Aspects of duality in 2-categories

Transcription

1 Aspects of duality in 2-categories Bruce Bartlett PhD student, supervisor Simon Willerton Sheffield University Categories, Logic and Foundations of Physics Imperial College 14th May 2008

2 Introduction Basic Principle (Baez, Dolan, Coecke, Abramsky,...) Spacetime and quantum theory make more sense when expressed in their natural language not the language of sets and functions, but the language of (higher) categories with duals.

3 Introduction Basic Principle (Baez, Dolan, Coecke, Abramsky,...) Spacetime and quantum theory make more sense when expressed in their natural language not the language of sets and functions, but the language of (higher) categories with duals. Category theory allows you to work on structures without the need first to pulverize them into set theoretic dust. To give an example from the field of architecture, when studying the Notre Dame cathedral in Paris, you try to understand how the building relates to other cathedrals of the day, and then to earlier and later cathedrals, and other kinds of ecclesiastical building. What you don t do is begin by imagining it reduced to a pile of mineral fragments. David Corfield

4 Introduction Basic Principle (Baez, Dolan, Coecke, Abramsky,...) Spacetime and quantum theory make more sense when expressed in their natural language not the language of sets and functions, but the language of (higher) categories with duals. Category theory allows you to work on structures without the need first to pulverize them into set theoretic dust. To give an example from the field of architecture, when studying the Notre Dame cathedral in Paris, you try to understand how the building relates to other cathedrals of the day, and then to earlier and later cathedrals, and other kinds of ecclesiastical building. What you don t do is begin by imagining it reduced to a pile of mineral fragments. David Corfield When in Rome, do as the Romans do. St Augustine

5 Introduction Category theory allows you to work on structures without the need first to pulverize them into set theoretic dust. To give an example from the field of architecture, when studying the Notre Dame cathedral in Paris, you try to understand how the building relates to other cathedrals of the day, and then to earlier and later cathedrals, and other kinds of ecclesiastical building. What you don t do is begin by imagining it reduced to a pile of mineral fragments. David Corfield When in Rome, do as the Romans do. St Augustine

6 Introduction Category theory allows you to work on structures without the need first to pulverize them into set theoretic dust. To give an example from the field of architecture, when studying the Notre Dame cathedral in Paris, you try to understand how the building relates to other cathedrals of the day, and then to earlier and later cathedrals, and other kinds of ecclesiastical building. What you don t do is begin by imagining it reduced to a pile of mineral fragments. David Corfield When in Rome, do as the Romans do. St Augustine Umntu ngumntu ngabantu. Nguni proverb, Southern Africa.

7 Introduction Category theory allows you to work on structures without the need first to pulverize them into set theoretic dust. To give an example from the field of architecture, when studying the Notre Dame cathedral in Paris, you try to understand how the building relates to other cathedrals of the day, and then to earlier and later cathedrals, and other kinds of ecclesiastical building. What you don t do is begin by imagining it reduced to a pile of mineral fragments. David Corfield When in Rome, do as the Romans do. St Augustine Umntu ngumntu ngabantu. Nguni proverb, Southern Africa. }{{} Um }{{} ntu ng }{{}}{{} um }{{} ntu ng }{{}}{{} aba }{{} ntu A person is a person plural person through

8 Introduction Category theory allows you to work on structures without the need first to pulverize them into set theoretic dust. To give an example from the field of architecture, when studying the Notre Dame cathedral in Paris, you try to understand how the building relates to other cathedrals of the day, and then to earlier and later cathedrals, and other kinds of ecclesiastical building. What you don t do is begin by imagining it reduced to a pile of mineral fragments. David Corfield When in Rome, do as the Romans do. St Augustine Umntu ngumntu ngabantu. Nguni proverb, Southern Africa. }{{} Um }{{} ntu ng }{{}}{{} um }{{} ntu ng }{{}}{{} aba }{{} ntu A person is a person through plural person (Related: Ubuntu lit. Person-ness ; basic human decency.)

9 Introduction Category theory allows you to work on structures without the need first to pulverize them into set theoretic dust. To give an example from the field of architecture, when studying the Notre Dame cathedral in Paris, you try to understand how the building relates to other cathedrals of the day, and then to earlier and later cathedrals, and other kinds of ecclesiastical building. What you don t do is begin by imagining it reduced to a pile of mineral fragments. David Corfield When in Rome, do as the Romans do. St Augustine Umntu ngumntu ngabantu. Nguni proverb, Southern Africa. }{{} Um }{{} ntu ng }{{}}{{} um }{{} ntu ng }{{}}{{} aba }{{} ntu A person is a person through plural person (Related: Ubuntu lit. Person-ness ; basic human decency.)!ke e: /karra //ke Khoisan proverb, Diverse people unite?

10 Introduction Category theory allows you to work on structures without the need first to pulverize them into set theoretic dust. To give an example from the field of architecture, when studying the Notre Dame cathedral in Paris, you try to understand how the building relates to other cathedrals of the day, and then to earlier and later cathedrals, and other kinds of ecclesiastical building. What you don t do is begin by imagining it reduced to a pile of mineral fragments. David Corfield When in Rome, do as the Romans do. St Augustine Umntu ngumntu ngabantu. Nguni proverb, Southern Africa. }{{} Um }{{} ntu ng }{{}}{{} um }{{} ntu ng }{{}}{{} aba }{{} ntu A person is a person through plural person (Related: Ubuntu lit. Person-ness ; basic human decency.)!ke e: /karra //ke Khoisan proverb, Diverse people unite? Stupid is as stupid does. Forrest Gump s mum.

11 Introduction ncob Hilb objects (n 1)-dim space H fin dim Hilbert space H 1 morphisms n-dim spacetime A H 2 linear map monoidal H H duals for objects H duals for morphisms H 2 A H 1 adjoint

12 Introduction ncob Hilb objects (n 1)-dim space H fin dim Hilbert space morphisms n-dim spacetime For instance, quantum entanglement: monoidal C duals for + ψ objects H H duals for morphisms C ψ 1 H H 1 A H 2 H H C + ψ 2 H H H 2 A H 1 linear map adjoint

13 Introduction: Reminder on Cobordism Hypothesis

14 Introduction: Reminder on Cobordism Hypothesis Baez and Dolan proposed that a unitary extended n-dimensional TQFT is a unitary representation of the cobordism n-category on the n-category of n-hilbert spaces: Z : ncob nhilb

15 Introduction: Reminder on Cobordism Hypothesis Baez and Dolan proposed that a unitary extended n-dimensional TQFT is a unitary representation of the cobordism n-category on the n-category of n-hilbert spaces: Z : ncob nhilb Cobordism Hyopthesis (Baez,Dolan) ncob is the free weak n-category with duals on one object.

16 Introduction: Reminder on Cobordism Hypothesis Baez and Dolan proposed that a unitary extended n-dimensional TQFT is a unitary representation of the cobordism n-category on the n-category of n-hilbert spaces: Z : ncob nhilb Cobordism Hyopthesis (Baez,Dolan) ncob is the free weak n-category with duals on one object. So understanding duality in higher categories will aid our understanding of spacetime and quantum theory! Skip extra stuff

17 Cobordism Hypothesis: Extra Cobordism Hyopthesis (Baez,Dolan) ncob is the free weak n-category with duals on one object.

18 Cobordism Hypothesis: Extra Cobordism Hyopthesis (Baez,Dolan) ncob is the free weak n-category with duals on one object. This means that an n-dimensional TQFT should be determined by Z(pt), the n-hilbert space assigned to a point.

19 Cobordism Hypothesis: Extra Cobordism Hyopthesis (Baez,Dolan) ncob is the free weak n-category with duals on one object. This means that an n-dimensional TQFT should be determined by Z(pt), the n-hilbert space assigned to a point. For example in the untwisted 3d Chern-Simons theory associated to a Lie group G, it seems that Z(pt) = 2Rep(G).

20 Cobordism Hypothesis: Extra Cobordism Hyopthesis (Baez,Dolan) ncob is the free weak n-category with duals on one object. This means that an n-dimensional TQFT should be determined by Z(pt), the n-hilbert space assigned to a point. For example in the untwisted 3d Chern-Simons theory associated to a Lie group G, it seems that Indeed, at least when G is finite: Theorem (BB, SW). Z(pt) = 2Rep(G). braided mon Bun G (G) Dim 2Rep(G). }{{}}{{} Z(S 1 ) category of weak transformations and modifications of identity 2-functor

21 2. String diagram notation for 2-categories

22 2. String diagram notation for 2-categories Note string diagrams work perfectly well for weak 2-categories, as long as the the parenthesis scheme of the input and output 1-morphisms are understood in each context, eg.: [ Fn (F n 1 F n 2 ) ] [F 2 F 1 ] unique interpretation by coherence G m [ G 3 (G 2 G 1 ) ]

23 Ambiijunction groupoid I

24 Ambiijunction groupoid I An ambidextrous adjoint of a morphism F : A B in a 2-category is a morphism F : B A equipped with unit and counit 2-morphisms expressing F as a right adjoint of F, and unit and counit 2-morphisms expressing F as a left adjoint of F : ( ) F,,,,

25 Ambiijunction groupoid I An ambidextrous adjoint of a morphism F : A B in a 2-category is a morphism F : B A equipped with unit and counit 2-morphisms expressing F as a right adjoint of F, and unit and counit 2-morphisms expressing F as a left adjoint of F : ( ) F,,,, Taken together these form the ambijunction groupoid Amb(F ) of F, where a morphism γ : F (F ) is defined to be an invertible 2-morphism such that (,,,, ) ( =,,,, ).

26 Ambijunction groupoid II We write [Amb(F )] for the isomorphism classes in Amb(F ). Properties of the ambijunction groupoid If Amb(F ) is not empty, then There is at most one arrow between any two ambidextrous adjunctions in Amb(F ). The group Aut(F ) of automorphisms of F acts freely and transitively on [Amb(F )].

27 Ambijunction groupoid II We write [Amb(F )] for the isomorphism classes in Amb(F ). Properties of the ambijunction groupoid If Amb(F ) is not empty, then There is at most one arrow between any two ambidextrous adjunctions in Amb(F ). The group Aut(F ) of automorphisms of F acts freely and transitively on [Amb(F )]. An automorphism α: F F acts on an ambidextrous adjoint by twisting the right unit and counit maps, [ ] α [F ] =,,,,.

28 Even-handed structures

29 Even-handed structures Suppose θ : F G is a 2-morphism, and that choices of ambidextrous adjoints F and G have been made. The right and left daggers of θ are: θ = θ = Whether θ = θ only depends on [F ] and [G ].

30 Even-handed structures Suppose θ : F G is a 2-morphism, and that choices of ambidextrous adjoints F and G have been made. The right and left daggers of θ are: θ = θ = Whether θ = θ only depends on [F ] and [G ]. We say that a 2-category C has ambidextrous adjoints if every morphism has an (unspecified) ambidextrous adjoint.

31 Even-handed structures Suppose θ : F G is a 2-morphism, and that choices of ambidextrous adjoints F and G have been made. The right and left daggers of θ are: θ = θ = Whether θ = θ only depends on [F ] and [G ]. We say that a 2-category C has ambidextrous adjoints if every morphism has an (unspecified) ambidextrous adjoint. Definition An even-handed structure on a 2-category with ambidextrous adjoints is a choice F [ ] [Amb(F )] for every morphism F, such that: 1 id [ ] = class of trivial ambidextrous adjunction for all identity 1-cells, 2 (G F ) [ ] = F [ ] G [ ] for all composable pairs of morphisms, and 3 θ = θ for every 2-morphism θ : F G, provided they are computed using ambidextrous adjoints from the classes F [ ] and G [ ].

32 Even-handed structures So... an even-handed structure on a 2-category is an even-handed trivialization of the ambijunction gerbe : Suppose θ : F G is a 2-morphism, and that choices of ambidextrous adjoints F and G have been made. The right and left daggers of θ are: θ = θ = Whether θ = θ only depends on [F ] and [G ]. We Analagous say thatoa Murray 2-category and CSinger s has ambidextrous refomulationadjoints of a spin if every structure morphism on a has manifold (unspecified) as a trivialization ambidextrous of the spin adjoint. gerbe. Definition An even-handed structure on a 2-category with ambidextrous adjoints is a choice F [ ] [Amb(F )] for every morphism F, such that: 1 id [ ] = class of trivial ambidextrous adjunction for all identity 1-cells, 2 (G F ) [ ] = F [ ] G [ ] for all composable pairs of morphisms, and 3 θ = θ for every 2-morphism θ : F G, provided they are computed using ambidextrous adjoints from the classes F [ ] and G [ ].

33 Examples: Monoidal categories, remarks

34 Examples: Monoidal categories, remarks Consider a monoidal category C (a one-object 2-category) with ambidextrous duals, eg. Rep(G). Some preliminary remarks:

35 Examples: Monoidal categories, remarks Consider a monoidal category C (a one-object 2-category) with ambidextrous duals, eg. Rep(G). Some preliminary remarks: The set of even-handed structures on C is a torsor for the group Aut (id C ), which acts by twisting the right unit and counit maps as before.

36 Examples: Monoidal categories, remarks Consider a monoidal category C (a one-object 2-category) with ambidextrous duals, eg. Rep(G). Some preliminary remarks: The set of even-handed structures on C is a torsor for the group Aut (id C ), which acts by twisting the right unit and counit maps as before. The conventional paradigm for duals in monoidal categories is to choose a specific right dual V for each object, and define a pivotal structure on C as a monoidal natural isomorphism γ : id.

37 Examples: Monoidal categories, remarks Consider a monoidal category C (a one-object 2-category) with ambidextrous duals, eg. Rep(G). Some preliminary remarks: The set of even-handed structures on C is a torsor for the group Aut (id C ), which acts by twisting the right unit and counit maps as before. The conventional paradigm for duals in monoidal categories is to choose a specific right dual V for each object, and define a pivotal structure on C as a monoidal natural isomorphism γ : id. Lemma For each choice of system of right duals on C, there is a canonical bijection { Even-handed structures on C, considered as a one object 2-category } = { Pivotal structures on C with respect to Moreover, this bijection is natural with respect to changing the system of right duals. }.

38 Examples: Monoidal categories, remarks Consider a monoidal category C (a one-object 2-category) with ambidextrous duals, eg. Rep(G). Some preliminary remarks: The set of even-handed structures on C is a torsor for the group Aut (id C ), which acts by twisting the right unit and counit maps as before. The conventional paradigm for duals in monoidal categories is to choose a specific right dual V for each object, and define a pivotal structure on C as a monoidal natural isomorphism γ : id.

39 Examples: Monoidal categories, remarks Consider a monoidal category C (a one-object 2-category) with ambidextrous duals, eg. Rep(G). Some preliminary remarks: The set of even-handed structures on C is a torsor for the group Aut (id C ), which acts by twisting the right unit and counit maps as before. The conventional paradigm for duals in monoidal categories is to choose a specific right dual V for each object, and define a pivotal structure on C as a monoidal natural isomorphism γ : id. In good cases, an even-handed structure [ ] gives rise to a ring homomorphism dim [ ] : K(C) End(1) [V ]. If C is a semisimple linear category, [ ] is characterized by dim [ ].

40 Examples: Monoidal categories (one-object 2-categories) 1 For the monoidal category (G, ω) coming from a 3-cocycle ω Z 3 (G, U(1)), { } { Even-handed structures Group homomorphisms = on (G, ω) f : G U(1) }.

41 Examples: Monoidal categories (one-object 2-categories) 1 For the monoidal category (G, ω) coming from a 3-cocycle ω Z 3 (G, U(1)), { } { Even-handed structures Group homomorphisms = on (G, ω) f : G U(1) }. 2 Suppose C is a monoidal category where every object has a right dual. If C can be equipped with a braiding σ, then { } { } Even-handed structures Pretwists on C =. on C with respect to σ

42 Examples: Monoidal categories (one-object 2-categories) 1 For the monoidal category (G, ω) coming from a 3-cocycle ω Z 3 (G, U(1)), θ Aut(id) is a pretwist with respect to σ if { } { } Even-handed structures Group homomorphisms =. on (G, ω) f : G U(1) =. 2 Suppose C is a monoidal category where every object has a right dual. If C can be equipped with a braiding σ, then { } { } Gives even-handed Even-handed structures Pretwists on C =. on C with respect to σ V [ ] =,,,,.

43 Examples: Monoidal categories (one-object 2-categories) 1 For the monoidal category (G, ω) coming from a 3-cocycle ω Z 3 (G, U(1)), { } { Even-handed structures Group homomorphisms = on (G, ω) f : G U(1) }. 2 Suppose C is a monoidal category where every object has a right dual. If C can be equipped with a braiding σ, then { } { } Even-handed structures Pretwists on C =. on C with respect to σ

44 Examples: Monoidal categories (one-object 2-categories) 1 For the monoidal category (G, ω) coming from a 3-cocycle ω Z 3 (G, U(1)), { } { Even-handed structures Group homomorphisms = on (G, ω) f : G U(1) }. 2 Suppose C is a monoidal category where every object has a right dual. If C can be equipped with a braiding σ, then { } { } Even-handed structures Pretwists on C =. on C with respect to σ 3 For fusion categories (semisimple rigid monoidal k-linear categories), some mysteries remain: What are the equations which have the symmetry group Aut (id) = {θ i k : θ i = θ j θ k whenever X i appears in X j X k }? Will they ensure that an even-handed structure always exists?

45 Examples: Monoidal categories (one-object 2-categories) The manifest invariants of a fusion category are the fusion ring K(C) 1 andfor Müger s the monoidal squaredcategory norms d(g, i ofω) thecoming simple from objects: a 3-cocycle ω Z 3 (G, U(1)), { } { } d i = Even-handed structures...this is independent Group homomorphisms of choice of Xi =.. on (G, ω) f : G U(1) 2If an Suppose even-handed C is a monoidal structure category exists, then where dim every [ ] : K(C) object has k satisfies: a right dual. It isifa Cring can homomorphism be equippedtaking with anonzero braiding values σ, then on simple objects, dim[x { } { } i ] dim[x Even-handed i ] = d i for structures all simple objects. Pretwists on C =. Do these equations have on CAut (id) as their symmetry with respect group? to σto make an even-handed structure, crucially need: 3 For fusion categories (semisimple rigid monoidal k-linear categories), some mysteries remain: What are the equations which have the symmetry group = δ Aut (id) = {θ i k q p : θ i = θ j θ k whenever X i appears in X j X k }? Will they ensure that an even-handed structure always exists?

46 Even-handed structures on sub-2-categories of Cat

47 Even-handed structures on sub-2-categories of Cat If F : A B is a functor between categories, express right and left adjunctions via φ: Hom(Fx, y) = Hom(x, F y) ψ : Hom(F y, x) = Hom(y, Fx)

48 Even-handed structures on sub-2-categories of Cat If F : A B is a functor between categories, express right and left adjunctions via φ: Hom(Fx, y) = Hom(x, F y) ψ : Hom(F y, x) = Hom(y, Fx) For θ : F G, can express θ, θ : G F as follows: post(θ y ) = φ F pre(θ x ) φ -1 G pre( θ y ) = ψg -1 post(θ x) ψ F

49 Even-handed structures on sub-2-categories of Cat If F : A B is a functor between categories, express right and left adjunctions via φ: Hom(Fx, y) = Hom(x, F y) ψ : Hom(F y, x) = Hom(y, Fx) For θ : F G, can express θ, θ : G F as follows: post(θ y ) = φ F pre(θ x ) φ -1 G pre( θ y ) = ψg -1 post(θ x) ψ F In this way, an even-handed structure on a sub-2-category C Cat with ambidextrous adjoints translates into a system of bijective maps Ψ F,F : Adj(F F ) Adj(F F ) which transform correctly under isomorphism 2-cells, respect composition, and satisfy the even-handed equation: post 1( φ F pre(θ) φ 1 ) G = pre 1 ( Ψ(φ G ) 1 post(θ) Ψ(φ F ) )

50 Even-handed structures from traces on linear categories

51 Even-handed structures from traces on linear categories Definition A trace on a k-linear category is a linear map Tr x : End(x) k for each object satisfying: Tr x (gf ) = Tr y (fg). Nondegeneracy: s : Hom(x, y) = Hom(y, x).

52 Even-handed structures from traces on linear categories Definition A trace on a k-linear category is a linear map Tr x : End(x) k for each object Remark satisfying: A trace Tr x (gf on) C= istr the y (fg). same thing as a symmetric monoidal functor Nondegeneracy: s : Hom(x, y) = PlanarCob Hom(y, x). Ob(C) Vect k Hom(x, y) Hom(x, y) Hom(y, z) Hom(x, z) Hom(x, x) Tr x k 1 i f i f i etc.

53 Even-handed structures from traces on linear categories Definition A trace on a k-linear category is a linear map Tr x : End(x) k for each object satisfying: Tr x (gf ) = Tr y (fg). Nondegeneracy: s : Hom(x, y) = Hom(y, x).

54 Even-handed structures from traces on linear categories Definition A trace on a k-linear category is a linear map Tr x : End(x) k for each object satisfying: Tr x (gf ) = Tr y (fg). Nondegeneracy: s : Hom(x, y) = Hom(y, x). Turn right adjoints into left adjoints: Hom(F y, x) φ T Hom(y, Fx) φ s A s B φ Hom(x, F y) Hom(Fx, y) T := s -1 B φ s A

55 Even-handed structures from traces on linear categories Definition A trace on a k-linear category is a linear map Tr x : End(x) k for each object satisfying: Tr x (gf ) = Tr y (fg). Nondegeneracy: s : Hom(x, y) = Hom(y, x). Turn right adjoints into left adjoints: Hom(F y, x) φ T Hom(y, Fx) φ s A s B φ Hom(x, F y) Hom(Fx, y) T := s -1 B φ s A Theorem If each category in C LinearCat k comes equipped with a trace, then sending φ Ψ φ T gives an even handed structure on C.

56 Example: The 2-category of 2-Hilbert spaces

57 Example: The 2-category of 2-Hilbert spaces Definition (Baez) A 2-Hilbert space is an abelian Hilb-category H equipped with antilinear maps : Hom(x, y) Hom(y, x) compatible with composition and inner products. They form a 2-category 2Hilb.

58 Example: The 2-category of 2-Hilbert spaces Definition (Baez) A 2-Hilbert space is an abelian Hilb-category H equipped with antilinear maps : Hom(x, y) Hom(y, x) compatible with composition and inner products. They form a 2-category 2Hilb. A good example is the category Rep(G) of unitary representations of a compact group.

59 Example: The 2-category of 2-Hilbert spaces Definition (Baez) A 2-Hilbert space is an abelian Hilb-category H equipped with antilinear maps : Hom(x, y) Hom(y, x) compatible with composition and inner products. They form a 2-category 2Hilb. A good example is the category Rep(G) of unitary representations of a compact group. Every 2-Hilbert space comes equipped with a canonical trace, f (id x, f ).

60 Example: The 2-category of 2-Hilbert spaces Definition (Baez) A 2-Hilbert space is an abelian Hilb-category H equipped with antilinear maps : Hom(x, y) Hom(y, x) compatible with composition and inner products. They form a 2-category 2Hilb. A good example is the category Rep(G) of unitary representations of a compact group. Every 2-Hilbert space comes equipped with a canonical trace, f (id x, f ). Lemma The resulting even-handed structure on 2Hilb can be expressed as φ T = φ. where φ is the adjoint linear map of φ.

61 Example: The 2-category of 2-Hilbert spaces Definition (Baez) A 2-Hilbert space is an abelian Hilb-category H equipped with antilinear maps : Hom(x, y) Hom(y, x) compatible with composition and inner products. They form a 2-category 2Hilb. A good example is the category Rep(G) of unitary representations of a compact group. Every 2-Hilbert space comes equipped Similar withtoa the canonical formulatrace, d on a f (id x, f ). compact Riemannian manifold, Lemma d = d The resulting even-handed structure on 2Hilb can be expressed as φ T = φ. where φ is the adjoint linear map of φ.

62 Example: Indeed, can prove The that 2-category in the semisimple of 2-Hilbert context, spaces Even-handed structures on Definition a full (Baez) sub-2-category Weightings on the = simple objects in S. A 2-HilbertSspace SLCat is an k abelian Hilb-category up toh aequipped global scale with factor antilinear maps : Hom(x, y) Hom(y, x) compatible with composition and inner products. This is similar Theytoform a 2-category 2Hilb. dim{harmonic spinors on a spin manifold} is a conformal invariant. A good example is the category Rep(G) of unitary representations of a compact group. Every 2-Hilbert space comes equipped Similar withtoa the canonical formulatrace, d on a f (id x, f ). compact Riemannian manifold, Lemma d = d The resulting even-handed structure on 2Hilb can be expressed as φ T = φ. where φ is the adjoint linear map of φ.

63 Example: The 2-category CYau

64 Example: The 2-category CYau X an n-dimensional Calabi-Yau manifold. Have the graded derived category D(X ): An object is a bounded complex E of coherent sheaves on X. Hom D(X ) (E, F ) := i Hom D(X ) (E, F [i]).

65 Example: The 2-category CYau X an n-dimensional Calabi-Yau manifold. Have the graded derived category D(X ): An object is a bounded complex E of coherent sheaves on X. Hom D(X ) (E, F ) := Hom D(X ) (E, F [i]). i Serre duality gives D(X ) a supertrace, Tr : Hom n D(X )(E, E ) C.

66 Example: The 2-category CYau X an n-dimensional Calabi-Yau manifold. Have the graded derived category D(X ): An object is a bounded complex E of coherent sheaves on X. Hom D(X ) (E, F ) := Hom D(X ) (E, F [i]). i Serre duality gives D(X ) a supertrace, Definition Tr : Hom n D(X )(E, E ) C. The 2-category CYau has graded derived categories D(X ) of Calabi-Yau manifolds as objects. A morphism is a functor F : D(X ) D(Y ) naturally isomorphic to an integral kernel. A 2-morphism is a natural transformation.

67 Example: The 2-category CYau X an n-dimensional Calabi-Yau manifold. Have the graded derived category D(X ): An object is a bounded complex E of coherent sheaves on X. Hom D(X ) (E, F ) := Hom D(X ) (E, F [i]). i Serre duality gives D(X ) a supertrace, Definition Tr : Hom n D(X )(E, E ) C. The 2-category CYau has graded derived categories D(X ) of Calabi-Yau manifolds as objects. A morphism is a functor F : D(X ) D(Y ) naturally isomorphic to an integral kernel. A 2-morphism is a natural transformation. Corollary The 2-category CYau comes equipped with a canonical even-handed structure arising from Serre duality on each D(X ).

68 Summary

69 Summary Investigating duality in 2-categories from the paradigm of TQFT.

70 Summary Investigating duality in 2-categories from the paradigm of TQFT. The concept of an even-handed structure

71 Summary Investigating duality in 2-categories from the paradigm of TQFT. The concept of an even-handed structure is economical

72 Summary Investigating duality in 2-categories from the paradigm of TQFT. The concept of an even-handed structure is economical is flexible

73 Summary Investigating duality in 2-categories from the paradigm of TQFT. The concept of an even-handed structure is economical is flexible requires no awkward fixed choices of adjoints, so fits examples like 2Hilb and CYau.

74 Summary Investigating duality in 2-categories from the paradigm of TQFT. The concept of an even-handed structure is economical is flexible requires no awkward fixed choices of adjoints, so fits examples like 2Hilb and CYau. goes hand in hand with a powerful string-diagram calculus

75 Summary Investigating duality in 2-categories from the paradigm of TQFT. The concept of an even-handed structure is economical is flexible requires no awkward fixed choices of adjoints, so fits examples like 2Hilb and CYau. goes hand in hand with a powerful string-diagram calculus The concept of an even-handed structure, and the moduli space of even-handed structures on a given 2-category, have geometric overtones.

76 Summary Investigating duality in 2-categories from the paradigm of TQFT. The concept of an even-handed structure is economical is flexible requires no awkward fixed choices of adjoints, so fits examples like 2Hilb and CYau. goes hand in hand with a powerful string-diagram calculus The concept of an even-handed structure, and the moduli space of even-handed structures on a given 2-category, have geometric overtones. Personal motivation: needed an even-handed structure on 2Hilb to make the 2-character into a functor χ: 2Rep(G) Bun }{{} G (G) homotopy category

77 Reference J. Baez and J. Dolan, Higher dimensional algebra and topological quantum field theory, arxiv:q-alg/ J. Baez, Higher dimensional algebra II: 2-Hilbert spaces, arxiv:q-alg/ M. Müger, From Subfactors to Categories and Topology I. Frobenius algebras in and Morita equivalence of tensor categories, arxiv:math/ P. Etingof, D. Nikshych and V. Ostrik, On fusion categories, arxiv:math/ v9. A. Caldararu and S. Willerton, The Mukai Pairing, I : A categorical approach, arxiv/ M. Boyarchenko, notes for the series of lectures Introduction to modular categories, N. Ganter, M. Kapranov, Representation and character theory in 2-categories, arxiv/math.kt/ J. Roberts and S. Willerton, On the Rozansky-Witten weight systems,