod and its tensor product ⊠ B . Dmitri Nikshych (University of New Hampshire) Braided extensions October 15, 2018 11 / 28

Module braiding = central structure Let N = ( N , σ N ) be a braided B -module category and M be any B -module category. Let us combine previously mentioned equivalences: transposition module braiding of N B M , N : M + ⊠ B N − − − − − − − → N − ⊠ B M − − − − − − − − − − − − − → N + ⊠ B M . Let us denote B− Mod + simply B− Mod and its tensor product ⊠ B . ∼ The above B M , N : M ⊠ B N − → N ⊠ B M equips M with a structure of an object in the Z ( B− Mod ) (= the 2-center of the monoidal 2-category B - Mod ) Dmitri Nikshych (University of New Hampshire) Braided extensions October 15, 2018 11 / 28

Module braiding = central structure Let N = ( N , σ N ) be a braided B -module category and M be any B -module category. Let us combine previously mentioned equivalences: transposition module braiding of N B M , N : M + ⊠ B N − − − − − − − → N − ⊠ B M − − − − − − − − − − − − − → N + ⊠ B M . Let us denote B− Mod + simply B− Mod and its tensor product ⊠ B . ∼ The above B M , N : M ⊠ B N − → N ⊠ B M equips M with a structure of an object in the Z ( B− Mod ) (= the 2-center of the monoidal 2-category B - Mod ) and vice versa. Dmitri Nikshych (University of New Hampshire) Braided extensions October 15, 2018 11 / 28

Module braiding = central structure Let N = ( N , σ N ) be a braided B -module category and M be any B -module category. Let us combine previously mentioned equivalences: transposition module braiding of N B M , N : M + ⊠ B N − − − − − − − → N − ⊠ B M − − − − − − − − − − − − − → N + ⊠ B M . Let us denote B− Mod + simply B− Mod and its tensor product ⊠ B . ∼ The above B M , N : M ⊠ B N − → N ⊠ B M equips M with a structure of an object in the Z ( B− Mod ) (= the 2-center of the monoidal 2-category B - Mod ) and vice versa. Dmitri Nikshych (University of New Hampshire) Braided extensions October 15, 2018 11 / 28

Module braiding = central structure Let N = ( N , σ N ) be a braided B -module category and M be any B -module category. Let us combine previously mentioned equivalences: transposition module braiding of N B M , N : M + ⊠ B N − − − − − − − → N − ⊠ B M − − − − − − − − − − − − − → N + ⊠ B M . Let us denote B− Mod + simply B− Mod and its tensor product ⊠ B . ∼ The above B M , N : M ⊠ B N − → N ⊠ B M equips M with a structure of an object in the Z ( B− Mod ) (= the 2-center of the monoidal 2-category B - Mod ) and vice versa. Thus, Dmitri Nikshych (University of New Hampshire) Braided extensions October 15, 2018 11 / 28

Module braiding = central structure Let N = ( N , σ N ) be a braided B -module category and M be any B -module category. Let us combine previously mentioned equivalences: transposition module braiding of N B M , N : M + ⊠ B N − − − − − − − → N − ⊠ B M − − − − − − − − − − − − − → N + ⊠ B M . Let us denote B− Mod + simply B− Mod and its tensor product ⊠ B . ∼ The above B M , N : M ⊠ B N − → N ⊠ B M equips M with a structure of an object in the Z ( B− Mod ) (= the 2-center of the monoidal 2-category B - Mod ) and vice versa. Thus, B− Mod br ≃ Z ( B− Mod ). Dmitri Nikshych (University of New Hampshire) Braided extensions October 15, 2018 11 / 28

Module braiding = central structure Let N = ( N , σ N ) be a braided B -module category and M be any B -module category. Let us combine previously mentioned equivalences: transposition module braiding of N B M , N : M + ⊠ B N − − − − − − − → N − ⊠ B M − − − − − − − − − − − − − → N + ⊠ B M . Let us denote B− Mod + simply B− Mod and its tensor product ⊠ B . ∼ The above B M , N : M ⊠ B N − → N ⊠ B M equips M with a structure of an object in the Z ( B− Mod ) (= the 2-center of the monoidal 2-category B - Mod ) and vice versa. Thus, B− Mod br ≃ Z ( B− Mod ). In particular, B− Mod br is a braided monoidal 2 -category . Dmitri Nikshych (University of New Hampshire) Braided extensions October 15, 2018 11 / 28

What is a braided monoidal 2-category? Dmitri Nikshych (University of New Hampshire) Braided extensions October 15, 2018 12 / 28

What is a braided monoidal 2-category? Defined by Kapranov-Voevodsky, Breen. Dmitri Nikshych (University of New Hampshire) Braided extensions October 15, 2018 12 / 28

What is a braided monoidal 2-category? Defined by Kapranov-Voevodsky, Breen. Just like usual braided category, but equalities now become isomorphisms (natural 2-cells): (id M ⊠ B B L , N )( B L , M ⊠ B id N ) ∼ − → B L , M ⊠ B N , β L , M , N : ( B L , N ⊠ B id K )(id L ⊠ B B K , N ) ∼ : − → B L ⊠ B K , N γ L , K , N for all braided B -module categories L , K , M , N . Dmitri Nikshych (University of New Hampshire) Braided extensions October 15, 2018 12 / 28

What is a braided monoidal 2-category? Defined by Kapranov-Voevodsky, Breen. Just like usual braided category, but equalities now become isomorphisms (natural 2-cells): (id M ⊠ B B L , N )( B L , M ⊠ B id N ) ∼ − → B L , M ⊠ B N , β L , M , N : ( B L , N ⊠ B id K )(id L ⊠ B B K , N ) ∼ : − → B L ⊠ B K , N γ L , K , N for all braided B -module categories L , K , M , N . These satisfy coherence of their own. Dmitri Nikshych (University of New Hampshire) Braided extensions October 15, 2018 12 / 28

� � � � � � � � � � � � � � � � � � � � � � � � � � � � K ⊠ B L ⊠ B M ⊠ B N K ⊠ B L ⊠ B M ⊠ B N L ⊠ B K ⊠ B M ⊠ B N L ⊠ B K ⊠ B M ⊠ B N β = β β L ⊠ B M ⊠ B K ⊠ B N L ⊠ B M ⊠ B K ⊠ B N β L ⊠ B M ⊠ B N ⊠ B K L ⊠ B M ⊠ B N ⊠ B K , K ⊠ B L ⊠ B M ⊠ B N K ⊠ B L ⊠ B M ⊠ B N K ⊠ B L ⊠ B N ⊠ B M K ⊠ B L ⊠ B N ⊠ B M γ = γ γ K ⊠ B N ⊠ B L ⊠ B M K ⊠ B N ⊠ B L ⊠ B M γ N ⊠ B K ⊠ B L ⊠ B M N ⊠ B K ⊠ B L ⊠ B M , Dmitri Nikshych (University of New Hampshire) Braided extensions October 15, 2018 13 / 28

Dmitri Nikshych (University of New Hampshire) Braided extensions October 15, 2018 14 / 28

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � K ⊠ B L ⊠ B M ⊠ B N K ⊠ B L ⊠ B M ⊠ B N γ � M ⊠ B K ⊠ B L ⊠ B N K ⊠ B M ⊠ B L ⊠ B N M ⊠ B K ⊠ B L ⊠ B N K ⊠ B M ⊠ B L ⊠ B N β β = γ γ can K ⊠ B M ⊠ B N ⊠ B L M ⊠ B N ⊠ B K ⊠ B L K ⊠ B M ⊠ B N ⊠ B L M ⊠ B N ⊠ B K ⊠ B L β M ⊠ B K ⊠ B N ⊠ B L M ⊠ B K ⊠ B N ⊠ B L , K ⊠ B L ⊠ B M K ⊠ B L ⊠ B M L ⊠ B K ⊠ B M K ⊠ B M ⊠ B L L ⊠ B K ⊠ B M K ⊠ B M ⊠ B L β γ = L ⊠ B M ⊠ B K M ⊠ B K ⊠ B L L ⊠ B M ⊠ B K M ⊠ B K ⊠ B L γ β M ⊠ B L ⊠ B K M ⊠ B L ⊠ B K . Dmitri Nikshych (University of New Hampshire) Braided extensions October 15, 2018 14 / 28

The braided 2-categorical Picard group Pic br ( B ) Dmitri Nikshych (University of New Hampshire) Braided extensions October 15, 2018 15 / 28

The braided 2-categorical Picard group Pic br ( B ) For our purposes we will need the “pointed part” of B− Mod br consisting of braided module categories invertible w.r.t. ⊠ B and equivalences between them: Pic br ( B ) = Inv ( B− Mod br ) . Dmitri Nikshych (University of New Hampshire) Braided extensions October 15, 2018 15 / 28

The braided 2-categorical Picard group Pic br ( B ) For our purposes we will need the “pointed part” of B− Mod br consisting of braided module categories invertible w.r.t. ⊠ B and equivalences between them: Pic br ( B ) = Inv ( B− Mod br ) . Dmitri Nikshych (University of New Hampshire) Braided extensions October 15, 2018 15 / 28

The braided 2-categorical Picard group Pic br ( B ) For our purposes we will need the “pointed part” of B− Mod br consisting of braided module categories invertible w.r.t. ⊠ B and equivalences between them: Pic br ( B ) = Inv ( B− Mod br ) . If we view Pic br ( B ) as a 3-categorical group with a single object, then the homotopy groups of the corresponding topological space are Dmitri Nikshych (University of New Hampshire) Braided extensions October 15, 2018 15 / 28

The braided 2-categorical Picard group Pic br ( B ) For our purposes we will need the “pointed part” of B− Mod br consisting of braided module categories invertible w.r.t. ⊠ B and equivalences between them: Pic br ( B ) = Inv ( B− Mod br ) . If we view Pic br ( B ) as a 3-categorical group with a single object, then the homotopy groups of the corresponding topological space are π 1 = 1, π 2 = Pic br ( B ), π 3 = Inv( Z sym ( B )), and π 4 = k × . Dmitri Nikshych (University of New Hampshire) Braided extensions October 15, 2018 15 / 28

The braided 2-categorical Picard group Pic br ( B ) For our purposes we will need the “pointed part” of B− Mod br consisting of braided module categories invertible w.r.t. ⊠ B and equivalences between them: Pic br ( B ) = Inv ( B− Mod br ) . If we view Pic br ( B ) as a 3-categorical group with a single object, then the homotopy groups of the corresponding topological space are π 1 = 1, π 2 = Pic br ( B ), π 3 = Inv( Z sym ( B )), and π 4 = k × . There is an exact sequence for the underlying group Pic br ( B ) of Pic br ( B ): 0 → Inv( Z sym ( B )) − → Inv( B ) − → Aut ⊗ (id B ) − → Pic br ( B ) − → Pic( B ) − → Aut br ( B ) . Dmitri Nikshych (University of New Hampshire) Braided extensions October 15, 2018 15 / 28

The braided 2-categorical Picard group Pic br ( B ) For our purposes we will need the “pointed part” of B− Mod br consisting of braided module categories invertible w.r.t. ⊠ B and equivalences between them: Pic br ( B ) = Inv ( B− Mod br ) . If we view Pic br ( B ) as a 3-categorical group with a single object, then the homotopy groups of the corresponding topological space are π 1 = 1, π 2 = Pic br ( B ), π 3 = Inv( Z sym ( B )), and π 4 = k × . There is an exact sequence for the underlying group Pic br ( B ) of Pic br ( B ): 0 → Inv( Z sym ( B )) − → Inv( B ) − → Aut ⊗ (id B ) − → Pic br ( B ) − → Pic( B ) − → Aut br ( B ) . Here Inv() denotes the group of invertible objects, Pic( B ) is the usual Picard group of B . Dmitri Nikshych (University of New Hampshire) Braided extensions October 15, 2018 15 / 28

Whitehead products π k × π l → π k + l − 1 Dmitri Nikshych (University of New Hampshire) Braided extensions October 15, 2018 16 / 28

Whitehead products π k × π l → π k + l − 1 Pic br ( B ) (the 1-categorical truncation of Pic br ( B )) is a braided categorical group. Dmitri Nikshych (University of New Hampshire) Braided extensions October 15, 2018 16 / 28

Whitehead products π k × π l → π k + l − 1 Pic br ( B ) (the 1-categorical truncation of Pic br ( B )) is a braided categorical group. So there is a canonical quadratic form Q B : Pic br ( B ) → Inv( Z sym ( B )) by [Joyal-Street]. Dmitri Nikshych (University of New Hampshire) Braided extensions October 15, 2018 16 / 28

Whitehead products π k × π l → π k + l − 1 Pic br ( B ) (the 1-categorical truncation of Pic br ( B )) is a braided categorical group. So there is a canonical quadratic form Q B : Pic br ( B ) → Inv( Z sym ( B )) by [Joyal-Street]. This comes from π 2 × π 2 → π 3 . Dmitri Nikshych (University of New Hampshire) Braided extensions October 15, 2018 16 / 28

Whitehead products π k × π l → π k + l − 1 Pic br ( B ) (the 1-categorical truncation of Pic br ( B )) is a braided categorical group. So there is a canonical quadratic form Q B : Pic br ( B ) → Inv( Z sym ( B )) by [Joyal-Street]. This comes from π 2 × π 2 → π 3 . Dmitri Nikshych (University of New Hampshire) Braided extensions October 15, 2018 16 / 28

Whitehead products π k × π l → π k + l − 1 Pic br ( B ) (the 1-categorical truncation of Pic br ( B )) is a braided categorical group. So there is a canonical quadratic form Q B : Pic br ( B ) → Inv( Z sym ( B )) by [Joyal-Street]. This comes from π 2 × π 2 → π 3 . There is a well-defined bilinear map P B : Inv( Z sym ( B )) × Pic br ( B ) → k × given by P B ( Z , M ) = σ Z , X ∈ Aut( Z ⊗ X ) = k × , X ∈ M . Dmitri Nikshych (University of New Hampshire) Braided extensions October 15, 2018 16 / 28

Whitehead products π k × π l → π k + l − 1 Pic br ( B ) (the 1-categorical truncation of Pic br ( B )) is a braided categorical group. So there is a canonical quadratic form Q B : Pic br ( B ) → Inv( Z sym ( B )) by [Joyal-Street]. This comes from π 2 × π 2 → π 3 . There is a well-defined bilinear map P B : Inv( Z sym ( B )) × Pic br ( B ) → k × given by P B ( Z , M ) = σ Z , X ∈ Aut( Z ⊗ X ) = k × , X ∈ M . This is π 3 × π 2 → π 4 . Dmitri Nikshych (University of New Hampshire) Braided extensions October 15, 2018 16 / 28

Outline Graded extensions of fusion categories 1 Braided module categories over braided fusion categories 2 Braided extensions 3 Dmitri Nikshych (University of New Hampshire) Braided extensions October 15, 2018 17 / 28

From extensions to braided monoidal 2-functors and back Dmitri Nikshych (University of New Hampshire) Braided extensions October 15, 2018 18 / 28

From extensions to braided monoidal 2-functors and back Let A be a finite Abelian group. Let B be a braided fusion category with braiding c . Dmitri Nikshych (University of New Hampshire) Braided extensions October 15, 2018 18 / 28

From extensions to braided monoidal 2-functors and back Let A be a finite Abelian group. Let B be a braided fusion category with braiding c . Given a braided extension � C = C x , C 1 = B , x ∈ A we have C x ∈ Pic br ( B ) , x ∈ X with the module braiding given by σ X , V = c X , V c V , X , V ∈ B , X ∈ C x . Dmitri Nikshych (University of New Hampshire) Braided extensions October 15, 2018 18 / 28

From extensions to braided monoidal 2-functors and back Let A be a finite Abelian group. Let B be a braided fusion category with braiding c . Given a braided extension � C = C x , C 1 = B , x ∈ A we have C x ∈ Pic br ( B ) , x ∈ X with the module braiding given by σ X , V = c X , V c V , X , V ∈ B , X ∈ C x . Dmitri Nikshych (University of New Hampshire) Braided extensions October 15, 2018 18 / 28

From extensions to braided monoidal 2-functors and back Let A be a finite Abelian group. Let B be a braided fusion category with braiding c . Given a braided extension � C = C x , C 1 = B , x ∈ A we have C x ∈ Pic br ( B ) , x ∈ X with the module braiding given by σ X , V = c X , V c V , X , V ∈ B , X ∈ C x . Furthermore, the tensor products ⊗ x , y : C x ⊠ B C y → C xy gives rise to ∼ B -module equivalences M x , y : C x × C y − → C xy . Dmitri Nikshych (University of New Hampshire) Braided extensions October 15, 2018 18 / 28

From extensions to braided monoidal 2-functors and back Let A be a finite Abelian group. Let B be a braided fusion category with braiding c . Given a braided extension � C = C x , C 1 = B , x ∈ A we have C x ∈ Pic br ( B ) , x ∈ X with the module braiding given by σ X , V = c X , V c V , X , V ∈ B , X ∈ C x . Furthermore, the tensor products ⊗ x , y : C x ⊠ B C y → C xy gives rise to ∼ B -module equivalences M x , y : C x × C y − → C xy . Dmitri Nikshych (University of New Hampshire) Braided extensions October 15, 2018 18 / 28

From extensions to braided monoidal 2-functors and back Let A be a finite Abelian group. Let B be a braided fusion category with braiding c . Given a braided extension � C = C x , C 1 = B , x ∈ A we have C x ∈ Pic br ( B ) , x ∈ X with the module braiding given by σ X , V = c X , V c V , X , V ∈ B , X ∈ C x . Furthermore, the tensor products ⊗ x , y : C x ⊠ B C y → C xy gives rise to ∼ B -module equivalences M x , y : C x × C y − → C xy . This gives a (usual) monoidal functor A → Pic br ( B ) : x �→ C x . Dmitri Nikshych (University of New Hampshire) Braided extensions October 15, 2018 18 / 28

From extensions to braided monoidal 2-functors and back Let A be a finite Abelian group. Let B be a braided fusion category with braiding c . Given a braided extension � C = C x , C 1 = B , x ∈ A we have C x ∈ Pic br ( B ) , x ∈ X with the module braiding given by σ X , V = c X , V c V , X , V ∈ B , X ∈ C x . Furthermore, the tensor products ⊗ x , y : C x ⊠ B C y → C xy gives rise to ∼ B -module equivalences M x , y : C x × C y − → C xy . This gives a (usual) monoidal functor A → Pic br ( B ) : x �→ C x . Dmitri Nikshych (University of New Hampshire) It upgrades to a braided monoidal 2-functor: Braided extensions October 15, 2018 18 / 28

From extensions to braided monoidal 2-functors and back Let A be a finite Abelian group. Let B be a braided fusion category with braiding c . Given a braided extension � C = C x , C 1 = B , x ∈ A we have C x ∈ Pic br ( B ) , x ∈ X with the module braiding given by σ X , V = c X , V c V , X , V ∈ B , X ∈ C x . Furthermore, the tensor products ⊗ x , y : C x ⊠ B C y → C xy gives rise to ∼ B -module equivalences M x , y : C x × C y − → C xy . This gives a (usual) monoidal functor A → Pic br ( B ) : x �→ C x . Dmitri Nikshych (University of New Hampshire) It upgrades to a braided monoidal 2-functor: Braided extensions October 15, 2018 18 / 28

Dmitri Nikshych (University of New Hampshire) Braided extensions October 15, 2018 19 / 28

Namely, the associativity and braiding constraints of C give rise to natural 2-cells involving M x , y , x , y ∈ A : Dmitri Nikshych (University of New Hampshire) Braided extensions October 15, 2018 19 / 28

� � � � � � � � Namely, the associativity and braiding constraints of C give rise to natural 2-cells involving M x , y , x , y ∈ A : M y , z C x ⊠ B C y ⊠ B C z C x ⊠ B C yz α x , y , z M x , y M x , yz � C xyz , C xy ⊠ B C z M xy , z and B x , y C x ⊠ B C y C y ⊠ B C x δ x , y M x , y M y , x C xy . Here B x , y is the braiding in Pic br ( B ). The moral: Structure morphisms in C ← → structure 2-cells in Pic br ( B ). Dmitri Nikshych (University of New Hampshire) Braided extensions October 15, 2018 19 / 28

Dmitri Nikshych (University of New Hampshire) Braided extensions October 15, 2018 20 / 28