Generalized Tannakian duality Daniel Sch appi University of - PowerPoint PPT Presentation

Generalized Tannakian duality Daniel Sch¨ appi University of Chicago 22 July, 2011 International Category Theory Conference University of British Columbia Daniel Sch¨ appi (University of Chicago) Generalized Tannakian duality CT 2011 Vancouver 1 / 17

Outline Introduction 1 A bicategorical interpretation 2 The Tannakian biadjunction 3 Applications 4 Daniel Sch¨ appi (University of Chicago) Generalized Tannakian duality CT 2011 Vancouver 2 / 17

Classical Tannaka duality Categories equipped with Group-like objects suitable structures Daniel Sch¨ appi (University of Chicago) Generalized Tannakian duality CT 2011 Vancouver 3 / 17

Classical Tannaka duality Categories equipped with Group-like objects suitable structures Reconstruction problem Can a group-like object be reconstructed from its category of representations? Daniel Sch¨ appi (University of Chicago) Generalized Tannakian duality CT 2011 Vancouver 3 / 17

Classical Tannaka duality Categories equipped with Group-like objects suitable structures Reconstruction problem Can a group-like object be reconstructed from its category of representations? Recognition problem Which categories are equivalent to categories of representations for some group-like object? Daniel Sch¨ appi (University of Chicago) Generalized Tannakian duality CT 2011 Vancouver 3 / 17

Tannaka duality for Hopf algebras over fields Daniel Sch¨ appi (University of Chicago) Generalized Tannakian duality CT 2011 Vancouver 4 / 17

Tannaka duality for Hopf algebras over fields Theorem Every Hopf algebra can be reconstructed from the category of finite dimensional comodules. Daniel Sch¨ appi (University of Chicago) Generalized Tannakian duality CT 2011 Vancouver 4 / 17

Tannaka duality for Hopf algebras over fields Theorem Every Hopf algebra can be reconstructed from the category of finite dimensional comodules. Theorem (Saavedra Rivano, Deligne) Let k be a field. If Daniel Sch¨ appi (University of Chicago) Generalized Tannakian duality CT 2011 Vancouver 4 / 17

Tannaka duality for Hopf algebras over fields Theorem Every Hopf algebra can be reconstructed from the category of finite dimensional comodules. Theorem (Saavedra Rivano, Deligne) Let k be a field. If A is an abelian autonomous symmetric monoidal k -linear category Daniel Sch¨ appi (University of Chicago) Generalized Tannakian duality CT 2011 Vancouver 4 / 17

Tannaka duality for Hopf algebras over fields Theorem Every Hopf algebra can be reconstructed from the category of finite dimensional comodules. Theorem (Saavedra Rivano, Deligne) Let k be a field. If A is an abelian autonomous symmetric monoidal k -linear category w : A → Vect k is a faithful exact symmetric strong monoidal k -linear functor Daniel Sch¨ appi (University of Chicago) Generalized Tannakian duality CT 2011 Vancouver 4 / 17

Tannaka duality for Hopf algebras over fields Theorem Every Hopf algebra can be reconstructed from the category of finite dimensional comodules. Theorem (Saavedra Rivano, Deligne) Let k be a field. If A is an abelian autonomous symmetric monoidal k -linear category w : A → Vect k is a faithful exact symmetric strong monoidal k -linear functor then there exists a Hopf algebra H such that A ≃ Rep( H ) . Daniel Sch¨ appi (University of Chicago) Generalized Tannakian duality CT 2011 Vancouver 4 / 17

The classical proof Deligne’s proof Daniel Sch¨ appi (University of Chicago) Generalized Tannakian duality CT 2011 Vancouver 5 / 17

The classical proof Deligne’s proof A abelian, w : A → Vect k faithful & exact A ≃ Comod( C ) � Daniel Sch¨ appi (University of Chicago) Generalized Tannakian duality CT 2011 Vancouver 5 / 17

The classical proof Deligne’s proof A abelian, w : A → Vect k faithful & exact A ≃ Comod( C ) � symmetric monoidal structure bialgebra structure on C � Daniel Sch¨ appi (University of Chicago) Generalized Tannakian duality CT 2011 Vancouver 5 / 17

The classical proof Deligne’s proof A abelian, w : A → Vect k faithful & exact A ≃ Comod( C ) � symmetric monoidal structure bialgebra structure on C � A autonomous Hopf algebra structure on C � Daniel Sch¨ appi (University of Chicago) Generalized Tannakian duality CT 2011 Vancouver 5 / 17

The classical proof Deligne’s proof A abelian, w : A → Vect k faithful & exact A ≃ Comod( C ) � symmetric monoidal structure bialgebra structure on C � A autonomous Hopf algebra structure on C � Theorem (Street) There is a biadjunction between k -linear categories over Vect k and coalgebras. Daniel Sch¨ appi (University of Chicago) Generalized Tannakian duality CT 2011 Vancouver 5 / 17

The classical proof Deligne’s proof A abelian, w : A → Vect k faithful & exact A ≃ Comod( C ) � symmetric monoidal structure bialgebra structure on C � A autonomous Hopf algebra structure on C � Theorem (Street) There is a biadjunction between k -linear categories over Vect k and coalgebras. Reconstruction problem: when is the counit an isomorphism? Daniel Sch¨ appi (University of Chicago) Generalized Tannakian duality CT 2011 Vancouver 5 / 17

The classical proof Deligne’s proof A abelian, w : A → Vect k faithful & exact A ≃ Comod( C ) � symmetric monoidal structure bialgebra structure on C � A autonomous Hopf algebra structure on C � Theorem (Street) There is a biadjunction between k -linear categories over Vect k and coalgebras. Reconstruction problem: when is the counit an isomorphism? Recognition problem: when is the unit an equivalence? Daniel Sch¨ appi (University of Chicago) Generalized Tannakian duality CT 2011 Vancouver 5 / 17

Finding the right environment Daniel Sch¨ appi (University of Chicago) Generalized Tannakian duality CT 2011 Vancouver 6 / 17

Finding the right environment Definition A cosmos is a complete and cocomplete symmetric monoidal closed category V . Daniel Sch¨ appi (University of Chicago) Generalized Tannakian duality CT 2011 Vancouver 6 / 17

� Finding the right environment Definition A cosmos is a complete and cocomplete symmetric monoidal closed category V . Definition A profunctor (also known as distributor or module) A − → B is a − cocontinuous functor [ A op , V ] → [ B op , V ] . The category of profunctors is denoted by Prof ( V ) . Daniel Sch¨ appi (University of Chicago) Generalized Tannakian duality CT 2011 Vancouver 6 / 17

� � Finding the right environment Definition A cosmos is a complete and cocomplete symmetric monoidal closed category V . Definition A profunctor (also known as distributor or module) A − → B is a − cocontinuous functor [ A op , V ] → [ B op , V ] . The category of profunctors is denoted by Prof ( V ) . Observation Coalgebras are precisely comonads I − → I in Prof ( V ) . − Daniel Sch¨ appi (University of Chicago) Generalized Tannakian duality CT 2011 Vancouver 6 / 17

Finding the right environment Definition A left adjoint 1-cell in a bicategory is called a map. Daniel Sch¨ appi (University of Chicago) Generalized Tannakian duality CT 2011 Vancouver 7 / 17

� Finding the right environment Definition A left adjoint 1-cell in a bicategory is called a map. Lemma Maps A − → B in Prof ( V ) are in bijection with V -functors A → B . − Daniel Sch¨ appi (University of Chicago) Generalized Tannakian duality CT 2011 Vancouver 7 / 17

� Finding the right environment Definition A left adjoint 1-cell in a bicategory is called a map. Lemma Maps A − → B in Prof ( V ) are in bijection with V -functors A → B . − Proof. Let L : [ A op , V ] → [ B op , V ] be cocontinuous. Daniel Sch¨ appi (University of Chicago) Generalized Tannakian duality CT 2011 Vancouver 7 / 17

� Finding the right environment Definition A left adjoint 1-cell in a bicategory is called a map. Lemma Maps A − → B in Prof ( V ) are in bijection with V -functors A → B . − Proof. Let L : [ A op , V ] → [ B op , V ] be cocontinuous. Then there exists w : A → [ B op , V ] such that L = Lan Y w . Daniel Sch¨ appi (University of Chicago) Generalized Tannakian duality CT 2011 Vancouver 7 / 17

� Finding the right environment Definition A left adjoint 1-cell in a bicategory is called a map. Lemma Maps A − → B in Prof ( V ) are in bijection with V -functors A → B . − Proof. Let L : [ A op , V ] → [ B op , V ] be cocontinuous. Then there exists w : A → [ B op , V ] such that L = Lan Y w . Lan Y w has a right adjoint X �→ Hom( w − , X ) . Daniel Sch¨ appi (University of Chicago) Generalized Tannakian duality CT 2011 Vancouver 7 / 17

� Finding the right environment Definition A left adjoint 1-cell in a bicategory is called a map. Lemma Maps A − → B in Prof ( V ) are in bijection with V -functors A → B . − Proof. Let L : [ A op , V ] → [ B op , V ] be cocontinuous. Then there exists w : A → [ B op , V ] such that L = Lan Y w . Lan Y w has a right adjoint X �→ Hom( w − , X ) . The right adjoint is cocontinuous ⇔ w ( A ) ∈ B for all A ∈ A . Daniel Sch¨ appi (University of Chicago) Generalized Tannakian duality CT 2011 Vancouver 7 / 17