172
Hopf (co)monads, tensor functors and exact sequences of tensor categories Alain Bruguières (Université Montpellier II) based on joint works with Alexis Virelizier and Steve Lack [BLV] and with Sonia Natale [BN] Conference ‘Quantum Groups’ Clermont-Ferrand August 30- September 3 2010 Conference of the ANR project GALOISINT Quantum Groups : Galois and integration techniques

Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

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Page 1: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf (co)monads, tensor functors and exactsequences of tensor categories

Alain Bruguières

(Université Montpellier II)

based on joint workswith Alexis Virelizier and Steve Lack [BLV]

and with Sonia Natale [BN]

Conference ‘Quantum Groups’

Clermont-Ferrand August 30- September 3 2010Conference of the ANR project GALOISINT Quantum Groups : Galois and integration techniques

Page 2: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Introduction

Motivation : Tannaka theory 2 / 35

Over k field:

H Hopf algebra −→a tensor category C = comodH+ a fiber functor C → vect

Reconstruction: given C tensor category + ω : C → vect fiber functor

H = Coend(ω) =

∫ X∈C

ω(X) ⊗ ω(X)∗ Hopf algebra

with commutative diagram:

Cω //

'⊗ $$

vect

comodH

99

A fiber functor is encoded by a Hopf algebra (in Vect)

Page 3: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Introduction

Motivation : Tannaka theory 2 / 35

Over k field:

H Hopf algebra −→a tensor category C = comodH+ a fiber functor C → vect

Reconstruction: given C tensor category + ω : C → vect fiber functor

H = Coend(ω) =

∫ X∈C

ω(X) ⊗ ω(X)∗ Hopf algebra

with commutative diagram:

Cω //

'⊗ $$

vect

comodH

99

A fiber functor is encoded by a Hopf algebra (in Vect)

Page 4: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Introduction

Motivation : Tannaka theory 2 / 35

Over k field:

H Hopf algebra −→a tensor category C = comodH+ a fiber functor C → vect

Reconstruction: given C tensor category + ω : C → vect fiber functor

H = Coend(ω) =

∫ X∈C

ω(X) ⊗ ω(X)∗ Hopf algebra

with commutative diagram:

Cω //

'⊗ $$

vect

comodH

99

A fiber functor is encoded by a Hopf algebra (in Vect)

Page 5: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Introduction

Motivation : Tannaka theory 2 / 35

Over k field:

H Hopf algebra −→a tensor category C = comodH+ a fiber functor C → vect

Reconstruction: given C tensor category + ω : C → vect fiber functor

H = Coend(ω) =

∫ X∈C

ω(X) ⊗ ω(X)∗ Hopf algebra

with commutative diagram:

Cω //

'⊗ $$

vect

comodH

99

A fiber functor is encoded by a Hopf algebra (in Vect)

Page 6: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Introduction

Motivation : Tannaka theory 2 / 35

Over k field:

H Hopf algebra −→a tensor category C = comodH+ a fiber functor C → vect

Reconstruction: given C tensor category + ω : C → vect fiber functor

H = Coend(ω) =

∫ X∈C

ω(X) ⊗ ω(X)∗ Hopf algebra

with commutative diagram:

Cω //

'⊗ $$

vect

comodH

99

A fiber functor is encoded by a Hopf algebra (in Vect)

Page 7: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Introduction

Motivation : Tannaka theory 2 / 35

Over k field:

H Hopf algebra −→a tensor category C = comodH+ a fiber functor C → vect

Reconstruction: given C tensor category + ω : C → vect fiber functor

H = Coend(ω) =

∫ X∈C

ω(X) ⊗ ω(X)∗ Hopf algebra

with commutative diagram:

Cω //

'⊗ $$

vect

comodH

99

A fiber functor is encoded by a Hopf algebra (in Vect)

Page 8: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Introduction

Motivation : Tannaka theory 2 / 35

Over k field:

H Hopf algebra −→a tensor category C = comodH+ a fiber functor C → vect

Reconstruction: given C tensor category + ω : C → vect fiber functor

H = Coend(ω) =

∫ X∈C

ω(X) ⊗ ω(X)∗ Hopf algebra

with commutative diagram:

Cω //

'⊗ $$

vect

comodH

99

A fiber functor is encoded by a Hopf algebra (in Vect)

Page 9: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Introduction

Motivation : Tannaka theory 2 / 35

Over k field:

H Hopf algebra −→a tensor category C = comodH+ a fiber functor C → vect

Reconstruction: given C tensor category + ω : C → vect fiber functor

H = Coend(ω) =

∫ X∈C

ω(X) ⊗ ω(X)∗ Hopf algebra

with commutative diagram:

Cω //

'⊗ $$

vect

comodH

99

A fiber functor is encoded by a Hopf algebra (in Vect)

Page 10: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Introduction

G affine group scheme/k = commutative Hopf algebra H = O(G). ThenC = comodH = repG and the fiber functor C → vect are both symmetric.

Converse: C symmetric tensor category + ω symmetric fiber functor H = Coend(ω) commutative Hopf algebra, G = SpecH affine groupscheme and C ' repG as symmetric tensor categories.Then there exists a commutative algebra A in C (or its Ind-completion)satisfying

∀X in C, A ⊗ X∼−→ An as left A - modules

Hom(1,A) = k

and we haveω(X) = Hom(1,A ⊗ X).

The proof of Deligne’s internal characterization of tannaka categoriesconsists in constructing such a trivializing algebra.

A symmetric fiber functor is encoded by a certain commutative algebra inC (or IndC)

Can we give similar encodings for arbitrary tensor functors?

Page 11: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Introduction

G affine group scheme/k = commutative Hopf algebra H = O(G).

ThenC = comodH = repG and the fiber functor C → vect are both symmetric.

Converse: C symmetric tensor category + ω symmetric fiber functor H = Coend(ω) commutative Hopf algebra, G = SpecH affine groupscheme and C ' repG as symmetric tensor categories.Then there exists a commutative algebra A in C (or its Ind-completion)satisfying

∀X in C, A ⊗ X∼−→ An as left A - modules

Hom(1,A) = k

and we haveω(X) = Hom(1,A ⊗ X).

The proof of Deligne’s internal characterization of tannaka categoriesconsists in constructing such a trivializing algebra.

A symmetric fiber functor is encoded by a certain commutative algebra inC (or IndC)

Can we give similar encodings for arbitrary tensor functors?

Page 12: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Introduction

G affine group scheme/k = commutative Hopf algebra H = O(G). ThenC = comodH = repG and the fiber functor C → vect are both symmetric.

Converse: C symmetric tensor category + ω symmetric fiber functor H = Coend(ω) commutative Hopf algebra, G = SpecH affine groupscheme and C ' repG as symmetric tensor categories.Then there exists a commutative algebra A in C (or its Ind-completion)satisfying

∀X in C, A ⊗ X∼−→ An as left A - modules

Hom(1,A) = k

and we haveω(X) = Hom(1,A ⊗ X).

The proof of Deligne’s internal characterization of tannaka categoriesconsists in constructing such a trivializing algebra.

A symmetric fiber functor is encoded by a certain commutative algebra inC (or IndC)

Can we give similar encodings for arbitrary tensor functors?

Page 13: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Introduction

G affine group scheme/k = commutative Hopf algebra H = O(G). ThenC = comodH = repG and the fiber functor C → vect are both symmetric.

Converse: C symmetric tensor category + ω symmetric fiber functor

H = Coend(ω) commutative Hopf algebra, G = SpecH affine groupscheme and C ' repG as symmetric tensor categories.Then there exists a commutative algebra A in C (or its Ind-completion)satisfying

∀X in C, A ⊗ X∼−→ An as left A - modules

Hom(1,A) = k

and we haveω(X) = Hom(1,A ⊗ X).

The proof of Deligne’s internal characterization of tannaka categoriesconsists in constructing such a trivializing algebra.

A symmetric fiber functor is encoded by a certain commutative algebra inC (or IndC)

Can we give similar encodings for arbitrary tensor functors?

Page 14: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Introduction

G affine group scheme/k = commutative Hopf algebra H = O(G). ThenC = comodH = repG and the fiber functor C → vect are both symmetric.

Converse: C symmetric tensor category + ω symmetric fiber functor H = Coend(ω) commutative Hopf algebra,

G = SpecH affine groupscheme and C ' repG as symmetric tensor categories.Then there exists a commutative algebra A in C (or its Ind-completion)satisfying

∀X in C, A ⊗ X∼−→ An as left A - modules

Hom(1,A) = k

and we haveω(X) = Hom(1,A ⊗ X).

The proof of Deligne’s internal characterization of tannaka categoriesconsists in constructing such a trivializing algebra.

A symmetric fiber functor is encoded by a certain commutative algebra inC (or IndC)

Can we give similar encodings for arbitrary tensor functors?

Page 15: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Introduction

G affine group scheme/k = commutative Hopf algebra H = O(G). ThenC = comodH = repG and the fiber functor C → vect are both symmetric.

Converse: C symmetric tensor category + ω symmetric fiber functor H = Coend(ω) commutative Hopf algebra, G = SpecH affine groupscheme

and C ' repG as symmetric tensor categories.Then there exists a commutative algebra A in C (or its Ind-completion)satisfying

∀X in C, A ⊗ X∼−→ An as left A - modules

Hom(1,A) = k

and we haveω(X) = Hom(1,A ⊗ X).

The proof of Deligne’s internal characterization of tannaka categoriesconsists in constructing such a trivializing algebra.

A symmetric fiber functor is encoded by a certain commutative algebra inC (or IndC)

Can we give similar encodings for arbitrary tensor functors?

Page 16: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Introduction

G affine group scheme/k = commutative Hopf algebra H = O(G). ThenC = comodH = repG and the fiber functor C → vect are both symmetric.

Converse: C symmetric tensor category + ω symmetric fiber functor H = Coend(ω) commutative Hopf algebra, G = SpecH affine groupscheme and C ' repG as symmetric tensor categories.

Then there exists a commutative algebra A in C (or its Ind-completion)satisfying

∀X in C, A ⊗ X∼−→ An as left A - modules

Hom(1,A) = k

and we haveω(X) = Hom(1,A ⊗ X).

The proof of Deligne’s internal characterization of tannaka categoriesconsists in constructing such a trivializing algebra.

A symmetric fiber functor is encoded by a certain commutative algebra inC (or IndC)

Can we give similar encodings for arbitrary tensor functors?

Page 17: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Introduction

G affine group scheme/k = commutative Hopf algebra H = O(G). ThenC = comodH = repG and the fiber functor C → vect are both symmetric.

Converse: C symmetric tensor category + ω symmetric fiber functor H = Coend(ω) commutative Hopf algebra, G = SpecH affine groupscheme and C ' repG as symmetric tensor categories.Then there exists a commutative algebra A in C (or its Ind-completion)satisfying

∀X in C, A ⊗ X∼−→ An as left A - modules

Hom(1,A) = k

and we haveω(X) = Hom(1,A ⊗ X).

The proof of Deligne’s internal characterization of tannaka categoriesconsists in constructing such a trivializing algebra.

A symmetric fiber functor is encoded by a certain commutative algebra inC (or IndC)

Can we give similar encodings for arbitrary tensor functors?

Page 18: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Introduction

G affine group scheme/k = commutative Hopf algebra H = O(G). ThenC = comodH = repG and the fiber functor C → vect are both symmetric.

Converse: C symmetric tensor category + ω symmetric fiber functor H = Coend(ω) commutative Hopf algebra, G = SpecH affine groupscheme and C ' repG as symmetric tensor categories.Then there exists a commutative algebra A in C (or its Ind-completion)satisfying

∀X in C, A ⊗ X∼−→ An as left A - modules

Hom(1,A) = k

and we haveω(X) = Hom(1,A ⊗ X).

The proof of Deligne’s internal characterization of tannaka categoriesconsists in constructing such a trivializing algebra.

A symmetric fiber functor is encoded by a certain commutative algebra inC (or IndC)

Can we give similar encodings for arbitrary tensor functors?

Page 19: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Introduction

G affine group scheme/k = commutative Hopf algebra H = O(G). ThenC = comodH = repG and the fiber functor C → vect are both symmetric.

Converse: C symmetric tensor category + ω symmetric fiber functor H = Coend(ω) commutative Hopf algebra, G = SpecH affine groupscheme and C ' repG as symmetric tensor categories.Then there exists a commutative algebra A in C (or its Ind-completion)satisfying

∀X in C, A ⊗ X∼−→ An as left A - modules

Hom(1,A) = k

and we haveω(X) = Hom(1,A ⊗ X).

The proof of Deligne’s internal characterization of tannaka categoriesconsists in constructing such a trivializing algebra.

A symmetric fiber functor is encoded by a certain commutative algebra inC (or IndC)

Can we give similar encodings for arbitrary tensor functors?

Page 20: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Introduction

G affine group scheme/k = commutative Hopf algebra H = O(G). ThenC = comodH = repG and the fiber functor C → vect are both symmetric.

Converse: C symmetric tensor category + ω symmetric fiber functor H = Coend(ω) commutative Hopf algebra, G = SpecH affine groupscheme and C ' repG as symmetric tensor categories.Then there exists a commutative algebra A in C (or its Ind-completion)satisfying

∀X in C, A ⊗ X∼−→ An as left A - modules

Hom(1,A) = k

and we haveω(X) = Hom(1,A ⊗ X).

The proof of Deligne’s internal characterization of tannaka categoriesconsists in constructing such a trivializing algebra.

A symmetric fiber functor is encoded by a certain commutative algebra inC (or IndC)

Can we give similar encodings for arbitrary tensor functors?

Page 21: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Introduction

Tensor categories and tensor functors 4 / 35

Let k be a field.

DefinitionIn this talk a tensor category is a k- linear abelian category with a structureof rigid category (=monoidal with duals) such that:

C is locally finite (Hom’s are finite dim’l and objects have finite length)

⊗ is k- bilinear and End(1) = k

C is finite if Ck'R mod for some finite dimensional k- algebra R.

DefinitionA tensor functor F : C → D is a k- linear exact strong monoidal functorbetween tensor categories.

A tensor functor F is faithful. It has a right adjoint iff it has a left adjoint; inthat case we say that F is finite.

Page 22: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Introduction

Examples1 vect is the initial tensor category

2 A fiber functor for C is a tensor functor C → vect3 A Hopf algebra morphism f : H → H′ induces a tensor functor

f∗ : comodH → comodH′

Tannaka duality asserts that we have an equivalence of categories

{{Hopf Algebras}} ' {{Tensor categories}} / vect

But many tensor categories do not come from Hopf algebras!

Page 23: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Introduction

Examples1 vect is the initial tensor category2 A fiber functor for C is a tensor functor C → vect

3 A Hopf algebra morphism f : H → H′ induces a tensor functor

f∗ : comodH → comodH′

Tannaka duality asserts that we have an equivalence of categories

{{Hopf Algebras}} ' {{Tensor categories}} / vect

But many tensor categories do not come from Hopf algebras!

Page 24: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Introduction

Examples1 vect is the initial tensor category2 A fiber functor for C is a tensor functor C → vect3 A Hopf algebra morphism f : H → H′ induces a tensor functor

f∗ : comodH → comodH′

Tannaka duality asserts that we have an equivalence of categories

{{Hopf Algebras}} ' {{Tensor categories}} / vect

But many tensor categories do not come from Hopf algebras!

Page 25: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Introduction

Examples1 vect is the initial tensor category2 A fiber functor for C is a tensor functor C → vect3 A Hopf algebra morphism f : H → H′ induces a tensor functor

f∗ : comodH → comodH′

Tannaka duality asserts that we have an equivalence of categories

{{Hopf Algebras}} ' {{Tensor categories}} / vect

But many tensor categories do not come from Hopf algebras!

Page 26: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Introduction

Examples1 vect is the initial tensor category2 A fiber functor for C is a tensor functor C → vect3 A Hopf algebra morphism f : H → H′ induces a tensor functor

f∗ : comodH → comodH′

Tannaka duality asserts that we have an equivalence of categories

{{Hopf Algebras}} ' {{Tensor categories}} / vect

But many tensor categories do not come from Hopf algebras!

Page 27: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Introduction

Examples1 vect is the initial tensor category2 A fiber functor for C is a tensor functor C → vect3 A Hopf algebra morphism f : H → H′ induces a tensor functor

f∗ : comodH → comodH′

Tannaka duality asserts that we have an equivalence of categories

{{Hopf Algebras}} ' {{Tensor categories}} / vect

But many tensor categories do not come from Hopf algebras!

Page 28: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Introduction

Let F : C → D be a tensor functor.

Question 1Can one encode F by algebraic data in D (or IndD)?

Yes. But this data cannot be a Hopf algebra, as D is not braided. It is aHopf (co)monad.

Question 2Can one encode F by an algebraic data in C (or IndC)?

Yes, if F is dominant.This data is a commutative algebra in the center of C (or IndC).

Page 29: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Introduction

Let F : C → D be a tensor functor.

Question 1Can one encode F by algebraic data in D (or IndD)?

Yes. But this data cannot be a Hopf algebra, as D is not braided. It is aHopf (co)monad.

Question 2Can one encode F by an algebraic data in C (or IndC)?

Yes, if F is dominant.This data is a commutative algebra in the center of C (or IndC).

Page 30: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Introduction

Let F : C → D be a tensor functor.

Question 1Can one encode F by algebraic data in D (or IndD)?

Yes. But this data cannot be a Hopf algebra, as D is not braided.

It is aHopf (co)monad.

Question 2Can one encode F by an algebraic data in C (or IndC)?

Yes, if F is dominant.This data is a commutative algebra in the center of C (or IndC).

Page 31: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Introduction

Let F : C → D be a tensor functor.

Question 1Can one encode F by algebraic data in D (or IndD)?

Yes. But this data cannot be a Hopf algebra, as D is not braided. It is aHopf (co)monad.

Question 2Can one encode F by an algebraic data in C (or IndC)?

Yes, if F is dominant.This data is a commutative algebra in the center of C (or IndC).

Page 32: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Introduction

Let F : C → D be a tensor functor.

Question 1Can one encode F by algebraic data in D (or IndD)?

Yes. But this data cannot be a Hopf algebra, as D is not braided. It is aHopf (co)monad.

Question 2Can one encode F by an algebraic data in C (or IndC)?

Yes, if F is dominant.

This data is a commutative algebra in the center of C (or IndC).

Page 33: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Introduction

Let F : C → D be a tensor functor.

Question 1Can one encode F by algebraic data in D (or IndD)?

Yes. But this data cannot be a Hopf algebra, as D is not braided. It is aHopf (co)monad.

Question 2Can one encode F by an algebraic data in C (or IndC)?

Yes, if F is dominant.This data is a commutative algebra

in the center of C (or IndC).

Page 34: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Introduction

Let F : C → D be a tensor functor.

Question 1Can one encode F by algebraic data in D (or IndD)?

Yes. But this data cannot be a Hopf algebra, as D is not braided. It is aHopf (co)monad.

Question 2Can one encode F by an algebraic data in C (or IndC)?

Yes, if F is dominant.This data is a commutative algebra in the center of C (or IndC).

Page 35: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Introduction

Outline of the talk 7 / 35

1 Introduction

2 Hopf Monads - a sketchy survey

3 Hopf (co)-monads applied to tensor functors

4 Exact sequences of tensor categories

Page 36: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Introduction

Outline of the talk 7 / 35

1 Introduction

2 Hopf Monads - a sketchy survey

3 Hopf (co)-monads applied to tensor functors

4 Exact sequences of tensor categories

Page 37: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Introduction

Outline of the talk 7 / 35

1 Introduction

2 Hopf Monads - a sketchy survey

3 Hopf (co)-monads applied to tensor functors

4 Exact sequences of tensor categories

Page 38: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Introduction

Outline of the talk 7 / 35

1 Introduction

2 Hopf Monads - a sketchy survey

3 Hopf (co)-monads applied to tensor functors

4 Exact sequences of tensor categories

Page 39: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey

1 Introduction

2 Hopf Monads - a sketchy surveyDefinitionExamplesSome aspects of the general theory

3 Hopf (co)-monads applied to tensor functors

4 Exact sequences of tensor categories

Page 40: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Definition

Monads 9 / 35

Let C be a category. The category EndoFun(C) is strict monoidal(⊗=composition, 1 = 1C)

A monad on C is an algebra (=monoid) in EndoFun(C) :

T : C → C, µ : T2 → T (product), η : 1C → T (unit)

A T -module is a pair (M, r), M ∈ Ob(C), r : T(M)→ M s. t.

rµM = rT(r) and rηM = idM .

CT category of T -modules.

ExampleA algebra in a monoidal category C T =? ⊗ A : X 7→ X ⊗ A is a monad on C and CT = Mod- A

T ′ = A⊗? is a monad on C and CT ′ = A - Mod

Page 41: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Definition

Monads 9 / 35

Let C be a category. The category EndoFun(C) is strict monoidal(⊗=composition, 1 = 1C)

A monad on C is an algebra (=monoid) in EndoFun(C) :

T : C → C, µ : T2 → T (product), η : 1C → T (unit)

A T -module is a pair (M, r), M ∈ Ob(C), r : T(M)→ M s. t.

rµM = rT(r) and rηM = idM .

CT category of T -modules.

ExampleA algebra in a monoidal category C T =? ⊗ A : X 7→ X ⊗ A is a monad on C and CT = Mod- A

T ′ = A⊗? is a monad on C and CT ′ = A - Mod

Page 42: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Definition

Monads 9 / 35

Let C be a category. The category EndoFun(C) is strict monoidal(⊗=composition, 1 = 1C)

A monad on C is an algebra (=monoid) in EndoFun(C) :

T : C → C, µ : T2 → T (product), η : 1C → T (unit)

A T -module is a pair (M, r), M ∈ Ob(C), r : T(M)→ M s. t.

rµM = rT(r) and rηM = idM .

CT category of T -modules.

ExampleA algebra in a monoidal category C T =? ⊗ A : X 7→ X ⊗ A is a monad on C and CT = Mod- A

T ′ = A⊗? is a monad on C and CT ′ = A - Mod

Page 43: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Definition

Monads 9 / 35

Let C be a category. The category EndoFun(C) is strict monoidal(⊗=composition, 1 = 1C)

A monad on C is an algebra (=monoid) in EndoFun(C) :

T : C → C, µ : T2 → T (product), η : 1C → T (unit)

A T -module is a pair (M, r), M ∈ Ob(C), r : T(M)→ M s. t.

rµM = rT(r) and rηM = idM .

CT category of T -modules.

ExampleA algebra in a monoidal category C T =? ⊗ A : X 7→ X ⊗ A is a monad on C and CT = Mod- A

T ′ = A⊗? is a monad on C and CT ′ = A - Mod

Page 44: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Definition

Monads 9 / 35

Let C be a category. The category EndoFun(C) is strict monoidal(⊗=composition, 1 = 1C)

A monad on C is an algebra (=monoid) in EndoFun(C) :

T : C → C, µ : T2 → T (product), η : 1C → T (unit)

A T -module is a pair (M, r), M ∈ Ob(C), r : T(M)→ M s. t.

rµM = rT(r) and rηM = idM .

CT category of T -modules.

ExampleA algebra in a monoidal category C T =? ⊗ A : X 7→ X ⊗ A is a monad on C and CT = Mod- A

T ′ = A⊗? is a monad on C and CT ′ = A - Mod

Page 45: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Definition

Monads 9 / 35

Let C be a category. The category EndoFun(C) is strict monoidal(⊗=composition, 1 = 1C)

A monad on C is an algebra (=monoid) in EndoFun(C) :

T : C → C, µ : T2 → T (product), η : 1C → T (unit)

A T -module is a pair (M, r), M ∈ Ob(C), r : T(M)→ M s. t.

rµM = rT(r) and rηM = idM .

CT category of T -modules.

ExampleA algebra in a monoidal category C T =? ⊗ A : X 7→ X ⊗ A is a monad on C and CT = Mod- A

T ′ = A⊗? is a monad on C and CT ′ = A - Mod

Page 46: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Definition

Monads and adjunctions 10 / 35

A monad T on a category C an adjunction

CT

UT

��C

FT

GG

where UT (M, r) = M and FT (X) = (T(X), µX ).

An adjunctionD

U��C

F

GG

a monad T = (UF , µ := U(εF ), η) on C

where η : 1C → UF and ε : FU → 1D are the adjunction morphisms

Page 47: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Definition

Monads and adjunctions 10 / 35

A monad T on a category C an adjunction

CT

UT

��C

FT

GG

where UT (M, r) = M and FT (X) = (T(X), µX ).

An adjunctionD

U��C

F

GG

a monad T = (UF , µ := U(εF ), η) on C

where η : 1C → UF and ε : FU → 1D are the adjunction morphisms

Page 48: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Definition

Monads and adjunctions 10 / 35

A monad T on a category C an adjunction

CT

UT

��C

FT

GG

where UT (M, r) = M and FT (X) = (T(X), µX ).

An adjunctionD

U��C

F

GG

a monad T = (UF , µ := U(εF ), η) on C

where η : 1C → UF and ε : FU → 1D are the adjunction morphisms

D

U

��

CT

UTvvCF

VVFT

77

Page 49: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Definition

Monads and adjunctions 10 / 35

A monad T on a category C an adjunction

CT

UT

��C

FT

GG

where UT (M, r) = M and FT (X) = (T(X), µX ).

An adjunctionD

U��C

F

GG

a monad T = (UF , µ := U(εF ), η) on C

where η : 1C → UF and ε : FU → 1D are the adjunction morphisms

D

U

��

K++CT

UTvvCF

VVFT

77

Page 50: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Definition

Monads and adjunctions 10 / 35

A monad T on a category C an adjunction

CT

UT

��C

FT

GG

where UT (M, r) = M and FT (X) = (T(X), µX ).

An adjunctionD

U��C

F

GG

a monad T = (UF , µ := U(εF ), η) on C

where η : 1C → UF and ε : FU → 1D are the adjunction morphisms

D

U

��

K++CT

UTvvCF

VVFT

77K : D 7→ (U(D),U(εD))(the comparison functor)

The adjunction (F ,U) ismonadic if K equivalence.

Page 51: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Definition

Monads and adjunctions 10 / 35

A monad T on a category C an adjunction

CT

UT

��C

FT

GG

where UT (M, r) = M and FT (X) = (T(X), µX ).

An adjunctionD

U��C

F

GG

a monad T = (UF , µ := U(εF ), η) on C

where η : 1C → UF and ε : FU → 1D are the adjunction morphisms

D

U

��

K++CT

UTvvCF

VVFT

77K : D 7→ (U(D),U(εD))(the comparison functor)

The adjunction (F ,U) ismonadic if K equivalence.

Page 52: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Definition

Bimonads [Moerdijk] 11 / 35

C monoidal category, (T , µ, η) monad on C

CT , UT : CT → C

T is a bimonad if and only if CT is monoidal and UT is strict monoidal. Thisis equivalent to:

T is comonoidal endofunctor(with ∆X ,Y : T(X ⊗ Y)→ TX ⊗ TY and ε : T1→ 1)

µ and η are comonoidal natural transformations.

Axioms similar to those of a bialgebra except the compatibility between µand ∆:

T2(X ⊗ Y)

µX⊗Y

��

T∆X ,Y// T(TX ⊗ TY)∆TX ,TY // T2X ⊗ T2Y

µX⊗µY

��T(X ⊗ Y)

∆X ,Y

// TX ⊗ TY

No braiding involved!

Page 53: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Definition

Bimonads [Moerdijk] 11 / 35

C monoidal category, (T , µ, η) monad on C CT , UT : CT → C

T is a bimonad if and only if CT is monoidal and UT is strict monoidal. Thisis equivalent to:

T is comonoidal endofunctor(with ∆X ,Y : T(X ⊗ Y)→ TX ⊗ TY and ε : T1→ 1)

µ and η are comonoidal natural transformations.

Axioms similar to those of a bialgebra except the compatibility between µand ∆:

T2(X ⊗ Y)

µX⊗Y

��

T∆X ,Y// T(TX ⊗ TY)∆TX ,TY // T2X ⊗ T2Y

µX⊗µY

��T(X ⊗ Y)

∆X ,Y

// TX ⊗ TY

No braiding involved!

Page 54: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Definition

Bimonads [Moerdijk] 11 / 35

C monoidal category, (T , µ, η) monad on C CT , UT : CT → C

T is a bimonad if and only if CT is monoidal and UT is strict monoidal.

Thisis equivalent to:

T is comonoidal endofunctor(with ∆X ,Y : T(X ⊗ Y)→ TX ⊗ TY and ε : T1→ 1)

µ and η are comonoidal natural transformations.

Axioms similar to those of a bialgebra except the compatibility between µand ∆:

T2(X ⊗ Y)

µX⊗Y

��

T∆X ,Y// T(TX ⊗ TY)∆TX ,TY // T2X ⊗ T2Y

µX⊗µY

��T(X ⊗ Y)

∆X ,Y

// TX ⊗ TY

No braiding involved!

Page 55: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Definition

Bimonads [Moerdijk] 11 / 35

C monoidal category, (T , µ, η) monad on C CT , UT : CT → C

T is a bimonad if and only if CT is monoidal and UT is strict monoidal. Thisis equivalent to:

T is comonoidal endofunctor(with ∆X ,Y : T(X ⊗ Y)→ TX ⊗ TY and ε : T1→ 1)

µ and η are comonoidal natural transformations.

Axioms similar to those of a bialgebra except the compatibility between µand ∆:

T2(X ⊗ Y)

µX⊗Y

��

T∆X ,Y// T(TX ⊗ TY)∆TX ,TY // T2X ⊗ T2Y

µX⊗µY

��T(X ⊗ Y)

∆X ,Y

// TX ⊗ TY

No braiding involved!

Page 56: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Definition

Bimonads [Moerdijk] 11 / 35

C monoidal category, (T , µ, η) monad on C CT , UT : CT → C

T is a bimonad if and only if CT is monoidal and UT is strict monoidal. Thisis equivalent to:

T is comonoidal endofunctor(with ∆X ,Y : T(X ⊗ Y)→ TX ⊗ TY and ε : T1→ 1)

µ and η are comonoidal natural transformations.

Axioms similar to those of a bialgebra except the compatibility between µand ∆:

T2(X ⊗ Y)

µX⊗Y

��

T∆X ,Y// T(TX ⊗ TY)∆TX ,TY // T2X ⊗ T2Y

µX⊗µY

��T(X ⊗ Y)

∆X ,Y

// TX ⊗ TY

No braiding involved!

Page 57: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Definition

Bimonads [Moerdijk] 11 / 35

C monoidal category, (T , µ, η) monad on C CT , UT : CT → C

T is a bimonad if and only if CT is monoidal and UT is strict monoidal. Thisis equivalent to:

T is comonoidal endofunctor(with ∆X ,Y : T(X ⊗ Y)→ TX ⊗ TY and ε : T1→ 1)

µ and η are comonoidal natural transformations.

Axioms similar to those of a bialgebra except the compatibility between µand ∆:

T2(X ⊗ Y)

µX⊗Y

��

T∆X ,Y// T(TX ⊗ TY)∆TX ,TY // T2X ⊗ T2Y

µX⊗µY

��T(X ⊗ Y)

∆X ,Y

// TX ⊗ TY

No braiding involved!

Page 58: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Definition

Bimonads [Moerdijk] 11 / 35

C monoidal category, (T , µ, η) monad on C CT , UT : CT → C

T is a bimonad if and only if CT is monoidal and UT is strict monoidal. Thisis equivalent to:

T is comonoidal endofunctor(with ∆X ,Y : T(X ⊗ Y)→ TX ⊗ TY and ε : T1→ 1)

µ and η are comonoidal natural transformations.

Axioms similar to those of a bialgebra except the compatibility between µand ∆:

T2(X ⊗ Y)

µX⊗Y

��

T∆X ,Y// T(TX ⊗ TY)∆TX ,TY // T2X ⊗ T2Y

µX⊗µY

��T(X ⊗ Y)

∆X ,Y

// TX ⊗ TY

No braiding involved!

Page 59: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Definition

Hopf monads 12 / 35

For a bimonad T define the (left and right) fusion morphismsHl(X ,Y) = (idTX ⊗ µY )∆X ,TY : T(X ⊗ TY)→ TX ⊗ TY ,Hr(X ,Y) = (µX ⊗ idTY )∆TX ,Y : T(TX ⊗ Y)→ TX ⊗ TY .

A bimonad T is a Hopf monad if the fusion morphisms are isomorphisms.

PropositionFor T bimonad on C rigid, equivalence:

(i) CT is rigid;

(ii) T is a Hopf monad;

(iii) (older definition) T admits a left and a right (unary) antipodes l

X : T(∨TX)→ ∨X and sr : T(TX∨)→ X∨.

There is a similar result for closed categories (monoidal categories withinternal Homs).

Page 60: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Definition

Hopf monads 12 / 35

For a bimonad T define the (left and right) fusion morphismsHl(X ,Y) = (idTX ⊗ µY )∆X ,TY : T(X ⊗ TY)→ TX ⊗ TY ,Hr(X ,Y) = (µX ⊗ idTY )∆TX ,Y : T(TX ⊗ Y)→ TX ⊗ TY .

A bimonad T is a Hopf monad if the fusion morphisms are isomorphisms.

PropositionFor T bimonad on C rigid, equivalence:

(i) CT is rigid;

(ii) T is a Hopf monad;

(iii) (older definition) T admits a left and a right (unary) antipodes l

X : T(∨TX)→ ∨X and sr : T(TX∨)→ X∨.

There is a similar result for closed categories (monoidal categories withinternal Homs).

Page 61: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Definition

Hopf monads 12 / 35

For a bimonad T define the (left and right) fusion morphismsHl(X ,Y) = (idTX ⊗ µY )∆X ,TY : T(X ⊗ TY)→ TX ⊗ TY ,Hr(X ,Y) = (µX ⊗ idTY )∆TX ,Y : T(TX ⊗ Y)→ TX ⊗ TY .

A bimonad T is a Hopf monad if the fusion morphisms are isomorphisms.

PropositionFor T bimonad on C rigid, equivalence:

(i) CT is rigid;

(ii) T is a Hopf monad;

(iii) (older definition) T admits a left and a right (unary) antipodes l

X : T(∨TX)→ ∨X and sr : T(TX∨)→ X∨.

There is a similar result for closed categories (monoidal categories withinternal Homs).

Page 62: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Definition

Hopf monads 12 / 35

For a bimonad T define the (left and right) fusion morphismsHl(X ,Y) = (idTX ⊗ µY )∆X ,TY : T(X ⊗ TY)→ TX ⊗ TY ,Hr(X ,Y) = (µX ⊗ idTY )∆TX ,Y : T(TX ⊗ Y)→ TX ⊗ TY .

A bimonad T is a Hopf monad if the fusion morphisms are isomorphisms.

PropositionFor T bimonad on C rigid, equivalence:

(i) CT is rigid;

(ii) T is a Hopf monad;

(iii) (older definition) T admits a left and a right (unary) antipodes l

X : T(∨TX)→ ∨X and sr : T(TX∨)→ X∨.

There is a similar result for closed categories (monoidal categories withinternal Homs).

Page 63: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Definition

Tannaka dictionary 13 / 35

There is a Tannaka dictionary relating properties of a monad T on amonoidal category C and properties of its category of modules CT .

T CT Structural morphism

bimonad monoidal ∆X ,Y : T(X ⊗ Y)→ T(X) ⊗ T(Y)

Hopf monad(C rigid)

rigids l

X : T(∨T(X))→ ∨Xsr

X : T(T(X)∨)→ X∨

quasitriangular braided RX ,Y : X ⊗ Y → T(Y) ⊗ T(X)

ribbon ribbon θX : X → T(X)

(M, r) ⊗ (N, s) = (M ⊗ N, (r ⊗ s)∆M,N) ∨(M, r) = (∨M, s lMT(∨r))

τ(M,r),(N,s) = (s ⊗ r)RM,N Θ(M,r) = rθM

Page 64: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Definition

Tannaka dictionary 13 / 35

There is a Tannaka dictionary relating properties of a monad T on amonoidal category C and properties of its category of modules CT .

T CT Structural morphism

bimonad monoidal ∆X ,Y : T(X ⊗ Y)→ T(X) ⊗ T(Y)

Hopf monad(C rigid)

rigids l

X : T(∨T(X))→ ∨Xsr

X : T(T(X)∨)→ X∨

quasitriangular braided RX ,Y : X ⊗ Y → T(Y) ⊗ T(X)

ribbon ribbon θX : X → T(X)

(M, r) ⊗ (N, s) = (M ⊗ N, (r ⊗ s)∆M,N)

∨(M, r) = (∨M, s lMT(∨r))

τ(M,r),(N,s) = (s ⊗ r)RM,N Θ(M,r) = rθM

Page 65: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Definition

Tannaka dictionary 13 / 35

There is a Tannaka dictionary relating properties of a monad T on amonoidal category C and properties of its category of modules CT .

T CT Structural morphism

bimonad monoidal ∆X ,Y : T(X ⊗ Y)→ T(X) ⊗ T(Y)

Hopf monad(C rigid)

rigids l

X : T(∨T(X))→ ∨Xsr

X : T(T(X)∨)→ X∨

quasitriangular braided RX ,Y : X ⊗ Y → T(Y) ⊗ T(X)

ribbon ribbon θX : X → T(X)

(M, r) ⊗ (N, s) = (M ⊗ N, (r ⊗ s)∆M,N)

∨(M, r) = (∨M, s lMT(∨r))

τ(M,r),(N,s) = (s ⊗ r)RM,N Θ(M,r) = rθM

Page 66: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Definition

Tannaka dictionary 13 / 35

There is a Tannaka dictionary relating properties of a monad T on amonoidal category C and properties of its category of modules CT .

T CT Structural morphism

bimonad monoidal ∆X ,Y : T(X ⊗ Y)→ T(X) ⊗ T(Y)

Hopf monad(C rigid)

rigid

s lX : T(∨T(X))→ ∨X

srX : T(T(X)∨)→ X∨

quasitriangular braided RX ,Y : X ⊗ Y → T(Y) ⊗ T(X)

ribbon ribbon θX : X → T(X)

(M, r) ⊗ (N, s) = (M ⊗ N, (r ⊗ s)∆M,N)

∨(M, r) = (∨M, s lMT(∨r))

τ(M,r),(N,s) = (s ⊗ r)RM,N Θ(M,r) = rθM

Page 67: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Definition

Tannaka dictionary 13 / 35

There is a Tannaka dictionary relating properties of a monad T on amonoidal category C and properties of its category of modules CT .

T CT Structural morphism

bimonad monoidal ∆X ,Y : T(X ⊗ Y)→ T(X) ⊗ T(Y)

Hopf monad(C rigid)

rigids l

X : T(∨T(X))→ ∨Xsr

X : T(T(X)∨)→ X∨

quasitriangular braided RX ,Y : X ⊗ Y → T(Y) ⊗ T(X)

ribbon ribbon θX : X → T(X)

(M, r) ⊗ (N, s) = (M ⊗ N, (r ⊗ s)∆M,N)

∨(M, r) = (∨M, s lMT(∨r))

τ(M,r),(N,s) = (s ⊗ r)RM,N Θ(M,r) = rθM

Page 68: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Definition

Tannaka dictionary 13 / 35

There is a Tannaka dictionary relating properties of a monad T on amonoidal category C and properties of its category of modules CT .

T CT Structural morphism

bimonad monoidal ∆X ,Y : T(X ⊗ Y)→ T(X) ⊗ T(Y)

Hopf monad(C rigid)

rigids l

X : T(∨T(X))→ ∨Xsr

X : T(T(X)∨)→ X∨

quasitriangular braided RX ,Y : X ⊗ Y → T(Y) ⊗ T(X)

ribbon ribbon θX : X → T(X)

(M, r) ⊗ (N, s) = (M ⊗ N, (r ⊗ s)∆M,N) ∨(M, r) = (∨M, s lMT(∨r))

τ(M,r),(N,s) = (s ⊗ r)RM,N Θ(M,r) = rθM

Page 69: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Definition

Tannaka dictionary 13 / 35

There is a Tannaka dictionary relating properties of a monad T on amonoidal category C and properties of its category of modules CT .

T CT Structural morphism

bimonad monoidal ∆X ,Y : T(X ⊗ Y)→ T(X) ⊗ T(Y)

Hopf monad(C rigid)

rigids l

X : T(∨T(X))→ ∨Xsr

X : T(T(X)∨)→ X∨

quasitriangular

braided RX ,Y : X ⊗ Y → T(Y) ⊗ T(X)

ribbon ribbon θX : X → T(X)

(M, r) ⊗ (N, s) = (M ⊗ N, (r ⊗ s)∆M,N) ∨(M, r) = (∨M, s lMT(∨r))

τ(M,r),(N,s) = (s ⊗ r)RM,N Θ(M,r) = rθM

Page 70: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Definition

Tannaka dictionary 13 / 35

There is a Tannaka dictionary relating properties of a monad T on amonoidal category C and properties of its category of modules CT .

T CT Structural morphism

bimonad monoidal ∆X ,Y : T(X ⊗ Y)→ T(X) ⊗ T(Y)

Hopf monad(C rigid)

rigids l

X : T(∨T(X))→ ∨Xsr

X : T(T(X)∨)→ X∨

quasitriangular braided

RX ,Y : X ⊗ Y → T(Y) ⊗ T(X)

ribbon ribbon θX : X → T(X)

(M, r) ⊗ (N, s) = (M ⊗ N, (r ⊗ s)∆M,N) ∨(M, r) = (∨M, s lMT(∨r))

τ(M,r),(N,s) = (s ⊗ r)RM,N Θ(M,r) = rθM

Page 71: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Definition

Tannaka dictionary 13 / 35

There is a Tannaka dictionary relating properties of a monad T on amonoidal category C and properties of its category of modules CT .

T CT Structural morphism

bimonad monoidal ∆X ,Y : T(X ⊗ Y)→ T(X) ⊗ T(Y)

Hopf monad(C rigid)

rigids l

X : T(∨T(X))→ ∨Xsr

X : T(T(X)∨)→ X∨

quasitriangular braided RX ,Y : X ⊗ Y → T(Y) ⊗ T(X)

ribbon ribbon θX : X → T(X)

(M, r) ⊗ (N, s) = (M ⊗ N, (r ⊗ s)∆M,N) ∨(M, r) = (∨M, s lMT(∨r))

τ(M,r),(N,s) = (s ⊗ r)RM,N Θ(M,r) = rθM

Page 72: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Definition

Tannaka dictionary 13 / 35

There is a Tannaka dictionary relating properties of a monad T on amonoidal category C and properties of its category of modules CT .

T CT Structural morphism

bimonad monoidal ∆X ,Y : T(X ⊗ Y)→ T(X) ⊗ T(Y)

Hopf monad(C rigid)

rigids l

X : T(∨T(X))→ ∨Xsr

X : T(T(X)∨)→ X∨

quasitriangular braided RX ,Y : X ⊗ Y → T(Y) ⊗ T(X)

ribbon ribbon θX : X → T(X)

(M, r) ⊗ (N, s) = (M ⊗ N, (r ⊗ s)∆M,N) ∨(M, r) = (∨M, s lMT(∨r))

τ(M,r),(N,s) = (s ⊗ r)RM,N

Θ(M,r) = rθM

Page 73: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Definition

Tannaka dictionary 13 / 35

There is a Tannaka dictionary relating properties of a monad T on amonoidal category C and properties of its category of modules CT .

T CT Structural morphism

bimonad monoidal ∆X ,Y : T(X ⊗ Y)→ T(X) ⊗ T(Y)

Hopf monad(C rigid)

rigids l

X : T(∨T(X))→ ∨Xsr

X : T(T(X)∨)→ X∨

quasitriangular braided RX ,Y : X ⊗ Y → T(Y) ⊗ T(X)

ribbon

ribbon θX : X → T(X)

(M, r) ⊗ (N, s) = (M ⊗ N, (r ⊗ s)∆M,N) ∨(M, r) = (∨M, s lMT(∨r))

τ(M,r),(N,s) = (s ⊗ r)RM,N

Θ(M,r) = rθM

Page 74: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Definition

Tannaka dictionary 13 / 35

There is a Tannaka dictionary relating properties of a monad T on amonoidal category C and properties of its category of modules CT .

T CT Structural morphism

bimonad monoidal ∆X ,Y : T(X ⊗ Y)→ T(X) ⊗ T(Y)

Hopf monad(C rigid)

rigids l

X : T(∨T(X))→ ∨Xsr

X : T(T(X)∨)→ X∨

quasitriangular braided RX ,Y : X ⊗ Y → T(Y) ⊗ T(X)

ribbon ribbon

θX : X → T(X)

(M, r) ⊗ (N, s) = (M ⊗ N, (r ⊗ s)∆M,N) ∨(M, r) = (∨M, s lMT(∨r))

τ(M,r),(N,s) = (s ⊗ r)RM,N

Θ(M,r) = rθM

Page 75: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Definition

Tannaka dictionary 13 / 35

There is a Tannaka dictionary relating properties of a monad T on amonoidal category C and properties of its category of modules CT .

T CT Structural morphism

bimonad monoidal ∆X ,Y : T(X ⊗ Y)→ T(X) ⊗ T(Y)

Hopf monad(C rigid)

rigids l

X : T(∨T(X))→ ∨Xsr

X : T(T(X)∨)→ X∨

quasitriangular braided RX ,Y : X ⊗ Y → T(Y) ⊗ T(X)

ribbon ribbon θX : X → T(X)

(M, r) ⊗ (N, s) = (M ⊗ N, (r ⊗ s)∆M,N) ∨(M, r) = (∨M, s lMT(∨r))

τ(M,r),(N,s) = (s ⊗ r)RM,N

Θ(M,r) = rθM

Page 76: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Definition

Tannaka dictionary 13 / 35

There is a Tannaka dictionary relating properties of a monad T on amonoidal category C and properties of its category of modules CT .

T CT Structural morphism

bimonad monoidal ∆X ,Y : T(X ⊗ Y)→ T(X) ⊗ T(Y)

Hopf monad(C rigid)

rigids l

X : T(∨T(X))→ ∨Xsr

X : T(T(X)∨)→ X∨

quasitriangular braided RX ,Y : X ⊗ Y → T(Y) ⊗ T(X)

ribbon ribbon θX : X → T(X)

(M, r) ⊗ (N, s) = (M ⊗ N, (r ⊗ s)∆M,N) ∨(M, r) = (∨M, s lMT(∨r))

τ(M,r),(N,s) = (s ⊗ r)RM,N Θ(M,r) = rθM

Page 77: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Definition

Hopf comonads 14 / 35

The notion of a Hopf monad is not self-dual, unlike that of a Hopf algebra:if you reverse the arrows in the definition, you obtain the notion of a Hopfcomonad. A Hopf comonad is a monoidal comonad such that the cofusionoperators are invertible.

All results about Hopf monads translate into results about Hopf comonads.In particular, if T is a Hopf comonad on C,

1 the category CT of comodules over T is monoidal,

2 we have a Hopf monoidal adjunction: DUT

77CFTvv

where UT is the forgetful functor and FT is its right adjoint, the cofreecomodule functor.

Page 78: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Definition

Hopf comonads 14 / 35

The notion of a Hopf monad is not self-dual, unlike that of a Hopf algebra:if you reverse the arrows in the definition, you obtain the notion of a Hopfcomonad. A Hopf comonad is a monoidal comonad such that the cofusionoperators are invertible.All results about Hopf monads translate into results about Hopf comonads.

In particular, if T is a Hopf comonad on C,1 the category CT of comodules over T is monoidal,

2 we have a Hopf monoidal adjunction: DUT

77CFTvv

where UT is the forgetful functor and FT is its right adjoint, the cofreecomodule functor.

Page 79: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Definition

Hopf comonads 14 / 35

The notion of a Hopf monad is not self-dual, unlike that of a Hopf algebra:if you reverse the arrows in the definition, you obtain the notion of a Hopfcomonad. A Hopf comonad is a monoidal comonad such that the cofusionoperators are invertible.All results about Hopf monads translate into results about Hopf comonads.In particular, if T is a Hopf comonad on C,

1 the category CT of comodules over T is monoidal,

2 we have a Hopf monoidal adjunction: DUT

77CFTvv

where UT is the forgetful functor and FT is its right adjoint, the cofreecomodule functor.

Page 80: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Definition

Hopf comonads 14 / 35

The notion of a Hopf monad is not self-dual, unlike that of a Hopf algebra:if you reverse the arrows in the definition, you obtain the notion of a Hopfcomonad. A Hopf comonad is a monoidal comonad such that the cofusionoperators are invertible.All results about Hopf monads translate into results about Hopf comonads.In particular, if T is a Hopf comonad on C,

1 the category CT of comodules over T is monoidal,

2 we have a Hopf monoidal adjunction: DUT

77CFTvv

where UT is the forgetful functor and FT is its right adjoint, the cofreecomodule functor.

Page 81: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Definition

Hopf comonads 14 / 35

The notion of a Hopf monad is not self-dual, unlike that of a Hopf algebra:if you reverse the arrows in the definition, you obtain the notion of a Hopfcomonad. A Hopf comonad is a monoidal comonad such that the cofusionoperators are invertible.All results about Hopf monads translate into results about Hopf comonads.In particular, if T is a Hopf comonad on C,

1 the category CT of comodules over T is monoidal,

2 we have a Hopf monoidal adjunction: DUT

77CFTvv

where UT is the forgetful functor and FT is its right adjoint, the cofreecomodule functor.

Page 82: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Examples

Hopf monads from adjunctions 15 / 35

Let DU''C

Fhh be a comonoidal adjunction (meaning C, D are monoidal

and U is strong monoidal)

Then F is comonoidal and T = UF is a bimonad on C.There are canonical morphisms:

F(c ⊗ Ud)→ Fc ⊗ dF(Ud ⊗ c)→ d ⊗ Fc

and (F ,U) is a Hopf adjunction if these morphisms are isos.

Proposition

If the adjunction is Hopf, T is a Hopf monad. Such is the case if either ofthe following hold:

C, D are rigid;

C, D and U are closed.

A bimonad is Hopf iff its adjunction is Hopf!

Page 83: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Examples

Hopf monads from adjunctions 15 / 35

Let DU''C

Fhh be a comonoidal adjunction (meaning C, D are monoidal

and U is strong monoidal)Then F is comonoidal and T = UF is a bimonad on C.

There are canonical morphisms:F(c ⊗ Ud)→ Fc ⊗ dF(Ud ⊗ c)→ d ⊗ Fc

and (F ,U) is a Hopf adjunction if these morphisms are isos.

Proposition

If the adjunction is Hopf, T is a Hopf monad. Such is the case if either ofthe following hold:

C, D are rigid;

C, D and U are closed.

A bimonad is Hopf iff its adjunction is Hopf!

Page 84: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Examples

Hopf monads from adjunctions 15 / 35

Let DU''C

Fhh be a comonoidal adjunction (meaning C, D are monoidal

and U is strong monoidal)Then F is comonoidal and T = UF is a bimonad on C.

There are canonical morphisms:F(c ⊗ Ud)→ Fc ⊗ dF(Ud ⊗ c)→ d ⊗ Fc

and (F ,U) is a Hopf adjunction if these morphisms are isos.

Proposition

If the adjunction is Hopf, T is a Hopf monad. Such is the case if either ofthe following hold:

C, D are rigid;

C, D and U are closed.

A bimonad is Hopf iff its adjunction is Hopf!

Page 85: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Examples

Hopf monads from adjunctions 15 / 35

Let DU''C

Fhh be a comonoidal adjunction (meaning C, D are monoidal

and U is strong monoidal)Then F is comonoidal and T = UF is a bimonad on C.There are canonical morphisms:

F(c ⊗ Ud)→ Fc ⊗ dF(Ud ⊗ c)→ d ⊗ Fc

and (F ,U) is a Hopf adjunction if these morphisms are isos.

Proposition

If the adjunction is Hopf, T is a Hopf monad. Such is the case if either ofthe following hold:

C, D are rigid;

C, D and U are closed.

A bimonad is Hopf iff its adjunction is Hopf!

Page 86: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Examples

Hopf monads from adjunctions 15 / 35

Let DU''C

Fhh be a comonoidal adjunction (meaning C, D are monoidal

and U is strong monoidal)Then F is comonoidal and T = UF is a bimonad on C.There are canonical morphisms:

F(c ⊗ Ud)→ Fc ⊗ dF(Ud ⊗ c)→ d ⊗ Fc

and (F ,U) is a Hopf adjunction if these morphisms are isos.

Proposition

If the adjunction is Hopf, T is a Hopf monad. Such is the case if either ofthe following hold:

C, D are rigid;

C, D and U are closed.

A bimonad is Hopf iff its adjunction is Hopf!

Page 87: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Examples

Hopf monads from adjunctions 15 / 35

Let DU''C

Fhh be a comonoidal adjunction (meaning C, D are monoidal

and U is strong monoidal)Then F is comonoidal and T = UF is a bimonad on C.There are canonical morphisms:

F(c ⊗ Ud)→ Fc ⊗ dF(Ud ⊗ c)→ d ⊗ Fc

and (F ,U) is a Hopf adjunction if these morphisms are isos.

Proposition

If the adjunction is Hopf, T is a Hopf monad. Such is the case if either ofthe following hold:

C, D are rigid;

C, D and U are closed.

A bimonad is Hopf iff its adjunction is Hopf!

Page 88: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Examples

Hopf monads from Hopf algebras 16 / 35

Hopf monads generalize Hopf algebras in braided categories

H Hopf algebra in B braided category with braiding τ T = H⊗? is a Hopf monad on BThe monad structure of T comes from the algebra structure of HThe comonoidal structure of T is

∆X ,Y = (H ⊗ τH,X ⊗ Y)(∆ ⊗ X ⊗ Y) : H ⊗ X ⊗ Y → H ⊗ X ⊗ H ⊗ Y

ε = counit of H : H → 1

We have BT =H Mod as monoidal categories.

Can we extend this construction to non-braided categories?

Page 89: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Examples

Hopf monads from Hopf algebras 16 / 35

Hopf monads generalize Hopf algebras in braided categories

H Hopf algebra in B braided category with braiding τ

T = H⊗? is a Hopf monad on BThe monad structure of T comes from the algebra structure of HThe comonoidal structure of T is

∆X ,Y = (H ⊗ τH,X ⊗ Y)(∆ ⊗ X ⊗ Y) : H ⊗ X ⊗ Y → H ⊗ X ⊗ H ⊗ Y

ε = counit of H : H → 1

We have BT =H Mod as monoidal categories.

Can we extend this construction to non-braided categories?

Page 90: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Examples

Hopf monads from Hopf algebras 16 / 35

Hopf monads generalize Hopf algebras in braided categories

H Hopf algebra in B braided category with braiding τ T = H⊗? is a Hopf monad on B

The monad structure of T comes from the algebra structure of HThe comonoidal structure of T is

∆X ,Y = (H ⊗ τH,X ⊗ Y)(∆ ⊗ X ⊗ Y) : H ⊗ X ⊗ Y → H ⊗ X ⊗ H ⊗ Y

ε = counit of H : H → 1

We have BT =H Mod as monoidal categories.

Can we extend this construction to non-braided categories?

Page 91: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Examples

Hopf monads from Hopf algebras 16 / 35

Hopf monads generalize Hopf algebras in braided categories

H Hopf algebra in B braided category with braiding τ T = H⊗? is a Hopf monad on BThe monad structure of T comes from the algebra structure of H

The comonoidal structure of T is

∆X ,Y = (H ⊗ τH,X ⊗ Y)(∆ ⊗ X ⊗ Y) : H ⊗ X ⊗ Y → H ⊗ X ⊗ H ⊗ Y

ε = counit of H : H → 1

We have BT =H Mod as monoidal categories.

Can we extend this construction to non-braided categories?

Page 92: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Examples

Hopf monads from Hopf algebras 16 / 35

Hopf monads generalize Hopf algebras in braided categories

H Hopf algebra in B braided category with braiding τ T = H⊗? is a Hopf monad on BThe monad structure of T comes from the algebra structure of HThe comonoidal structure of T is

∆X ,Y = (H ⊗ τH,X ⊗ Y)(∆ ⊗ X ⊗ Y) : H ⊗ X ⊗ Y → H ⊗ X ⊗ H ⊗ Y

ε = counit of H : H → 1

We have BT =H Mod as monoidal categories.

Can we extend this construction to non-braided categories?

Page 93: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Examples

Hopf monads from Hopf algebras 16 / 35

Hopf monads generalize Hopf algebras in braided categories

H Hopf algebra in B braided category with braiding τ T = H⊗? is a Hopf monad on BThe monad structure of T comes from the algebra structure of HThe comonoidal structure of T is

∆X ,Y = (H ⊗ τH,X ⊗ Y)(∆ ⊗ X ⊗ Y) : H ⊗ X ⊗ Y → H ⊗ X ⊗ H ⊗ Y

ε = counit of H : H → 1

We have BT =H Mod as monoidal categories.

Can we extend this construction to non-braided categories?

Page 94: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Examples

Hopf monads from Hopf algebras 16 / 35

Hopf monads generalize Hopf algebras in braided categories

H Hopf algebra in B braided category with braiding τ T = H⊗? is a Hopf monad on BThe monad structure of T comes from the algebra structure of HThe comonoidal structure of T is

∆X ,Y = (H ⊗ τH,X ⊗ Y)(∆ ⊗ X ⊗ Y) : H ⊗ X ⊗ Y → H ⊗ X ⊗ H ⊗ Y

ε = counit of H : H → 1

We have BT =H Mod as monoidal categories.

Can we extend this construction to non-braided categories?

Page 95: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Examples

The Joyal-Street Center 17 / 35

C monoidal categoryJoyal-Street

Center//Z(C) braided category

Objects of Z(C) = half-braidings of C :

pair (X , σ) with σY : X ⊗ Y∼→ Y ⊗ X natural in Y s. t.

σY⊗Z = (idY ⊗ σZ )(σY ⊗ idZ )

Morphisms f : (X , σ)→ (X ′, σ′) in Z(C) are morphisms f : X → X ′ inC s. t. σ′(f ⊗ id) = (id ⊗ f)σ

(X , σ) ⊗Z(C) (X ′, σ′) =(X ⊗ X ′, (σY ⊗ id)(id ⊗ σ′)

)Braiding: c(X ,σ),(X ′,σ′) = σX ′

Page 96: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Examples

The Joyal-Street Center 17 / 35

C monoidal categoryJoyal-Street

Center//Z(C) braided category

Objects of Z(C) = half-braidings of C :

pair (X , σ) with σY : X ⊗ Y∼→ Y ⊗ X natural in Y s. t.

σY⊗Z = (idY ⊗ σZ )(σY ⊗ idZ )

Morphisms f : (X , σ)→ (X ′, σ′) in Z(C) are morphisms f : X → X ′ inC s. t. σ′(f ⊗ id) = (id ⊗ f)σ

(X , σ) ⊗Z(C) (X ′, σ′) =(X ⊗ X ′, (σY ⊗ id)(id ⊗ σ′)

)Braiding: c(X ,σ),(X ′,σ′) = σX ′

Page 97: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Examples

The Joyal-Street Center 17 / 35

C monoidal categoryJoyal-Street

Center//Z(C) braided category

Objects of Z(C) = half-braidings of C :

pair (X , σ) with σY : X ⊗ Y∼→ Y ⊗ X natural in Y s. t.

σY⊗Z = (idY ⊗ σZ )(σY ⊗ idZ )

Morphisms f : (X , σ)→ (X ′, σ′) in Z(C) are morphisms f : X → X ′ inC s. t. σ′(f ⊗ id) = (id ⊗ f)σ

(X , σ) ⊗Z(C) (X ′, σ′) =(X ⊗ X ′, (σY ⊗ id)(id ⊗ σ′)

)Braiding: c(X ,σ),(X ′,σ′) = σX ′

Page 98: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Examples

The Joyal-Street Center 17 / 35

C monoidal categoryJoyal-Street

Center//Z(C) braided category

Objects of Z(C) = half-braidings of C :

pair (X , σ) with σY : X ⊗ Y∼→ Y ⊗ X natural in Y s. t.

σY⊗Z = (idY ⊗ σZ )(σY ⊗ idZ )

Morphisms f : (X , σ)→ (X ′, σ′) in Z(C) are morphisms f : X → X ′ inC s. t. σ′(f ⊗ id) = (id ⊗ f)σ

(X , σ) ⊗Z(C) (X ′, σ′) =(X ⊗ X ′, (σY ⊗ id)(id ⊗ σ′)

)Braiding: c(X ,σ),(X ′,σ′) = σX ′

Page 99: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Examples

The Joyal-Street Center 17 / 35

C monoidal categoryJoyal-Street

Center//Z(C) braided category

Objects of Z(C) = half-braidings of C :

pair (X , σ) with σY : X ⊗ Y∼→ Y ⊗ X natural in Y s. t.

σY⊗Z = (idY ⊗ σZ )(σY ⊗ idZ )

Morphisms f : (X , σ)→ (X ′, σ′) in Z(C) are morphisms f : X → X ′ inC s. t. σ′(f ⊗ id) = (id ⊗ f)σ

(X , σ) ⊗Z(C) (X ′, σ′) =(X ⊗ X ′, (σY ⊗ id)(id ⊗ σ′)

)

Braiding: c(X ,σ),(X ′,σ′) = σX ′

Page 100: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Examples

The Joyal-Street Center 17 / 35

C monoidal categoryJoyal-Street

Center//Z(C) braided category

Objects of Z(C) = half-braidings of C :

pair (X , σ) with σY : X ⊗ Y∼→ Y ⊗ X natural in Y s. t.

σY⊗Z = (idY ⊗ σZ )(σY ⊗ idZ )

Morphisms f : (X , σ)→ (X ′, σ′) in Z(C) are morphisms f : X → X ′ inC s. t. σ′(f ⊗ id) = (id ⊗ f)σ

(X , σ) ⊗Z(C) (X ′, σ′) =(X ⊗ X ′, (σY ⊗ id)(id ⊗ σ′)

)Braiding: c(X ,σ),(X ′,σ′) = σX ′

Page 101: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Examples

The Joyal-Street Center 17 / 35

C monoidal categoryJoyal-Street

Center//Z(C) braided category

Objects of Z(C) = half-braidings of C :

pair (X , σ) with σY : X ⊗ Y∼→ Y ⊗ X natural in Y s. t.

σY⊗Z = (idY ⊗ σZ )(σY ⊗ idZ )

Morphisms f : (X , σ)→ (X ′, σ′) in Z(C) are morphisms f : X → X ′ inC s. t. σ′(f ⊗ id) = (id ⊗ f)σ

(X , σ) ⊗Z(C) (X ′, σ′) =(X ⊗ X ′, (σY ⊗ id)(id ⊗ σ′)

)Braiding: c(X ,σ),(X ′,σ′) = σX ′

Page 102: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Examples

Representable Hopf monads 18 / 35

C monoidal category, (H, σ) a Hopf algebra in Z(C) (which is braided) a Hopf monad T = H⊗σ? on C, defined by X 7→ H ⊗ X . Thecomonoidal structure of T is

∆X ,Y = (H ⊗ σX ⊗ Y)(∆ ⊗ X ⊗ Y)

ε = counit of H

Moreover T is equipped with a Hopf monad morphism

e = (ε⊗?) : T → idC

Theorem (BVL)

This construction defines an equivalence of categories

{{Hopf algebras in Z(C)}}'−→ {{Hopf monads on C}} / idC

If H is a Hopf algebra and T = H⊗ we recover Sweedler’s Theorem.

Page 103: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Examples

Representable Hopf monads 18 / 35

C monoidal category, (H, σ) a Hopf algebra in Z(C) (which is braided) a Hopf monad T = H⊗σ? on C, defined by X 7→ H ⊗ X .

Thecomonoidal structure of T is

∆X ,Y = (H ⊗ σX ⊗ Y)(∆ ⊗ X ⊗ Y)

ε = counit of H

Moreover T is equipped with a Hopf monad morphism

e = (ε⊗?) : T → idC

Theorem (BVL)

This construction defines an equivalence of categories

{{Hopf algebras in Z(C)}}'−→ {{Hopf monads on C}} / idC

If H is a Hopf algebra and T = H⊗ we recover Sweedler’s Theorem.

Page 104: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Examples

Representable Hopf monads 18 / 35

C monoidal category, (H, σ) a Hopf algebra in Z(C) (which is braided) a Hopf monad T = H⊗σ? on C, defined by X 7→ H ⊗ X . Thecomonoidal structure of T is

∆X ,Y = (H ⊗ σX ⊗ Y)(∆ ⊗ X ⊗ Y)

ε = counit of H

Moreover T is equipped with a Hopf monad morphism

e = (ε⊗?) : T → idC

Theorem (BVL)

This construction defines an equivalence of categories

{{Hopf algebras in Z(C)}}'−→ {{Hopf monads on C}} / idC

If H is a Hopf algebra and T = H⊗ we recover Sweedler’s Theorem.

Page 105: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Examples

Representable Hopf monads 18 / 35

C monoidal category, (H, σ) a Hopf algebra in Z(C) (which is braided) a Hopf monad T = H⊗σ? on C, defined by X 7→ H ⊗ X . Thecomonoidal structure of T is

∆X ,Y = (H ⊗ σX ⊗ Y)(∆ ⊗ X ⊗ Y)

ε = counit of H

Moreover T is equipped with a Hopf monad morphism

e = (ε⊗?) : T → idC

Theorem (BVL)

This construction defines an equivalence of categories

{{Hopf algebras in Z(C)}}'−→ {{Hopf monads on C}} / idC

If H is a Hopf algebra and T = H⊗ we recover Sweedler’s Theorem.

Page 106: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Examples

Representable Hopf monads 18 / 35

C monoidal category, (H, σ) a Hopf algebra in Z(C) (which is braided) a Hopf monad T = H⊗σ? on C, defined by X 7→ H ⊗ X . Thecomonoidal structure of T is

∆X ,Y = (H ⊗ σX ⊗ Y)(∆ ⊗ X ⊗ Y)

ε = counit of H

Moreover T is equipped with a Hopf monad morphism

e = (ε⊗?) : T → idC

Theorem (BVL)

This construction defines an equivalence of categories

{{Hopf algebras in Z(C)}}'−→ {{Hopf monads on C}} / idC

If H is a Hopf algebra and T = H⊗ we recover Sweedler’s Theorem.

Page 107: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Examples

Monadicity of the center 19 / 35

Let C be a rigid category, with center Z(C).

Using duality, interpret a half-braiding σY : X ⊗ Y → Y ⊗ X as a dinaturaltransformation ∨Y ⊗ X ⊗ Y → XWe say that C is centralizable if Z(X) =

∫ Y∈C ∨Y ⊗ X ⊗ Y exists for allX ∈ C (note that Z(1) is the coend of C). Then a half braiding σcorresponds with σ : Z(X)→ X

Theorem (BV)

If C is centralizable, then Z : X 7→ Z(X) is a quasitriangular Hopf monadon C and we have a braided isomorphism of categories

Z(C)→ CZ

(X , σ) 7→ (X , σ)

Remark: In general the Hopf monad Z is not augmented, i e. notrepresentable by a Hopf algebra: e. g. C = {{G-graded vector spaces}},for G non abelian finite group.

Page 108: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Examples

Monadicity of the center 19 / 35

Let C be a rigid category, with center Z(C).Using duality, interpret a half-braiding σY : X ⊗ Y → Y ⊗ X as a dinaturaltransformation ∨Y ⊗ X ⊗ Y → X

We say that C is centralizable if Z(X) =∫ Y∈C ∨Y ⊗ X ⊗ Y exists for all

X ∈ C (note that Z(1) is the coend of C). Then a half braiding σcorresponds with σ : Z(X)→ X

Theorem (BV)

If C is centralizable, then Z : X 7→ Z(X) is a quasitriangular Hopf monadon C and we have a braided isomorphism of categories

Z(C)→ CZ

(X , σ) 7→ (X , σ)

Remark: In general the Hopf monad Z is not augmented, i e. notrepresentable by a Hopf algebra: e. g. C = {{G-graded vector spaces}},for G non abelian finite group.

Page 109: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Examples

Monadicity of the center 19 / 35

Let C be a rigid category, with center Z(C).Using duality, interpret a half-braiding σY : X ⊗ Y → Y ⊗ X as a dinaturaltransformation ∨Y ⊗ X ⊗ Y → XWe say that C is centralizable if Z(X) =

∫ Y∈C ∨Y ⊗ X ⊗ Y exists for allX ∈ C

(note that Z(1) is the coend of C). Then a half braiding σcorresponds with σ : Z(X)→ X

Theorem (BV)

If C is centralizable, then Z : X 7→ Z(X) is a quasitriangular Hopf monadon C and we have a braided isomorphism of categories

Z(C)→ CZ

(X , σ) 7→ (X , σ)

Remark: In general the Hopf monad Z is not augmented, i e. notrepresentable by a Hopf algebra: e. g. C = {{G-graded vector spaces}},for G non abelian finite group.

Page 110: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Examples

Monadicity of the center 19 / 35

Let C be a rigid category, with center Z(C).Using duality, interpret a half-braiding σY : X ⊗ Y → Y ⊗ X as a dinaturaltransformation ∨Y ⊗ X ⊗ Y → XWe say that C is centralizable if Z(X) =

∫ Y∈C ∨Y ⊗ X ⊗ Y exists for allX ∈ C (note that Z(1) is the coend of C). Then a half braiding σcorresponds with σ : Z(X)→ X

Theorem (BV)

If C is centralizable, then Z : X 7→ Z(X) is a quasitriangular Hopf monadon C and we have a braided isomorphism of categories

Z(C)→ CZ

(X , σ) 7→ (X , σ)

Remark: In general the Hopf monad Z is not augmented, i e. notrepresentable by a Hopf algebra: e. g. C = {{G-graded vector spaces}},for G non abelian finite group.

Page 111: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Examples

Monadicity of the center 19 / 35

Let C be a rigid category, with center Z(C).Using duality, interpret a half-braiding σY : X ⊗ Y → Y ⊗ X as a dinaturaltransformation ∨Y ⊗ X ⊗ Y → XWe say that C is centralizable if Z(X) =

∫ Y∈C ∨Y ⊗ X ⊗ Y exists for allX ∈ C (note that Z(1) is the coend of C). Then a half braiding σcorresponds with σ : Z(X)→ X

Theorem (BV)

If C is centralizable, then Z : X 7→ Z(X) is a quasitriangular Hopf monadon C and we have a braided isomorphism of categories

Z(C)→ CZ

(X , σ) 7→ (X , σ)

Remark: In general the Hopf monad Z is not augmented, i e. notrepresentable by a Hopf algebra: e. g. C = {{G-graded vector spaces}},for G non abelian finite group.

Page 112: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Examples

Monadicity of the center 19 / 35

Let C be a rigid category, with center Z(C).Using duality, interpret a half-braiding σY : X ⊗ Y → Y ⊗ X as a dinaturaltransformation ∨Y ⊗ X ⊗ Y → XWe say that C is centralizable if Z(X) =

∫ Y∈C ∨Y ⊗ X ⊗ Y exists for allX ∈ C (note that Z(1) is the coend of C). Then a half braiding σcorresponds with σ : Z(X)→ X

Theorem (BV)

If C is centralizable, then Z : X 7→ Z(X) is a quasitriangular Hopf monadon C and we have a braided isomorphism of categories

Z(C)→ CZ

(X , σ) 7→ (X , σ)

Remark: In general the Hopf monad Z is not augmented, i e. notrepresentable by a Hopf algebra: e. g. C = {{G-graded vector spaces}},for G non abelian finite group.

Page 113: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Examples

The centralizer of a Hopf monad 20 / 35

Let C be a monoidal rigid category

A Hopf monad T : C → C is centralizable if

ZT (X) =

∫ Y∈C∨T(Y) ⊗ X ⊗ Y exists for all X ∈ Ob(X)

Proposition (BV)

If T is a centralizable Hopf monad, ZT : X 7→ ZT (X) is a Hopf monadcalled the centralizer of T .

In particular the monad Z of the previous slide is the centralizer of 1C.In a sense the centralizer plays the role of the dual of the Hopf monad T .

Page 114: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Examples

The centralizer of a Hopf monad 20 / 35

Let C be a monoidal rigid categoryA Hopf monad T : C → C is centralizable if

ZT (X) =

∫ Y∈C∨T(Y) ⊗ X ⊗ Y exists for all X ∈ Ob(X)

Proposition (BV)

If T is a centralizable Hopf monad, ZT : X 7→ ZT (X) is a Hopf monadcalled the centralizer of T .

In particular the monad Z of the previous slide is the centralizer of 1C.In a sense the centralizer plays the role of the dual of the Hopf monad T .

Page 115: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Examples

The centralizer of a Hopf monad 20 / 35

Let C be a monoidal rigid categoryA Hopf monad T : C → C is centralizable if

ZT (X) =

∫ Y∈C∨T(Y) ⊗ X ⊗ Y exists for all X ∈ Ob(X)

Proposition (BV)

If T is a centralizable Hopf monad, ZT : X 7→ ZT (X) is a Hopf monadcalled the centralizer of T .

In particular the monad Z of the previous slide is the centralizer of 1C.In a sense the centralizer plays the role of the dual of the Hopf monad T .

Page 116: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Examples

The centralizer of a Hopf monad 20 / 35

Let C be a monoidal rigid categoryA Hopf monad T : C → C is centralizable if

ZT (X) =

∫ Y∈C∨T(Y) ⊗ X ⊗ Y exists for all X ∈ Ob(X)

Proposition (BV)

If T is a centralizable Hopf monad, ZT : X 7→ ZT (X) is a Hopf monadcalled the centralizer of T .

In particular the monad Z of the previous slide is the centralizer of 1C.

In a sense the centralizer plays the role of the dual of the Hopf monad T .

Page 117: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Examples

The centralizer of a Hopf monad 20 / 35

Let C be a monoidal rigid categoryA Hopf monad T : C → C is centralizable if

ZT (X) =

∫ Y∈C∨T(Y) ⊗ X ⊗ Y exists for all X ∈ Ob(X)

Proposition (BV)

If T is a centralizable Hopf monad, ZT : X 7→ ZT (X) is a Hopf monadcalled the centralizer of T .

In particular the monad Z of the previous slide is the centralizer of 1C.In a sense the centralizer plays the role of the dual of the Hopf monad T .

Page 118: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Examples

Hopf monads as ‘quantum groupoids’ 21 / 35

Let R be a unitary ring a monoidal category (RModR ,⊗R ,R RR).

Factslinear bimonads on RModR with a right adjoint is are bialgebroids inthe sense of Takeuchi [Szlacháni]

linear Hopf monads on RModR with a right adjoints are a Hopfalgebroids in the sense of Schauenburg.

Hopf algebroids are non-commutative avatars of groupoids. Complicatedaxioms a Hopf adjunction a Hopf monad (much easier tomanipulate). Using Hopf monads one shows:

Theorem (BVL)

A finite tensor category C over a field k is tensor equivalent to the categoryof A -modules for some bialgebroid A .

Given a k- equivalence Ck'R mod for some finite dimensional k- algebra

R, one constructs a canonical Hopf algebroid A over R.

Page 119: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Examples

Hopf monads as ‘quantum groupoids’ 21 / 35

Let R be a unitary ring a monoidal category (RModR ,⊗R ,R RR).

Factslinear bimonads on RModR with a right adjoint is are bialgebroids inthe sense of Takeuchi [Szlacháni]

linear Hopf monads on RModR with a right adjoints are a Hopfalgebroids in the sense of Schauenburg.

Hopf algebroids are non-commutative avatars of groupoids. Complicatedaxioms a Hopf adjunction a Hopf monad (much easier tomanipulate). Using Hopf monads one shows:

Theorem (BVL)

A finite tensor category C over a field k is tensor equivalent to the categoryof A -modules for some bialgebroid A .

Given a k- equivalence Ck'R mod for some finite dimensional k- algebra

R, one constructs a canonical Hopf algebroid A over R.

Page 120: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Examples

Hopf monads as ‘quantum groupoids’ 21 / 35

Let R be a unitary ring a monoidal category (RModR ,⊗R ,R RR).

Factslinear bimonads on RModR with a right adjoint is are bialgebroids inthe sense of Takeuchi [Szlacháni]

linear Hopf monads on RModR with a right adjoints are a Hopfalgebroids in the sense of Schauenburg.

Hopf algebroids are non-commutative avatars of groupoids. Complicatedaxioms a Hopf adjunction a Hopf monad (much easier tomanipulate). Using Hopf monads one shows:

Theorem (BVL)

A finite tensor category C over a field k is tensor equivalent to the categoryof A -modules for some bialgebroid A .

Given a k- equivalence Ck'R mod for some finite dimensional k- algebra

R, one constructs a canonical Hopf algebroid A over R.

Page 121: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Examples

Hopf monads as ‘quantum groupoids’ 21 / 35

Let R be a unitary ring a monoidal category (RModR ,⊗R ,R RR).

Factslinear bimonads on RModR with a right adjoint is are bialgebroids inthe sense of Takeuchi [Szlacháni]

linear Hopf monads on RModR with a right adjoints are a Hopfalgebroids in the sense of Schauenburg.

Hopf algebroids are non-commutative avatars of groupoids.

Complicatedaxioms a Hopf adjunction a Hopf monad (much easier tomanipulate). Using Hopf monads one shows:

Theorem (BVL)

A finite tensor category C over a field k is tensor equivalent to the categoryof A -modules for some bialgebroid A .

Given a k- equivalence Ck'R mod for some finite dimensional k- algebra

R, one constructs a canonical Hopf algebroid A over R.

Page 122: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Examples

Hopf monads as ‘quantum groupoids’ 21 / 35

Let R be a unitary ring a monoidal category (RModR ,⊗R ,R RR).

Factslinear bimonads on RModR with a right adjoint is are bialgebroids inthe sense of Takeuchi [Szlacháni]

linear Hopf monads on RModR with a right adjoints are a Hopfalgebroids in the sense of Schauenburg.

Hopf algebroids are non-commutative avatars of groupoids. Complicatedaxioms a Hopf adjunction a Hopf monad (much easier tomanipulate).

Using Hopf monads one shows:

Theorem (BVL)

A finite tensor category C over a field k is tensor equivalent to the categoryof A -modules for some bialgebroid A .

Given a k- equivalence Ck'R mod for some finite dimensional k- algebra

R, one constructs a canonical Hopf algebroid A over R.

Page 123: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Examples

Hopf monads as ‘quantum groupoids’ 21 / 35

Let R be a unitary ring a monoidal category (RModR ,⊗R ,R RR).

Factslinear bimonads on RModR with a right adjoint is are bialgebroids inthe sense of Takeuchi [Szlacháni]

linear Hopf monads on RModR with a right adjoints are a Hopfalgebroids in the sense of Schauenburg.

Hopf algebroids are non-commutative avatars of groupoids. Complicatedaxioms a Hopf adjunction a Hopf monad (much easier tomanipulate). Using Hopf monads one shows:

Theorem (BVL)

A finite tensor category C over a field k is tensor equivalent to the categoryof A -modules for some bialgebroid A .

Given a k- equivalence Ck'R mod for some finite dimensional k- algebra

R, one constructs a canonical Hopf algebroid A over R.

Page 124: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Examples

Hopf monads as ‘quantum groupoids’ 21 / 35

Let R be a unitary ring a monoidal category (RModR ,⊗R ,R RR).

Factslinear bimonads on RModR with a right adjoint is are bialgebroids inthe sense of Takeuchi [Szlacháni]

linear Hopf monads on RModR with a right adjoints are a Hopfalgebroids in the sense of Schauenburg.

Hopf algebroids are non-commutative avatars of groupoids. Complicatedaxioms a Hopf adjunction a Hopf monad (much easier tomanipulate). Using Hopf monads one shows:

Theorem (BVL)

A finite tensor category C over a field k is tensor equivalent to the categoryof A -modules for some bialgebroid A .

Given a k- equivalence Ck'R mod for some finite dimensional k- algebra

R, one constructs a canonical Hopf algebroid A over R.

Page 125: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Some aspects of the general theory

Outlook of General Theory of Hopf monads 22 / 35

Tannaka dictionary

Hopf modules and Sweedler decomposition theorem

Existence of universal integrals (with values in a certainautoequivalence of C)

Semisimplicity, Maschke criterion

The drinfeld double of a Hopf monad

Cross-products

Bosonization for Hopf monads

Applications to construction and comparison of quantum invariants(non-braided setting)

Page 126: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Some aspects of the general theory

Outlook of General Theory of Hopf monads 22 / 35

Tannaka dictionary

Hopf modules and Sweedler decomposition theorem

Existence of universal integrals (with values in a certainautoequivalence of C)

Semisimplicity, Maschke criterion

The drinfeld double of a Hopf monad

Cross-products

Bosonization for Hopf monads

Applications to construction and comparison of quantum invariants(non-braided setting)

Page 127: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Some aspects of the general theory

Outlook of General Theory of Hopf monads 22 / 35

Tannaka dictionary

Hopf modules and Sweedler decomposition theorem

Existence of universal integrals (with values in a certainautoequivalence of C)

Semisimplicity, Maschke criterion

The drinfeld double of a Hopf monad

Cross-products

Bosonization for Hopf monads

Applications to construction and comparison of quantum invariants(non-braided setting)

Page 128: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Some aspects of the general theory

Outlook of General Theory of Hopf monads 22 / 35

Tannaka dictionary

Hopf modules and Sweedler decomposition theorem

Existence of universal integrals (with values in a certainautoequivalence of C)

Semisimplicity, Maschke criterion

The drinfeld double of a Hopf monad

Cross-products

Bosonization for Hopf monads

Applications to construction and comparison of quantum invariants(non-braided setting)

Page 129: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Some aspects of the general theory

Outlook of General Theory of Hopf monads 22 / 35

Tannaka dictionary

Hopf modules and Sweedler decomposition theorem

Existence of universal integrals (with values in a certainautoequivalence of C)

Semisimplicity, Maschke criterion

The drinfeld double of a Hopf monad

Cross-products

Bosonization for Hopf monads

Applications to construction and comparison of quantum invariants(non-braided setting)

Page 130: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Some aspects of the general theory

Outlook of General Theory of Hopf monads 22 / 35

Tannaka dictionary

Hopf modules and Sweedler decomposition theorem

Existence of universal integrals (with values in a certainautoequivalence of C)

Semisimplicity, Maschke criterion

The drinfeld double of a Hopf monad

Cross-products

Bosonization for Hopf monads

Applications to construction and comparison of quantum invariants(non-braided setting)

Page 131: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Some aspects of the general theory

Outlook of General Theory of Hopf monads 22 / 35

Tannaka dictionary

Hopf modules and Sweedler decomposition theorem

Existence of universal integrals (with values in a certainautoequivalence of C)

Semisimplicity, Maschke criterion

The drinfeld double of a Hopf monad

Cross-products

Bosonization for Hopf monads

Applications to construction and comparison of quantum invariants(non-braided setting)

Page 132: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Some aspects of the general theory

Outlook of General Theory of Hopf monads 22 / 35

Tannaka dictionary

Hopf modules and Sweedler decomposition theorem

Existence of universal integrals (with values in a certainautoequivalence of C)

Semisimplicity, Maschke criterion

The drinfeld double of a Hopf monad

Cross-products

Bosonization for Hopf monads

Applications to construction and comparison of quantum invariants(non-braided setting)

Page 133: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Some aspects of the general theory

Hopf modules and Sweedler’s Theorem for HopfMonads 23 / 35

T Hopf monad on C T1 is a coalgebra in C (coproduct ∆1,1, counit ε)

lifts to a coalgebra C = FT (1) in CT . Moreover we have a naturalisomorphism

σ : C⊗?→? ⊗ C .

Proposition (BVL)

σ is a half-braiding and (C , σ) is a cocommutative coalgebra in Z(CT )called the induced central coalgebra of T .

A (right) T - Hopf module is a (right) C-comodule in CT , i. e. a data (M, r , ∂)with (M, r) a T - module, (M, ∂) a T1- comodule + T - linearity of ∂.

Page 134: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Some aspects of the general theory

Hopf modules and Sweedler’s Theorem for HopfMonads 23 / 35

T Hopf monad on C T1 is a coalgebra in C (coproduct ∆1,1, counit ε) lifts to a coalgebra C = FT (1) in CT . Moreover we have a naturalisomorphism

σ : C⊗?→? ⊗ C .

Proposition (BVL)

σ is a half-braiding and (C , σ) is a cocommutative coalgebra in Z(CT )called the induced central coalgebra of T .

A (right) T - Hopf module is a (right) C-comodule in CT , i. e. a data (M, r , ∂)with (M, r) a T - module, (M, ∂) a T1- comodule + T - linearity of ∂.

Page 135: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Some aspects of the general theory

Hopf modules and Sweedler’s Theorem for HopfMonads 23 / 35

T Hopf monad on C T1 is a coalgebra in C (coproduct ∆1,1, counit ε) lifts to a coalgebra C = FT (1) in CT . Moreover we have a naturalisomorphism

σ : C⊗?→? ⊗ C .

Proposition (BVL)

σ is a half-braiding

and (C , σ) is a cocommutative coalgebra in Z(CT )called the induced central coalgebra of T .

A (right) T - Hopf module is a (right) C-comodule in CT , i. e. a data (M, r , ∂)with (M, r) a T - module, (M, ∂) a T1- comodule + T - linearity of ∂.

Page 136: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Some aspects of the general theory

Hopf modules and Sweedler’s Theorem for HopfMonads 23 / 35

T Hopf monad on C T1 is a coalgebra in C (coproduct ∆1,1, counit ε) lifts to a coalgebra C = FT (1) in CT . Moreover we have a naturalisomorphism

σ : C⊗?→? ⊗ C .

Proposition (BVL)

σ is a half-braiding and (C , σ) is a cocommutative coalgebra in Z(CT )called the induced central coalgebra of T .

A (right) T - Hopf module is a (right) C-comodule in CT , i. e. a data (M, r , ∂)with (M, r) a T - module, (M, ∂) a T1- comodule + T - linearity of ∂.

Page 137: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Some aspects of the general theory

Hopf modules and Sweedler’s Theorem for HopfMonads 23 / 35

T Hopf monad on C T1 is a coalgebra in C (coproduct ∆1,1, counit ε) lifts to a coalgebra C = FT (1) in CT . Moreover we have a naturalisomorphism

σ : C⊗?→? ⊗ C .

Proposition (BVL)

σ is a half-braiding and (C , σ) is a cocommutative coalgebra in Z(CT )called the induced central coalgebra of T .

A (right) T - Hopf module is a (right) C-comodule in CT

, i. e. a data (M, r , ∂)with (M, r) a T - module, (M, ∂) a T1- comodule + T - linearity of ∂.

Page 138: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Some aspects of the general theory

Hopf modules and Sweedler’s Theorem for HopfMonads 23 / 35

T Hopf monad on C T1 is a coalgebra in C (coproduct ∆1,1, counit ε) lifts to a coalgebra C = FT (1) in CT . Moreover we have a naturalisomorphism

σ : C⊗?→? ⊗ C .

Proposition (BVL)

σ is a half-braiding and (C , σ) is a cocommutative coalgebra in Z(CT )called the induced central coalgebra of T .

A (right) T - Hopf module is a (right) C-comodule in CT , i. e. a data (M, r , ∂)with (M, r) a T - module, (M, ∂) a T1- comodule + T - linearity of ∂.

Page 139: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Some aspects of the general theory

Under suitable exactness conditions (T is conservative, C hascoequalizers and T preserves them):

Theorem (BVL)

The assignment X 7→ (TX , µX ,∆X ,1) is an equivalence of categories

Q : C'−→ {{T - Hopf modules}}

with quasi-inverse the functor coinvariant part.Moreover if C has equalizers and T preserves them, Q is a monoidalequivalence, the category of Hopf modules (i.e. C- comodules) beingendowed with the cotensor product over C.

Page 140: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Some aspects of the general theory

Under suitable exactness conditions (T is conservative, C hascoequalizers and T preserves them):

Theorem (BVL)

The assignment X 7→ (TX , µX ,∆X ,1) is an equivalence of categories

Q : C'−→ {{T - Hopf modules}}

with quasi-inverse the functor coinvariant part.

Moreover if C has equalizers and T preserves them, Q is a monoidalequivalence, the category of Hopf modules (i.e. C- comodules) beingendowed with the cotensor product over C.

Page 141: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Some aspects of the general theory

Under suitable exactness conditions (T is conservative, C hascoequalizers and T preserves them):

Theorem (BVL)

The assignment X 7→ (TX , µX ,∆X ,1) is an equivalence of categories

Q : C'−→ {{T - Hopf modules}}

with quasi-inverse the functor coinvariant part.Moreover if C has equalizers and T preserves them, Q is a monoidalequivalence, the category of Hopf modules (i.e. C- comodules) beingendowed with the cotensor product over C.

Page 142: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Some aspects of the general theory

Proof of Sweedler’s theorem for Hopf monads 25 / 35

An adjunctionD

U��C

F

GG

a comonad T = (FU,F(ηU), ε) on D.

Denoting DT the category of T - comodules we have a cocomparison

functor K :

D

��wwC

88

K// DT

XXThe adjunction (F ,U) iscomonadic if K equiva-lence.

If T is a monad on C, its adjunction is comonadic under suitable exactnessassumptions (descent), i. e. K : C → (CT )T is an equivalence.For T Hopf monad, we have an isomorphism of comonads on CT

φ : T∼−→? ⊗ C

defined by φ(M,r) = (r ⊗ idT(1))TM,1 : TM → M ⊗ T1.

Hence CTT∼−→ {{right T - Hopf modules}} �

Page 143: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Some aspects of the general theory

Proof of Sweedler’s theorem for Hopf monads 25 / 35

An adjunctionD

U��C

F

GG

a comonad T = (FU,F(ηU), ε) on D.

Denoting DT the category of T - comodules we have a cocomparison

functor K :

D

��wwC

88

K// DT

XXThe adjunction (F ,U) iscomonadic if K equiva-lence.

If T is a monad on C, its adjunction is comonadic under suitable exactnessassumptions (descent), i. e. K : C → (CT )T is an equivalence.For T Hopf monad, we have an isomorphism of comonads on CT

φ : T∼−→? ⊗ C

defined by φ(M,r) = (r ⊗ idT(1))TM,1 : TM → M ⊗ T1.

Hence CTT∼−→ {{right T - Hopf modules}} �

Page 144: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Some aspects of the general theory

Proof of Sweedler’s theorem for Hopf monads 25 / 35

An adjunctionD

U��C

F

GG

a comonad T = (FU,F(ηU), ε) on D.

Denoting DT the category of T - comodules we have a cocomparison

functor K :

D

��wwC

88

K// DT

XXThe adjunction (F ,U) iscomonadic if K equiva-lence.

If T is a monad on C, its adjunction is comonadic under suitable exactnessassumptions (descent), i. e. K : C → (CT )T is an equivalence.For T Hopf monad, we have an isomorphism of comonads on CT

φ : T∼−→? ⊗ C

defined by φ(M,r) = (r ⊗ idT(1))TM,1 : TM → M ⊗ T1.

Hence CTT∼−→ {{right T - Hopf modules}} �

Page 145: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Some aspects of the general theory

Proof of Sweedler’s theorem for Hopf monads 25 / 35

An adjunctionD

U��C

F

GG

a comonad T = (FU,F(ηU), ε) on D.

Denoting DT the category of T - comodules we have a cocomparison

functor K :

D

��wwC

88

K// DT

XXThe adjunction (F ,U) iscomonadic if K equiva-lence.

If T is a monad on C, its adjunction is comonadic under suitable exactnessassumptions (descent), i. e. K : C → (CT )T is an equivalence.

For T Hopf monad, we have an isomorphism of comonads on CT

φ : T∼−→? ⊗ C

defined by φ(M,r) = (r ⊗ idT(1))TM,1 : TM → M ⊗ T1.

Hence CTT∼−→ {{right T - Hopf modules}} �

Page 146: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf Monads - a sketchy survey Some aspects of the general theory

Proof of Sweedler’s theorem for Hopf monads 25 / 35

An adjunctionD

U��C

F

GG

a comonad T = (FU,F(ηU), ε) on D.

Denoting DT the category of T - comodules we have a cocomparison

functor K :

D

��wwC

88

K// DT

XXThe adjunction (F ,U) iscomonadic if K equiva-lence.

If T is a monad on C, its adjunction is comonadic under suitable exactnessassumptions (descent), i. e. K : C → (CT )T is an equivalence.For T Hopf monad, we have an isomorphism of comonads on CT

φ : T∼−→? ⊗ C

defined by φ(M,r) = (r ⊗ idT(1))TM,1 : TM → M ⊗ T1.

Hence CTT∼−→ {{right T - Hopf modules}} �

Page 147: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf (co)-monads applied to tensor functors

1 Introduction

2 Hopf Monads - a sketchy survey

3 Hopf (co)-monads applied to tensor functors

4 Exact sequences of tensor categories

Page 148: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf (co)-monads applied to tensor functors

We now consider tensor categories over a field k.

If C is a tensor category, its Ind-completion IndC is a monoidal abeliancategory containing C as a full subcategory and whose objects are formalfiltering colimits of objects of C. For instance Ind vect = Vect, andInd comodH = ComodH. Note that these are no longer rigid.

TheoremLet F : C → D be a tensor functor. There exists a k- linear left exactcomonad on IndC such that we have a commutative diagram:

CF //

'⊗

vect

DT

<<

where CT is the category of T -comodule whose underlying object is in C.

Page 149: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf (co)-monads applied to tensor functors

We now consider tensor categories over a field k.

If C is a tensor category, its Ind-completion IndC is a monoidal abeliancategory containing C as a full subcategory and whose objects are formalfiltering colimits of objects of C.

For instance Ind vect = Vect, andInd comodH = ComodH. Note that these are no longer rigid.

TheoremLet F : C → D be a tensor functor. There exists a k- linear left exactcomonad on IndC such that we have a commutative diagram:

CF //

'⊗

vect

DT

<<

where CT is the category of T -comodule whose underlying object is in C.

Page 150: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf (co)-monads applied to tensor functors

We now consider tensor categories over a field k.

If C is a tensor category, its Ind-completion IndC is a monoidal abeliancategory containing C as a full subcategory and whose objects are formalfiltering colimits of objects of C. For instance Ind vect = Vect, andInd comodH = ComodH.

Note that these are no longer rigid.

TheoremLet F : C → D be a tensor functor. There exists a k- linear left exactcomonad on IndC such that we have a commutative diagram:

CF //

'⊗

vect

DT

<<

where CT is the category of T -comodule whose underlying object is in C.

Page 151: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf (co)-monads applied to tensor functors

We now consider tensor categories over a field k.

If C is a tensor category, its Ind-completion IndC is a monoidal abeliancategory containing C as a full subcategory and whose objects are formalfiltering colimits of objects of C. For instance Ind vect = Vect, andInd comodH = ComodH. Note that these are no longer rigid.

TheoremLet F : C → D be a tensor functor. There exists a k- linear left exactcomonad on IndC such that we have a commutative diagram:

CF //

'⊗

vect

DT

<<

where CT is the category of T -comodule whose underlying object is in C.

Page 152: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf (co)-monads applied to tensor functors

We now consider tensor categories over a field k.

If C is a tensor category, its Ind-completion IndC is a monoidal abeliancategory containing C as a full subcategory and whose objects are formalfiltering colimits of objects of C. For instance Ind vect = Vect, andInd comodH = ComodH. Note that these are no longer rigid.

TheoremLet F : C → D be a tensor functor. There exists a k- linear left exactcomonad on IndC such that we have a commutative diagram:

CF //

'⊗

vect

DT

<<

where CT is the category of T -comodule whose underlying object is in C.

Page 153: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf (co)-monads applied to tensor functors

We now consider tensor categories over a field k.

If C is a tensor category, its Ind-completion IndC is a monoidal abeliancategory containing C as a full subcategory and whose objects are formalfiltering colimits of objects of C. For instance Ind vect = Vect, andInd comodH = ComodH. Note that these are no longer rigid.

TheoremLet F : C → D be a tensor functor. There exists a k- linear left exactcomonad on IndC such that we have a commutative diagram:

CF //

'⊗

vect

DT

<<

where CT is the category of T -comodule whose underlying object is in C.

Page 154: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf (co)-monads applied to tensor functors

Proof 28 / 35

The functor F : C → D extends to a linear faithful exact functorIndF : IndC → IndD which preserves colimits and is strong monoidal.

IndF has a right adjoint, denoted by R.It is also a monoidal adjunction, which is Hopf. Its comonad T = IndFR isa Hopf comonad on IndC.IndF being faithful exact, the adjunction (IndF ,R) is comonadic by Beck,hence the theorem.

Example

If D = vect, a linear Hopf comonad on Vect is of the form H⊗? for someHopf algebra H, so we recover the classical tannakian result.

Page 155: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf (co)-monads applied to tensor functors

Proof 28 / 35

The functor F : C → D extends to a linear faithful exact functorIndF : IndC → IndD which preserves colimits and is strong monoidal.IndF has a right adjoint, denoted by R.

It is also a monoidal adjunction, which is Hopf. Its comonad T = IndFR isa Hopf comonad on IndC.IndF being faithful exact, the adjunction (IndF ,R) is comonadic by Beck,hence the theorem.

Example

If D = vect, a linear Hopf comonad on Vect is of the form H⊗? for someHopf algebra H, so we recover the classical tannakian result.

Page 156: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf (co)-monads applied to tensor functors

Proof 28 / 35

The functor F : C → D extends to a linear faithful exact functorIndF : IndC → IndD which preserves colimits and is strong monoidal.IndF has a right adjoint, denoted by R.It is also a monoidal adjunction, which is Hopf.

Its comonad T = IndFR isa Hopf comonad on IndC.IndF being faithful exact, the adjunction (IndF ,R) is comonadic by Beck,hence the theorem.

Example

If D = vect, a linear Hopf comonad on Vect is of the form H⊗? for someHopf algebra H, so we recover the classical tannakian result.

Page 157: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf (co)-monads applied to tensor functors

Proof 28 / 35

The functor F : C → D extends to a linear faithful exact functorIndF : IndC → IndD which preserves colimits and is strong monoidal.IndF has a right adjoint, denoted by R.It is also a monoidal adjunction, which is Hopf. Its comonad T = IndFR isa Hopf comonad on IndC.

IndF being faithful exact, the adjunction (IndF ,R) is comonadic by Beck,hence the theorem.

Example

If D = vect, a linear Hopf comonad on Vect is of the form H⊗? for someHopf algebra H, so we recover the classical tannakian result.

Page 158: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf (co)-monads applied to tensor functors

Proof 28 / 35

The functor F : C → D extends to a linear faithful exact functorIndF : IndC → IndD which preserves colimits and is strong monoidal.IndF has a right adjoint, denoted by R.It is also a monoidal adjunction, which is Hopf. Its comonad T = IndFR isa Hopf comonad on IndC.IndF being faithful exact, the adjunction (IndF ,R) is comonadic by Beck,hence the theorem.

Example

If D = vect, a linear Hopf comonad on Vect is of the form H⊗? for someHopf algebra H, so we recover the classical tannakian result.

Page 159: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf (co)-monads applied to tensor functors

Proof 28 / 35

The functor F : C → D extends to a linear faithful exact functorIndF : IndC → IndD which preserves colimits and is strong monoidal.IndF has a right adjoint, denoted by R.It is also a monoidal adjunction, which is Hopf. Its comonad T = IndFR isa Hopf comonad on IndC.IndF being faithful exact, the adjunction (IndF ,R) is comonadic by Beck,hence the theorem.

Example

If D = vect, a linear Hopf comonad on Vect is of the form H⊗? for someHopf algebra H, so we recover the classical tannakian result.

Page 160: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Hopf (co)-monads applied to tensor functors

Let F : C → D be a tensor functor. We say that F is dominant if the rightadjoint R of IndF is faithful exact.Then applying the classification theorem for Hopf modules in its dual formwe obtain:

Theorem

If F is dominant, there exists a commutative algebra (A , σ) in Z(IndC) -the induced central algebra of T - such that we have a commutativediagram

CFA //

F ��

A- mod C

D

'⊗

99

where A- mod is the category of ‘finite type’ A-modules in IndC(=quotients of A ⊗ X, X ∈ C), with tensor product ⊗A ,σ, and FA is thetensor functor X 7→ A ⊗ X.

If D = vectk and C, F are symmetric, then A is Deligne’s trivializingalgebra.

Page 161: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Exact sequences of tensor categories

1 Introduction

2 Hopf Monads - a sketchy survey

3 Hopf (co)-monads applied to tensor functors

4 Exact sequences of tensor categories

Page 162: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Exact sequences of tensor categories

An exact sequence of Hopf algebras in the sense of Schneider is asequence

K i // Hp // H′

of Hopf algebras such that1 p−1(0) is a normal Hopf ideal of H;2 H is right faithfully coflat over H′;3 i is a categorical kernel of p.

We extend this notion to tensor categories.Let F : C → D be a tensor functor. We denote by kF the full tensorsubcategory of C

kF = {X ∈ C | F(X)is trivial}

Note that F induces a fiber functor KF → vect, X 7→ Hom(1,F(X).We say that F is normal if the right adjoint R of IndF satisfiesR(1) ∈ Ind(KF ).This means that the subcategory < 1 > of IndC generated by 1 is stableunder the Hopf comonad T = UR which encodes F .

Page 163: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Exact sequences of tensor categories

An exact sequence of Hopf algebras in the sense of Schneider is asequence

K i // Hp // H′

of Hopf algebras such that1 p−1(0) is a normal Hopf ideal of H;2 H is right faithfully coflat over H′;3 i is a categorical kernel of p.

We extend this notion to tensor categories.Let F : C → D be a tensor functor. We denote by kF the full tensorsubcategory of C

kF = {X ∈ C | F(X)is trivial}

Note that F induces a fiber functor KF → vect, X 7→ Hom(1,F(X).We say that F is normal if the right adjoint R of IndF satisfiesR(1) ∈ Ind(KF ).This means that the subcategory < 1 > of IndC generated by 1 is stableunder the Hopf comonad T = UR which encodes F .

Page 164: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Exact sequences of tensor categories

An exact sequence of Hopf algebras in the sense of Schneider is asequence

K i // Hp // H′

of Hopf algebras such that1 p−1(0) is a normal Hopf ideal of H;2 H is right faithfully coflat over H′;3 i is a categorical kernel of p.

We extend this notion to tensor categories.Let F : C → D be a tensor functor. We denote by kF the full tensorsubcategory of C

kF = {X ∈ C | F(X)is trivial}

Note that F induces a fiber functor KF → vect, X 7→ Hom(1,F(X).We say that F is normal if the right adjoint R of IndF satisfiesR(1) ∈ Ind(KF ).This means that the subcategory < 1 > of IndC generated by 1 is stableunder the Hopf comonad T = UR which encodes F .

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Exact sequences of tensor categories

An exact sequence of tensor categories is a sequence

C′f // C

F // C′′

of tensor categories such that:1 F is normal and dominant;2 f induces a tensor equivalence C′ → KF .

If H′ → H → H′′ is an exact sequence of Hopf algebras, then

comodH′ → comodH → comodH′′

is an exact sequence of tensor categories, and, if H is finite dimensional,

mod H′′ → mod H → mod H′

is also an exact sequence of tensor categories.

Page 166: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Exact sequences of tensor categories

An exact sequence of tensor categories is a sequence

C′f // C

F // C′′

of tensor categories such that:1 F is normal and dominant;2 f induces a tensor equivalence C′ → KF .

If H′ → H → H′′ is an exact sequence of Hopf algebras, then

comodH′ → comodH → comodH′′

is an exact sequence of tensor categories, and, if H is finite dimensional,

mod H′′ → mod H → mod H′

is also an exact sequence of tensor categories.

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Exact sequences of tensor categories

Exact sequences of tensor categories are classified by certain Hopf(co)-monads.

A linear exact Hopf comonad T on tensor category C is normal ifT(1) ∈< 1 >. We have < 1 >' Vect, so if T is normal it restricts to a Hopfalgebra H on Vect. If in addition T is faithful, we have an exact sequenceof tensor categories

comodH → CT → C

and ‘all extensions of C by comodH’ are of this form up to tensorequivalence [one has to be more precise].

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Exact sequences of tensor categories

Exact sequences of tensor categories are classified by certain Hopf(co)-monads.A linear exact Hopf comonad T on tensor category C is normal ifT(1) ∈< 1 >. We have < 1 >' Vect, so if T is normal it restricts to a Hopfalgebra H on Vect. If in addition T is faithful, we have an exact sequenceof tensor categories

comodH → CT → C

and ‘all extensions of C by comodH’ are of this form up to tensorequivalence [one has to be more precise].

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Exact sequences of tensor categories

Examples 34 / 35

Equivariantization

Let G be a finite group acting on a tensor category C by tensorautomorphisms (Tg)g∈G . Then we have an exact sequence

repG → CG → C

where CG → C is the equivariantization functor.The endofunctor T =

⊕Tg admits a structure of Hopf comonad TG (it

admits also a structure of Hopf monad), and CG is just CTG. The Hopf

comonad TG is normal faithful exact, and its associated Hopf algebra isk G . It has a certain commutativity property. These conditions characterizeHopf comonads corresponding with equivariantizations (at least over C).

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Exact sequences of tensor categories

Examples 34 / 35

EquivariantizationLet G be a finite group acting on a tensor category C by tensorautomorphisms (Tg)g∈G . Then we have an exact sequence

repG → CG → C

where CG → C is the equivariantization functor.

The endofunctor T =⊕

Tg admits a structure of Hopf comonad TG (itadmits also a structure of Hopf monad), and CG is just CTG

. The Hopfcomonad TG is normal faithful exact, and its associated Hopf algebra isk G . It has a certain commutativity property. These conditions characterizeHopf comonads corresponding with equivariantizations (at least over C).

Page 171: Hopf (co)monads, tensor functors and exact sequences of ...bruguieres/docs/clermont.pdf · Tensor categories and tensor functors 4/35 Let | be a field. Definition In this talk a

Exact sequences of tensor categories

Examples 34 / 35

EquivariantizationLet G be a finite group acting on a tensor category C by tensorautomorphisms (Tg)g∈G . Then we have an exact sequence

repG → CG → C

where CG → C is the equivariantization functor.The endofunctor T =

⊕Tg admits a structure of Hopf comonad TG (it

admits also a structure of Hopf monad), and CG is just CTG. The Hopf

comonad TG is normal faithful exact, and its associated Hopf algebra isk G . It has a certain commutativity property. These conditions characterizeHopf comonads corresponding with equivariantizations (at least over C).

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Exact sequences of tensor categories

24. More on Hopf monads 35 / 35

BV1. Hopf Diagrams and Quantum Invariants, AGT 5 (2005) 1677-1710.Where Hopf diagram are introduced as a means for computing the Reshetikhin-Turaevinvariant in terms of the coend of a ribbon category and its structural morphisms.

BV2. Hopf Monads, Advances in Math. 215 (2007), 679-733.Where the notion of Hopf monad is introduced, and several fundamental results of thetheory of finite dimensional Hopf algebras are extended thereto.

BV3. Categorical Centers and Reshetikhin-Turaev Invariants, Acta MathematicaVietnamica 33 3, 255-279Where the coend of the center of a fusion spherical category over a ring is described, themodularity of the center, proven, and the corresponding Reshetikhin-Turaev invariant,constructed.

BV4. Quantum Double of Hopf monads and Categorical Centers, arXiv:0812.2443, toappear in Transactions of the American Mathematical Society (2010)Where the general theory of centralizers and doubles of Hopf monads is expounded.

BLV. Hopf Monads on Monoidal Categories, arXiv:1003.1920.Where Hopf monads are defined anew in the monoidal worldBN. Exact sequences of tensor categories, arXiv:1006.0569.

See also: http://www.math.univ-montp2.fr/∼bruguieres/recherche.html