Equivalences and iterations for weak crossed products Ramn Gonzlez - - PowerPoint PPT Presentation

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products

Equivalences and iterations for weak crossed products

Ramón González Rodríguez

http://www.dma.uvigo.es/˜rgon/ Departamento de Matemática Aplicada II. Universidade de Vigo Based in a joint work with A.B. Rodríguez Raposo and J.M. Fernández Vilaboa Research supported by Ministerio de Economía y Competitividad of Spain (European Feder support included). Grant MTM2013-43687-P Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products

Outline

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Weak crossed products

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Equivalences between weak crossed products

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Iterations for weak crossed products

Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products

Some notation and conventions.

From now on C denotes a strict monoidal category with tensor product denoted by ⊗ and unit object K. We assume that every idempotent morphism q : Y → Y splits, i.e., there exist an object Z and morphisms i : Z → Y and p : Y → Z such that q = i ◦ p and p ◦ i = idZ.

Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products

Some notation and conventions.

From now on C denotes a strict monoidal category with tensor product denoted by ⊗ and unit object K. We assume that every idempotent morphism q : Y → Y splits, i.e., there exist an object Z and morphisms i : Z → Y and p : Y → Z such that q = i ◦ p and p ◦ i = idZ. (A, ηA, µA) is a monoid with product µA and unit ηA.

Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products

Some notation and conventions.

From now on C denotes a strict monoidal category with tensor product denoted by ⊗ and unit object K. We assume that every idempotent morphism q : Y → Y splits, i.e., there exist an object Z and morphisms i : Z → Y and p : Y → Z such that q = i ◦ p and p ◦ i = idZ. (A, ηA, µA) is a monoid with product µA and unit ηA. For simplicity of notation, given three objects V , U, B in C and a morphism f : V → U, we write B ⊗ f for idB ⊗ f and f ⊗ B for f ⊗ idB.

Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products

Weak crossed products

1

Weak crossed products

2

Equivalences between weak crossed products

3

Iterations for weak crossed products

Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products

J.M. Fernández Vilaboa, R. González Rodríguez, R., A.B. Rodríguez Raposo, Preunits and weak crossed products, J. Pure Appl. Algebra (2009).

Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products

J.M. Fernández Vilaboa, R. González Rodríguez, R., A.B. Rodríguez Raposo, Preunits and weak crossed products, J. Pure Appl. Algebra (2009). Let A be a monoid and let V be an object in C. Suppose that there exists a morphism ψA

V : V ⊗ A → A ⊗ V

such that the following equality holds (1) (µA ⊗ V ) ◦ (A ⊗ ψA

V ) ◦ (ψA V ⊗ A) = ψA V ◦ (V ⊗ µA).

Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products

The morphism ∇A⊗V = (µA ⊗ V ) ◦ (A ⊗ ψA

V ) ◦ (A ⊗ V ⊗ ηA) : A ⊗ V → A ⊗ V

is idempotent.

Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products

The morphism ∇A⊗V = (µA ⊗ V ) ◦ (A ⊗ ψA

V ) ◦ (A ⊗ V ⊗ ηA) : A ⊗ V → A ⊗ V

is idempotent. For the idempotent morphism ∇A⊗V we denote by A × V the object such that pA⊗V ◦ iA⊗V = idA×V , where iA⊗V : A × V → A ⊗ V , pA⊗V : A ⊗ V → A × V the injection and the projection associated to ∇A⊗V .

Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products

The morphism ∇A⊗V = (µA ⊗ V ) ◦ (A ⊗ ψA

V ) ◦ (A ⊗ V ⊗ ηA) : A ⊗ V → A ⊗ V

is idempotent. For the idempotent morphism ∇A⊗V we denote by A × V the object such that pA⊗V ◦ iA⊗V = idA×V , where iA⊗V : A × V → A ⊗ V , pA⊗V : A ⊗ V → A × V the injection and the projection associated to ∇A⊗V . From now on we consider quadruples (A, V , ψA

V , σA V ) where A is an algebra, V an object,

ψA

V : V ⊗ A → A ⊗ V satisfies (1) and

σA

V : V ⊗ V → A ⊗ V

is a morphism in C.

Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products

For a quadruple (A, V , ψA

V , σA V ) define the product

µA⊗V = (µA ⊗ V ) ◦ (µA ⊗ σA

V ) ◦ (A ⊗ ψA V ⊗ V ) : A ⊗ V ⊗ A ⊗ V → A ⊗ V

and let µA×V be the product µA×V = pA⊗V ◦ µA⊗V ◦ (iA⊗V ⊗ iA⊗V ) : A × V ⊗ A × V → A × V .

Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products

For a quadruple (A, V , ψA

V , σA V ) define the product

µA⊗V = (µA ⊗ V ) ◦ (µA ⊗ σA

V ) ◦ (A ⊗ ψA V ⊗ V ) : A ⊗ V ⊗ A ⊗ V → A ⊗ V

and let µA×V be the product µA×V = pA⊗V ◦ µA⊗V ◦ (iA⊗V ⊗ iA⊗V ) : A × V ⊗ A × V → A × V . What conditions are needed to make µA⊗V and µA×V associative?

Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products

Twisted condition We say that (A, V , ψA

V , σA V ) satisfies the twisted condition if

(µA ⊗ V ) ◦ (A ⊗ ψA

V ) ◦ (σA V ⊗ A) = (µA ⊗ V ) ◦ (A ⊗ σA V ) ◦ (ψA V ⊗ V ) ◦ (V ⊗ ψA V )

Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products

Twisted condition We say that (A, V , ψA

V , σA V ) satisfies the twisted condition if

(µA ⊗ V ) ◦ (A ⊗ ψA

V ) ◦ (σA V ⊗ A) = (µA ⊗ V ) ◦ (A ⊗ σA V ) ◦ (ψA V ⊗ V ) ◦ (V ⊗ ψA V )

Cocycle condition We say that (A, V , ψA

V , σA V ) satisfies the the cocycle condition if

(µA ⊗ V ) ◦ (A ⊗ σA

V ) ◦ (σA V ⊗ V ) = (µA ⊗ V ) ◦ (A ⊗ σA V ) ◦ (ψA V ⊗ V ) ◦ (V ⊗ σA V ).

Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products

Theorem Let (A, V , ψA

V , σA V ) be a quadruple satisfying the twisted and the cocycle conditions.

The product µA⊗V is associative and normalized with respect to ∇A⊗V .

Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products

Theorem Let (A, V , ψA

V , σA V ) be a quadruple satisfying the twisted and the cocycle conditions.

The product µA⊗V is associative and normalized with respect to ∇A⊗V . Normal condition Normalized with respect to ∇A⊗V : ∇A⊗V ◦ µA⊗V = µA⊗V = µA⊗V ◦ (∇A⊗V ⊗ ∇A⊗V ).

Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products

Theorem Let (A, V , ψA

V , σA V ) be a quadruple satisfying the twisted and the cocycle conditions.

The product µA⊗V is associative and normalized with respect to ∇A⊗V . Normal condition Normalized with respect to ∇A⊗V : ∇A⊗V ◦ µA⊗V = µA⊗V = µA⊗V ◦ (∇A⊗V ⊗ ∇A⊗V ). Corollary Let (A, V , ψA

V , σA V ) be a quadruple satisfying the twisted and the cocycle conditions.

The product µA×V is associative.

Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products

Theorem Let (A, V , ψA

V , σA V ) be a quadruple satisfying the twisted and the cocycle conditions.

The product µA⊗V is associative and normalized with respect to ∇A⊗V . Normal condition Normalized with respect to ∇A⊗V : ∇A⊗V ◦ µA⊗V = µA⊗V = µA⊗V ◦ (∇A⊗V ⊗ ∇A⊗V ). Corollary Let (A, V , ψA

V , σA V ) be a quadruple satisfying the twisted and the cocycle conditions.

The product µA×V is associative. It is possible to assume without loss of generality that ∇A⊗V ◦ σA

V = σA V .

Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products

Definition If the quadruple (A, V , ψA

V , σA V ) satisfies the twisted condition and the cocycle condition

we say that (A ⊗ V , µA⊗V ) is a weak crossed product.

Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products

Definition If the quadruple (A, V , ψA

V , σA V ) satisfies the twisted condition and the cocycle condition

we say that (A ⊗ V , µA⊗V ) is a weak crossed product. What are the conditions under which µA×V admits an unit?

Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products

Definition If the quadruple (A, V , ψA

V , σA V ) satisfies the twisted condition and the cocycle condition

we say that (A ⊗ V , µA⊗V ) is a weak crossed product. What are the conditions under which µA×V admits an unit? Definition If mA⊗V is an associative product defined in A ⊗ V , a preunit ν : K → A ⊗ V is a morphism satisfying mA⊗V ◦ (A ⊗ V ⊗ ν) = mA⊗V ◦ (ν ⊗ A ⊗ V ) = mA⊗V ◦ (A ⊗ V ⊗ (mA⊗V ◦ (ν ⊗ ν))).

Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products

Associated to a preunit we obtain an idempotent morphism ∇ν

A⊗V = mA⊗V ◦ (A ⊗ V ⊗ ν) : A ⊗ V → A ⊗ V .

Take A ×ν V the image of ∇ν

A⊗V , pν A⊗V the projection and iν A⊗V the injection.

Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products

Associated to a preunit we obtain an idempotent morphism ∇ν

A⊗V = mA⊗V ◦ (A ⊗ V ⊗ ν) : A ⊗ V → A ⊗ V .

Take A ×ν V the image of ∇ν

A⊗V , pν A⊗V the projection and iν A⊗V the injection.

It is possible to endow A ×ν V with an associative product mA×νV = pν

A⊗V ◦ mA⊗V ◦ (iν A⊗V ⊗ iν A⊗V )

whose unit is ηA×νV = pν

A⊗V ◦ ν.

Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products

Theorem Let A be a monoid and mA⊗V : A⊗V ⊗A⊗V → A⊗V a morphism of left A-modules for the trivial actions. Then the following statements are equivalent: (i) The product mA⊗V is associative with preunit ν and normalized with respect to ∇ν

A⊗V .

(ii) There exist morphisms ψA

V : V ⊗ A → A ⊗ V , σA V : V ⊗ V → A ⊗ V and

ν : K → A ⊗ V such that the pair (A ⊗ V , µA⊗V ) is a weak crossed product with mA⊗V = µA⊗V satisfying: (µA ⊗ V ) ◦ (A ⊗ σA

V ) ◦ (ψA V ⊗ V ) ◦ (V ⊗ ν) = ∇A⊗V ◦ (ηA ⊗ V ),

(µA ⊗ V ) ◦ (A ⊗ σA

V ) ◦ (ν ⊗ V ) = ∇A⊗V ◦ (ηA ⊗ V ),

(µA ⊗ V ) ◦ (A ⊗ ψA

V ) ◦ (ν ⊗ A) = (µA ⊗ V ) ◦ (A ⊗ ν).

In this case ν is a preunit for µA⊗V and ∇A⊗V = ∇ν

A⊗V .

Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products

Examples with ∇A⊗V = idA⊗V . Brzeziński, Commun. in Algebra (1997).

Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products

Examples with ∇A⊗V = idA⊗V . Brzeziński, Commun. in Algebra (1997). Blattner, Cohen and Montgomery, (crossed products for Hopf algebras), T. Am.

• Math. Soc. Soc. (1986)

Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products

Examples with ∇A⊗V = idA⊗V . Brzeziński, Commun. in Algebra (1997). Blattner, Cohen and Montgomery, (crossed products for Hopf algebras), T. Am.

• Math. Soc. Soc. (1986)

Doi and Takeuchi, (crossed products for Hopf algebras), Commun. Algebra (1986)

Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products

Examples with ∇A⊗V = idA⊗V . Brzeziński, Commun. in Algebra (1997). Blattner, Cohen and Montgomery, (crossed products for Hopf algebras), T. Am.

• Math. Soc. Soc. (1986)

Doi and Takeuchi, (crossed products for Hopf algebras), Commun. Algebra (1986) Majid, (crossed products for braided Hopf algebras), J. Algebra (1994)

Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products

Examples with ∇A⊗V = idA⊗V . Brzeziński, Commun. in Algebra (1997). Blattner, Cohen and Montgomery, (crossed products for Hopf algebras), T. Am.

• Math. Soc. Soc. (1986)

Doi and Takeuchi, (crossed products for Hopf algebras), Commun. Algebra (1986) Majid, (crossed products for braided Hopf algebras), J. Algebra (1994) Bespalov and Drabant, (crossed products for braided Hopf algebras), J. Algebra (1999)-(2001)

Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products

Examples with ∇A⊗V = idA⊗V . Brzeziński, Commun. in Algebra (1997). Blattner, Cohen and Montgomery, (crossed products for Hopf algebras), T. Am.

• Math. Soc. Soc. (1986)

Doi and Takeuchi, (crossed products for Hopf algebras), Commun. Algebra (1986) Majid, (crossed products for braided Hopf algebras), J. Algebra (1994) Bespalov and Drabant, (crossed products for braided Hopf algebras), J. Algebra (1999)-(2001) Lack and Street (wreath products associated to distributive laws), J. Pure Appl. Algebra (2002).

Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products

Examples with ∇A⊗V = idA⊗V . Brzeziński, Commun. in Algebra (1997). Blattner, Cohen and Montgomery, (crossed products for Hopf algebras), T. Am.

• Math. Soc. Soc. (1986)

Doi and Takeuchi, (crossed products for Hopf algebras), Commun. Algebra (1986) Majid, (crossed products for braided Hopf algebras), J. Algebra (1994) Bespalov and Drabant, (crossed products for braided Hopf algebras), J. Algebra (1999)-(2001) Lack and Street (wreath products associated to distributive laws), J. Pure Appl. Algebra (2002). Agore and Militaru (unified products for Hopf algebras), Contemp. Math. (2013).

Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products

Examples with ∇A⊗V = idA⊗V . Caenepeel and De Groot, (weak smash products) Contemp. Math. (2000).

Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products

Examples with ∇A⊗V = idA⊗V . Caenepeel and De Groot, (weak smash products) Contemp. Math. (2000). Alonso Álvarez, Fernández Vilaboa, González Rodríguez and Rodríguez Raposo (weak crossed products for weak Hopf algebras), Commun. Algebra (2009), Ho-

• mol. Homotopy Appl. (2014).

Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products

Examples with ∇A⊗V = idA⊗V . Caenepeel and De Groot, (weak smash products) Contemp. Math. (2000). Alonso Álvarez, Fernández Vilaboa, González Rodríguez and Rodríguez Raposo (weak crossed products for weak Hopf algebras), Commun. Algebra (2009), Ho-

• mol. Homotopy Appl. (2014).

Street (weak wreath products associated to weak distributive laws), Theor. Appl.

• Categ. (2009).

Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products

Examples with ∇A⊗V = idA⊗V . Caenepeel and De Groot, (weak smash products) Contemp. Math. (2000). Alonso Álvarez, Fernández Vilaboa, González Rodríguez and Rodríguez Raposo (weak crossed products for weak Hopf algebras), Commun. Algebra (2009), Ho-

• mol. Homotopy Appl. (2014).

Street (weak wreath products associated to weak distributive laws), Theor. Appl.

• Categ. (2009).

Alves, Batista, Dokuchaev and Paques, (partial crossed products associated to partial actions of Hopf algebras), Isr. J. Math. (2013).

Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products

• G. Böhm, The weak theory of monads, Adv. Math. (2010).

Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products

• G. Böhm, The weak theory of monads, Adv. Math. (2010).

Let K be a 2-category. In the previous paper, the 2-category EMw(K) was introduced and the following result was proved: If K is the one-object 2-category corresponding to the category C, every weak crossed product with preunit is a monad in EMw(K).

Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products

Equivalences between weak crossed products

1

Weak crossed products

2

Equivalences between weak crossed products

3

Iterations for weak crossed products

Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products

Definition Let (A⊗V , µA⊗V ) and (A⊗W , µA⊗W ) be weak crossed products with preunits νV and νW respectively. We will say that (A ⊗ V , µA⊗V ) and (A ⊗ W , µA⊗W ) are equivalent if there exists a monoid isomorphism α : A × V → A × W

• f left A-modules for the actions

ϕA×V = pA⊗V ◦ (µA ⊗ V ) ◦ (A ⊗ iA⊗V ), ϕA×W = pA⊗W ◦ (µA ⊗ W ) ◦ (A ⊗ iA⊗W ).

Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products

Theorem Let (A ⊗ V , µA⊗V ) and (A ⊗ W , µA⊗W ) be weak crossed products with preunits νV and νW respectively. The following assertions are equivalent: (i) The weak crossed products (A ⊗ V , µA⊗V ) and (A ⊗ W , µA⊗W ) are equivalent. (ii) There exist two morphisms T : A ⊗ V → A ⊗ W , S : A ⊗ W → A ⊗ V of left A-modules for the actions ϕA⊗V , ϕA⊗W satisfying the conditions T ◦ νV = νW , T ◦ µA⊗V = µA⊗W ◦ (T ⊗ T), S ◦ T = ∇A⊗V , T ◦ S = ∇A⊗W , (iii) There exist two morphisms γ : V → A ⊗ W , θ : W → A ⊗ V satisfying the conditions νV = (µA ⊗ V ) ◦ (A ⊗ θ) ◦ νW , θ = ∇A⊗V ◦ θ, ψA

W = (µA ⊗ W ) ◦ (µA ⊗ γ) ◦ (A ⊗ ψA V ) ◦ (θ ⊗ A),

σA

W = (µA ⊗ W ) ◦ (A ⊗ γ) ◦ µA⊗V ◦ (θ ⊗ θ),

(µA ⊗ V ) ◦ (A ⊗ θ) ◦ γ = ∇A⊗V ◦ (ηA ⊗ V ).

Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products

Theorem Let (A ⊗ V , µA⊗V ) and (A ⊗ W , µA⊗W ) be weak crossed products with preunits νV and νW respectively. The following assertions are equivalent: (i) The weak crossed products (A ⊗ V , µA⊗V ) and (A ⊗ W , µA⊗W ) are equivalent. (ii) There exist two morphisms T : A ⊗ V → A ⊗ W , S : A ⊗ W → A ⊗ V of left A-modules for the actions ϕA⊗V , ϕA⊗W satisfying the conditions T ◦ νV = νW , T ◦ µA⊗V = µA⊗W ◦ (T ⊗ T), S ◦ T = ∇A⊗V , T ◦ S = ∇A⊗W , (iii) There exist two morphisms γ : V → A ⊗ W , θ : W → A ⊗ V satisfying the conditions νV = (µA ⊗ V ) ◦ (A ⊗ θ) ◦ νW , θ = ∇A⊗V ◦ θ, ψA

W = (µA ⊗ W ) ◦ (µA ⊗ γ) ◦ (A ⊗ ψA V ) ◦ (θ ⊗ A),

σA

W = (µA ⊗ W ) ◦ (A ⊗ γ) ◦ µA⊗V ◦ (θ ⊗ θ),

(µA ⊗ V ) ◦ (A ⊗ θ) ◦ γ = ∇A⊗V ◦ (ηA ⊗ V ). Note that for S and T we have T ◦ S ◦ T = T and S ◦ T ◦ S = S.

Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products

The case of Brzeziński crossed products (Brzeziński Commun. in Algebra (1997)) Let (A, ηA, µA) be a monoid and V an object equipped with a distinguished morphism ηV : K → V . The object A ⊗ V is a monoid with unit ηA ⊗ ηV and whose product has the property µA⊗V ◦ (A ⊗ ηV ⊗ A ⊗ V ) = µA ⊗ V , if and only if there exists two morphisms ψA

V : V ⊗ A → A ⊗ V , σA V : V ⊗ V → A ⊗ V satisfying (1), the twisted condition, the

cocycle condition and ψA

V ◦ (ηV ⊗ A) = A ⊗ ηV , ψA V ◦ (V ⊗ ηA) = ηA ⊗ V ,

σA

V ◦ (ηV ⊗ V ) = σA V ◦ (V ⊗ ηV ) = ηA ⊗ V .

If this is the case, the product of A ⊗ V is the one defined in the first section of this talk. Note that Brzeziński’s crossed products are examples of weak crossed products where the associated idempotent is ∇A⊗V = idA⊗V . Also, in this case the preunit νA⊗V = ηA ⊗ ηV is a unit.

Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products

Theorem (our result for Brzeziński’s crossed products) Let (A ⊗ V , µA⊗V ) and (A ⊗ W , µA⊗W ) be Brzeziński’s crossed products with dis- tinguished morphism ηV and ηW respectively. The following assertions are equivalent: (i) The crossed products (A ⊗ V , µA⊗V ) and (A ⊗ W , µA⊗W ) are equivalent. (ii) There exist two morphisms γ : V → A ⊗ W , θ : W → A ⊗ V satisfying the conditions ψA

W = (µA ⊗ W ) ◦ (µA ⊗ γ) ◦ (A ⊗ ψA V ) ◦ (θ ⊗ A),

σA

W = (µA ⊗ W ) ◦ (A ⊗ γ) ◦ µA⊗V ◦ (θ ⊗ θ),

(µA ⊗ V ) ◦ (A ⊗ θ) ◦ γ = ηA ⊗ V .

Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products

Improving other results: Equivalences for Brzeziński’s crossed products

• F. Panaite, Invariance under twisting for crossed products, P. Am. Math. Soc.

(2012).

• F. Panaite, Equivalent crossed products and cross product bialgebras, Commun.

Algebra (2014).

Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products

Panaite considered two Brzeziński’s crossed products (A ⊗ V , µ1

A⊗V ), (A ⊗ V , µ2 A⊗V ),

associated to the quadruples A1

V = (A, V , ψA,1 V

, σA,1

V

), A2

V = (A, V , ψA,2 V

, σA,2

V

), with distinguished morphism η1

V and η2 V .

Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products

Panaite considered two Brzeziński’s crossed products (A ⊗ V , µ1

A⊗V ), (A ⊗ V , µ2 A⊗V ),

associated to the quadruples A1

V = (A, V , ψA,1 V

, σA,1

V

), A2

V = (A, V , ψA,2 V

, σA,2

V

), with distinguished morphism η1

V and η2 V .

He proved that they are equivalent if, and only if, there exist morphisms γ, θ : V → A⊗V such that the following equalities are satisfied: ψA,2

V

= (µA ⊗ V ) ◦ (µA ⊗ γ) ◦ (A ⊗ ψA,1

V

) ◦ (θ ⊗ A), σA,2

V

= (µA ⊗ V ) ◦ (A ⊗ γ) ◦ µ1

A⊗V ◦ (θ ⊗ θ),

θ ◦ η2

V = ηA ⊗ η1 V , γ ◦ η1 V = ηA ⊗ η2 V ,

(µA ⊗ V ) ◦ (A ⊗ θ) ◦ γ = ηA ⊗ V , (µA ⊗ V ) ◦ (A ⊗ γ) ◦ θ = ηA ⊗ V , (µA ⊗V )◦(µA ⊗σA,2

V

)◦(A⊗γ ⊗V )◦(ψA,1

V

⊗V )◦(V ⊗γ) = (µA ⊗V )◦(A⊗γ)◦σA,1

V

.

Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products

Panaite considered two Brzeziński’s crossed products (A ⊗ V , µ1

A⊗V ), (A ⊗ V , µ2 A⊗V ),

associated to the quadruples A1

V = (A, V , ψA,1 V

, σA,1

V

), A2

V = (A, V , ψA,2 V

, σA,2

V

), with distinguished morphism η1

V and η2 V .

He proved that they are equivalent if, and only if, there exist morphisms γ, θ : V → A⊗V such that the following equalities are satisfied: ψA,2

V

= (µA ⊗ V ) ◦ (µA ⊗ γ) ◦ (A ⊗ ψA,1

V

) ◦ (θ ⊗ A), σA,2

V

= (µA ⊗ V ) ◦ (A ⊗ γ) ◦ µ1

A⊗V ◦ (θ ⊗ θ),

θ ◦ η2

V = ηA ⊗ η1 V , γ ◦ η1 V = ηA ⊗ η2 V ,

(µA ⊗ V ) ◦ (A ⊗ θ) ◦ γ = ηA ⊗ V , (µA ⊗ V ) ◦ (A ⊗ γ) ◦ θ = ηA ⊗ V , (µA ⊗V )◦(µA ⊗σA,2

V

)◦(A⊗γ ⊗V )◦(ψA,1

V

⊗V )◦(V ⊗γ) = (µA ⊗V )◦(A⊗γ)◦σA,1

V

.

Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products

Iterations for weak crossed products

1

Weak crossed products

2

Equivalences between weak crossed products

3

Iterations for weak crossed products

Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products

Definition Let AV = (A, V , ψA

V , σA V ) and AW = (A, W , ψA W , σA W ) be two quadruples. We say that

∆V ⊗W : V ⊗ W → V ⊗ W is a link morphism between AV and AW if the following conditions hold: ψA

V ⊗W = (A ⊗ ∆V ⊗W ) ◦ ψA V ⊗W ,

ψA

V ⊗W = ∇A⊗V ⊗W ◦ (ψA V ⊗ W ) ◦ (V ⊗ ψA W ),

where ψA

V ⊗W = (ψA V ⊗ W ) ◦ (V ⊗ ψA W ) ◦ (∆V ⊗W ⊗ A).

and ∇A⊗V ⊗W : A ⊗ V ⊗ W → A ⊗ V ⊗ W is the morphism defined by ∇A⊗V ⊗W = (µA ⊗ V ⊗ W ) ◦ (A ⊗ ψA

V ⊗W ) ◦ (A ⊗ V ⊗ W ⊗ ηA).

Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products

Proposition Let AV = (A, V , ψA

V , σA V ) and AW = (A, W , ψA W , σA W ) be two quadruples. If there

exists a link morphism ∆V ⊗W : V ⊗W → V ⊗W between them, the morphism ψA

V ⊗W

introduced in the previous definition satisfies (1) and, as a consequence, ∇A⊗V ⊗W is an idempotent morphism and the following identity holds: ψA

V ⊗W = ∇A⊗V ⊗W ◦ ψA V ⊗W .

Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products

Definition Let AV = (A, V , ψA

V , σA V ) and AW = (A, W , ψA W , σA W ) be two quadruples. We say that

τ V

W : W ⊗ V → V ⊗ W

is a twisting morphism between AV and AW if the following conditions hold: (i) (ψA

V ⊗ W ) ◦ (V ⊗ ψA W ) ◦ (τ V W ⊗ A) = (A ⊗ τ V W ) ◦ (ψA W ⊗ V ) ◦ (W ⊗ ψA V ).

(ii) (µA ⊗ V ⊗ W ) ◦ (A ⊗ σA

V ⊗ W ) ◦ (ψA V ⊗ τ V W ) ◦ (V ⊗ σA W ⊗ V ) ◦ (τ V W ⊗ W ⊗ V ) =

(µA ⊗ V ⊗ W ) ◦ (A ⊗ ψA

V ⊗ W ) ◦ (A ⊗ V ⊗ σA W ) ◦ (A ⊗ τ V W ⊗ W ) ◦ (ψA W ⊗ V ⊗ W )

• (V ⊗ σA

V ⊗ W ) ◦ (W ⊗ V ⊗ τ V W ).

Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products

Theorem Let AV = (A, V , ψA

V , σA V ), AW = (A, V , ψA W , σA W ) be two quadruples satisfying the

twisted condition, the cocycle condition, with a link morphism ∆V ⊗W : V ⊗ W → V ⊗ W , and with a twisting morphism τ V

W : W ⊗ V → V ⊗ W between them. Then if

we define σA

V ⊗W : V ⊗ W ⊗ V ⊗ W → A ⊗ V ⊗ W by

σA

V ⊗W = (µA ⊗ V ⊗ W ) ◦ (A ⊗ ψA V ⊗ W ) ◦ (σA V ⊗ σA W ) ◦ (V ⊗ τ V W ⊗ W )

and the equalities σA

V ⊗W = σA V ⊗W ◦ (∆V ⊗W ⊗ V ⊗ W ) = σA V ⊗W ◦ (V ⊗ W ⊗ ∆V ⊗W )

= (A ⊗ ∆V ⊗W ) ◦ σA

V ⊗W ,

hold, AV ⊗W = (A, V ⊗ W , ψA

V ⊗W , σA V ⊗W ), is a quadruple that satisfies the twisted

and cocycle conditions, and ∇A⊗V ⊗W ◦ σA

V ⊗W = σA V ⊗W . As a consequence,

(A ⊗ V ⊗ W , µA⊗V ⊗W ) is a weak crossed product with µA⊗V ⊗W = (µA ⊗ V ⊗ W ) ◦ (µA ⊗ σA

V ⊗W ) ◦ (A ⊗ ψA V ⊗W ⊗ V ).

Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products

Definition The weak crossed product (A ⊗ V ⊗ W , µA⊗V ⊗W ) will be called the iterated weak crossed product of (A⊗V , µA⊗V ) and (A⊗W , µA⊗W ).

Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products

Proposition Let AV = (A, V , ψA

V , σA V ), AW = (A, W , ψA W , σA W ) be two quadruples with preunits

νV : K → A ⊗ V and νW : K → A ⊗ W . Assume that the conditions of the previous theorem hold. If the following equalities hold (µA ⊗ V ⊗ W ) ◦ (A ⊗ σA

V ⊗ W ) ◦ (ψA V ⊗ τ V W ) ◦ (V ⊗ ψA W ⊗ V ) ◦ (∆V ⊗W ⊗ νV )

= ∇A⊗V ⊗W ◦ (ηA ⊗ V ⊗ W ), (µA ⊗ V ⊗ W ) ◦ (A ⊗ ψA

V ⊗ W ) ◦ (A ⊗ V ⊗ σA W ) ◦ (A ⊗ τ V W ⊗ W ) ◦ (νW ⊗ V ⊗ W )

= ∇A⊗V ⊗W ◦ (ηA ⊗ V ⊗ W ), the iterated weak crossed product of (A⊗V , µA⊗V ) and (A⊗W , µA⊗W ) has a preunit defined by νV ⊗W = ∇A⊗V ⊗W ◦ (µA ⊗ V ⊗ W ) ◦ (A ⊗ ψA

V ⊗ W ) ◦ (νV ⊗ νW ).

Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products

Example: Wreath products In End(C) we will denote the composition of functors by ⊚. The morphisms in End(C) are natural transformations and we denote the composition (the vertical composition)

• f these morphisms by ◦. The tensor product of morphisms in End(C) is defined by

the horizontal composition of natural transformations and will be denoted by the same symbol ⊚ used for the composition of functors. The notion of wreath was introduced by Lack and Street (J. Pure Appl. Algebra (2002)). A monad S in C is a wreath if there exist an object in T ∈ End(C) and morphisms in End(C), λ : T ⊚ S → S ⊚ T, τ : idC → S ⊚ T, v : T ⊚ T → S ⊚ T satisfying the following conditions: (µS ⊚ T) ◦ (S ⊚ λ) ◦ (λ ⊚ S) = λ ◦ (T ⊚ µS), λ ◦ (T ⊚ ηS) = ηS ⊚ T, (µS ⊚ T) ◦ (S ⊚ τ) = (µS ⊚ T) ◦ (S ⊚ λ) ◦ (τ ⊚ S), (µS ⊚ T) ◦ (S ⊚ v) ◦ (λ ⊚ T) ◦ (T ⊚ λ) = (µS ⊚ T) ◦ (S ⊚ λ) ◦ (v ⊚ S), (µS ⊚ T) ◦ (S ⊚ v) ◦ (v ⊚ T) = (µS ⊚ T) ◦ (S ⊚ v) ◦ (λ ⊚ T) ◦ (T ⊚ v), (µS ⊚ T) ◦ (S ⊚ v) ◦ (τ ⊚ T) = ηS ⊚ T = (µS ⊚ T) ◦ (S ⊚ v) ◦ (λ ⊚ T) ◦ (T ⊚ τ).

Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products

If we put ψS

T = λ and σS T = v, we obtain that

ST = (S, T, ψS

T , σS T )

is a quadruple satisfying the twisted and the cocycle conditions, where the associated idempotent is ∇S⊚T = idS⊚T . Then, the product induced by a wreath (wreath product) defined by µS⊚T = (µS ⊚ T) ◦ (µS ⊚ v) ◦ (S ⊚ λ ⊚ T) is an example of weak crossed product. Also, S ⊚ T is a monad with unit ηS⊚T = τ.

Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products

An example of wreath products cames from the notion of distributive law introduced by Beck (LNM (1969)). Suppose that T and S are two monads on C. A distributive law

• f the monad S over the monad T is a natural transformation

λ : T ⊚ S → S ⊚ T such that λ ◦ (µT ⊚ S) = (S ⊚ µT ) ◦ (λ ⊚ T) ◦ (T ⊚ λ), λ ◦ (ηT ⊚ S) = S ⊚ ηT , λ ◦ (T ⊚ µS) = (µS ⊚ T) ◦ (S ⊚ λ) ◦ (λ ⊚ S), λ ◦ (T ⊚ ηS) = ηS ⊚ T.

Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products

An example of wreath products cames from the notion of distributive law introduced by Beck (LNM (1969)). Suppose that T and S are two monads on C. A distributive law

• f the monad S over the monad T is a natural transformation

λ : T ⊚ S → S ⊚ T such that λ ◦ (µT ⊚ S) = (S ⊚ µT ) ◦ (λ ⊚ T) ◦ (T ⊚ λ), λ ◦ (ηT ⊚ S) = S ⊚ ηT , λ ◦ (T ⊚ µS) = (µS ⊚ T) ◦ (S ⊚ λ) ◦ (λ ⊚ S), λ ◦ (T ⊚ ηS) = ηS ⊚ T. Then, if τ = ηS ⊚ ηT and v = ηS ⊚ µT we obtain a wreath for the monad S and also a weak crossed product associated to the quadruple ST = (S, T, ψS

T , σS T ) where ψS T = λ,

σS

T = ηS ⊚ µT and

µS⊚T = (µS ⊚ µT ) ◦ (S ⊚ λ ⊚ T).

Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products

Suppose that S, T and D are monads in C such that there exists the following distributive laws between them λ1 : T ⊚ S → S ⊚ T, λ2 : D ⊚ T → T ⊚ D, λ3 : D ⊚ S → S ⊚ D, satisfying the compatibility identity (called the Yang-Baxter relation or the hexagon equation) (S ⊚ λ2) ◦ (λ3 ⊚ T) ◦ (D ⊚ λ1) = (λ1 ⊚ D) ◦ (T ⊚ λ3) ◦ (λ2 ⊚ S).

Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products

Suppose that S, T and D are monads in C such that there exists the following distributive laws between them λ1 : T ⊚ S → S ⊚ T, λ2 : D ⊚ T → T ⊚ D, λ3 : D ⊚ S → S ⊚ D, satisfying the compatibility identity (called the Yang-Baxter relation or the hexagon equation) (S ⊚ λ2) ◦ (λ3 ⊚ T) ◦ (D ⊚ λ1) = (λ1 ⊚ D) ◦ (T ⊚ λ3) ◦ (λ2 ⊚ S). Then, under these conditions we have two quadruples ST = (S, T, ψS

T = λ1, σS T = ηS ⊚ µT ),

SD = (S, D, ψS

D = λ3, σS D = ηS ⊚ µD),

satisfying the twisted and cocycle conditions. If we put ∆T⊚D = idT⊚D, τ T

D = λ2

we have that ∆T⊚D is a link morphism and τ T

D = λ2 is a twisting morphism. Also the

conditions of the main theorem of this section hold.

Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products

Then, the pair, (S ⊚ T ⊚ D, µS⊚T⊚D) is the iterated weak crossed of (S ⊚ T, µS⊚T ) and (S ⊚ D, µS⊚D) with associated product µS⊚T⊚D = (µS ⊚ T ⊚ D) ◦ (µS ⊚ σS

T⊚D) ◦ (S ⊚ ψS T⊚D ⊚ T ⊚ D) =

(µS ⊚ µT ⊚ µD) ◦ (S ⊚ ((λ1 ⊚ λ2) ◦ (T ⊚ λ3 ⊚ T)) ⊚ D).

Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products

Then, the pair, (S ⊚ T ⊚ D, µS⊚T⊚D) is the iterated weak crossed of (S ⊚ T, µS⊚T ) and (S ⊚ D, µS⊚D) with associated product µS⊚T⊚D = (µS ⊚ T ⊚ D) ◦ (µS ⊚ σS

T⊚D) ◦ (S ⊚ ψS T⊚D ⊚ T ⊚ D) =

(µS ⊚ µT ⊚ µD) ◦ (S ⊚ ((λ1 ⊚ λ2) ◦ (T ⊚ λ3 ⊚ T)) ⊚ D). In this case the preunits are units. The object S ⊚ T ⊚ D is a monad with unit ηS⊚T⊚D = ηS ⊚ ηT ⊚ ηD because S⊚T and S⊚D are also monads with unit ηS⊚T = ηS⊚ηT and ηS⊚D = ηS⊚ηD respectively.

Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products

Then, the pair, (S ⊚ T ⊚ D, µS⊚T⊚D) is the iterated weak crossed of (S ⊚ T, µS⊚T ) and (S ⊚ D, µS⊚D) with associated product µS⊚T⊚D = (µS ⊚ T ⊚ D) ◦ (µS ⊚ σS

T⊚D) ◦ (S ⊚ ψS T⊚D ⊚ T ⊚ D) =

(µS ⊚ µT ⊚ µD) ◦ (S ⊚ ((λ1 ⊚ λ2) ◦ (T ⊚ λ3 ⊚ T)) ⊚ D). In this case the preunits are units. The object S ⊚ T ⊚ D is a monad with unit ηS⊚T⊚D = ηS ⊚ ηT ⊚ ηD because S⊚T and S⊚D are also monads with unit ηS⊚T = ηS⊚ηT and ηS⊚D = ηS⊚ηD respectively. Particular cases

• J. López Peña, F. Panaite, F. van Oystaeyen, General twisting of algebras, Adv.
• Math. (2007).
• P. Jara Martínez, J. López Peña, F. Panaite, F. van Oystaeyen, On iterated tensor

product of algebras, Internat. J. Math. (2008).

Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products

Example: Weak wreath products Given to monads S and T, the notion of weak distributive law of the monad S over the monad T was introduced by Street (Theor. Appl. Categ. (2009)) as follows. It consists of a natural transformation λ : T ⊚ S → S ⊚ T such that satisfies λ ◦ (µT ⊚ S) = (S ⊚ µT ) ◦ (λ ⊚ T) ◦ (T ⊚ λ), λ ◦ (ηT ⊚ S) = (µS ⊚ T) ◦ (S ⊚ (λ ◦ (ηT ⊚ ηS))), λ ◦ (T ⊚ µS) = (µS ⊚ T) ◦ (S ⊚ λ) ◦ (λ ⊚ S), λ ◦ (T ⊚ ηS) = (S ⊚ µT ) ◦ ((λ ◦ (ηT ⊚ ηS)) ⊚ T).

Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products

Example: Weak wreath products Given to monads S and T, the notion of weak distributive law of the monad S over the monad T was introduced by Street (Theor. Appl. Categ. (2009)) as follows. It consists of a natural transformation λ : T ⊚ S → S ⊚ T such that satisfies λ ◦ (µT ⊚ S) = (S ⊚ µT ) ◦ (λ ⊚ T) ◦ (T ⊚ λ), λ ◦ (ηT ⊚ S) = (µS ⊚ T) ◦ (S ⊚ (λ ◦ (ηT ⊚ ηS))), λ ◦ (T ⊚ µS) = (µS ⊚ T) ◦ (S ⊚ λ) ◦ (λ ⊚ S), λ ◦ (T ⊚ ηS) = (S ⊚ µT ) ◦ ((λ ◦ (ηT ⊚ ηS)) ⊚ T). For a weak distributive law, the weak wreath product is defined by µS⊚T = (µS ⊚ µT ) ◦ (S ⊚ λ ⊚ T).

Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products

If we take the quadruple ST = (S, T, ψS

T = λ, σS T = (S ⊚ µT ) ◦ ((λ ◦ (T ⊚ ηS)) ⊚ T)),

we obtain that ST satisfies the twisted and cocycle conditions and ∇S⊚T ◦ σS

T = σS T ,

where the associated idempotent is ∇S⊚T = (µS ⊚ T) ◦ (S ⊚ (λ ◦ (T ⊚ ηS))).

Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products

If we take the quadruple ST = (S, T, ψS

T = λ, σS T = (S ⊚ µT ) ◦ ((λ ◦ (T ⊚ ηS)) ⊚ T)),

we obtain that ST satisfies the twisted and cocycle conditions and ∇S⊚T ◦ σS

T = σS T ,

where the associated idempotent is ∇S⊚T = (µS ⊚ T) ◦ (S ⊚ (λ ◦ (T ⊚ ηS))). Then, the weak wreath product defined by the weak distributive law is the one induced by the quadruple ST . Therefore, every weak wreath product is a weak crossed product. In this setting the morphism νT = ∇S⊚T ◦(ηS ⊚ηT ) is a preunit and S ×T is a monoid with unit ηS×T = pS⊚T ◦ νT

Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products

Suppose that S, T and D are monads in C such that there exists tree weak distributive laws between them λ1 : T ⊚ S → S ⊚ T, λ2 : D ⊚ T → T ⊚ D, λ3 : D ⊚ S → S ⊚ D, satisfying the Yang-Baxter relation of the previous example. Then, under these condi- tions we have two quadruples ST = (S, T, ψS

T = λ1, σS T = (S ⊚ µT ) ◦ ((λ1 ◦ (T ⊚ ηS)) ⊚ T)),

SD = (S, D, ψS

D = λ3, σS D = (S ⊚ µD) ◦ ((λ3 ◦ (D ⊚ ηS)) ⊚ D)),

satisfying the twisted and cocycle conditions and ∇S⊚T ◦ σS

T = σS T , ∇S⊚D ◦ σS D = σS D

Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products

Suppose that S, T and D are monads in C such that there exists tree weak distributive laws between them λ1 : T ⊚ S → S ⊚ T, λ2 : D ⊚ T → T ⊚ D, λ3 : D ⊚ S → S ⊚ D, satisfying the Yang-Baxter relation of the previous example. Then, under these condi- tions we have two quadruples ST = (S, T, ψS

T = λ1, σS T = (S ⊚ µT ) ◦ ((λ1 ◦ (T ⊚ ηS)) ⊚ T)),

SD = (S, D, ψS

D = λ3, σS D = (S ⊚ µD) ◦ ((λ3 ◦ (D ⊚ ηS)) ⊚ D)),

satisfying the twisted and cocycle conditions and ∇S⊚T ◦ σS

T = σS T , ∇S⊚D ◦ σS D = σS D

If we put ∆T⊚D = ∇T⊚D we obtain a link morphism. Also, τ T

D = λ2 is a twisting

morphism and the quadruple ST⊚D = (S, T ⊚ D, ψS

T⊚D, σS T⊚D),

where ψS

T⊚D = (λ1 ⊚ D) ◦ (T ⊚ λ3) ◦ (∇T⊚D ⊚ S),

σS

T⊚D = (µS ⊚ T ⊚ D) ◦ (S ⊚ ψS T ⊚ D) ◦ (σS T ⊚ σS D) ◦ (T ⊚ τ T D ⊗ D)

Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products

induces the iterated weak crossed product of (S ⊚ T, µS⊚T ) and (S ⊚ D, µS⊚D), (S ⊚ T ⊚ D, µS⊚T⊚D), with associated product µS⊚T⊚D = (µS ⊚ T ⊚ D) ◦ (µS ⊚ σS

T⊚D) ◦ (S ⊚ ψS T⊚D ⊚ T ⊚ D),

and equivalently µS⊚T⊚D = (µS ⊚ µT ⊚ µD) ◦ (S ⊚ ((λ1 ⊚ λ2) ◦ (T ⊚ λ3 ⊚ T) ◦ (∇T⊚D ⊚ ∇S⊚T )) ⊚ D).

Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products

induces the iterated weak crossed product of (S ⊚ T, µS⊚T ) and (S ⊚ D, µS⊚D), (S ⊚ T ⊚ D, µS⊚T⊚D), with associated product µS⊚T⊚D = (µS ⊚ T ⊚ D) ◦ (µS ⊚ σS

T⊚D) ◦ (S ⊚ ψS T⊚D ⊚ T ⊚ D),

and equivalently µS⊚T⊚D = (µS ⊚ µT ⊚ µD) ◦ (S ⊚ ((λ1 ⊚ λ2) ◦ (T ⊚ λ3 ⊚ T) ◦ (∇T⊚D ⊚ ∇S⊚T )) ⊚ D). Also the preunit conditions hold. Therefore νT⊚D = ∇S⊚T⊚D ◦ (µS ⊚ T ⊚ D) ◦ (S ⊚ λ1 ⊚ D) ◦ (νT ⊚ νD) is a preunit for µS⊚T⊚D and we have that νT⊚D = (λ1 ⊚ D) ◦ (T ⊚ λ3) ◦ (λ2 ⊚ S) ◦ (ηD ⊚ ηT ⊚ ηS).

Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products

Example: Iterations for Brzeziński’s crossed products Given two Brzeziński’s crossed products for A ⊗ V and A ⊗ W , Dăuş and Panaite (arXiv:1502.00031 (2015)) defined an iterated crossed product in A ⊗ V ⊗ W if there exists a morphism τ V

W : W ⊗ V → V ⊗ W satisfying the conditions

(ψA

V ⊗ W ) ◦ (V ⊗ ψA W ) ◦ (τ V W ⊗ A) = (A ⊗ τ V W ) ◦ (ψA W ⊗ V ) ◦ (W ⊗ ψA V ),

(A ⊗ τ V

W ) ◦ (ψA W ⊗ V ) ◦ (W ⊗ σA V ) = (σA V ⊗ W ) ◦ (V ⊗ τ V W ) ◦ (τ V W ⊗ V ),

(ψA

V ⊗ W ) ◦ (V ⊗ σA W ) ◦ (τ V W ⊗ W ) ◦ (W ⊗ τ V W ) = (A ⊗ τ V W ) ◦ (σA W ⊗ V ),

τ V

W ◦ (ηW ⊗ V ) = V ⊗ ηW ,

τ V

W ◦ (W ⊗ ηV ) = ηV ⊗ W .

Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products

Example: Iterations for Brzeziński’s crossed products Given two Brzeziński’s crossed products for A ⊗ V and A ⊗ W , Dăuş and Panaite (arXiv:1502.00031 (2015)) defined an iterated crossed product in A ⊗ V ⊗ W if there exists a morphism τ V

W : W ⊗ V → V ⊗ W satisfying the conditions

(ψA

V ⊗ W ) ◦ (V ⊗ ψA W ) ◦ (τ V W ⊗ A) = (A ⊗ τ V W ) ◦ (ψA W ⊗ V ) ◦ (W ⊗ ψA V ),

(A ⊗ τ V

W ) ◦ (ψA W ⊗ V ) ◦ (W ⊗ σA V ) = (σA V ⊗ W ) ◦ (V ⊗ τ V W ) ◦ (τ V W ⊗ V ),

(ψA

V ⊗ W ) ◦ (V ⊗ σA W ) ◦ (τ V W ⊗ W ) ◦ (W ⊗ τ V W ) = (A ⊗ τ V W ) ◦ (σA W ⊗ V ),

τ V

W ◦ (ηW ⊗ V ) = V ⊗ ηW ,

τ V

W ◦ (W ⊗ ηV ) = ηV ⊗ W .

The morphism ∆V ⊗W = idV ⊗W is a link morphism and τ V

W is a twisting morphism.

Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products

In this case, the iterated crossed product is the one induced by the quadruple AV ⊗W = (A, V ⊗ W , ψA

V ⊗W , σA V ⊗W ),

where ψA

V ⊗W = (ψA V ⊗ W ) ◦ (V ⊗ ψA W )

σA

V ⊗W = (µA ⊗ V ⊗ W ) ◦ (A ⊗ ψA V ⊗ W ) ◦ (σA V ⊗ σA W ) ◦ (V ⊗ τ V W ⊗ W )

ηV ⊗W = ηV ⊗ ηW

Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products

In this case, the iterated crossed product is the one induced by the quadruple AV ⊗W = (A, V ⊗ W , ψA

V ⊗W , σA V ⊗W ),

where ψA

V ⊗W = (ψA V ⊗ W ) ◦ (V ⊗ ψA W )

σA

V ⊗W = (µA ⊗ V ⊗ W ) ◦ (A ⊗ ψA V ⊗ W ) ◦ (σA V ⊗ σA W ) ◦ (V ⊗ τ V W ⊗ W )

ηV ⊗W = ηV ⊗ ηW Also, all conditions of the result about iterations of weak crossed products and preunits for them hold.Therefore, the iteration process proposed by Dăuş and Panaite is a par- ticular instance of the general iteration for weak crossed products.

Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products

References J.M. Fernández Vilaboa, R. González Rodríguez, R., A.B. Rodríguez Raposo, Ite- rated weak crossed products, arXiv:1503.01585 (2015) J.M. Fernández Vilaboa, R. González Rodríguez, R., A.B. Rodríguez Raposo, Equi- valences for weak crossed products, arXiv:1505.05532 (2015)

Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products

References J.M. Fernández Vilaboa, R. González Rodríguez, R., A.B. Rodríguez Raposo, Ite- rated weak crossed products, arXiv:1503.01585 (2015) J.M. Fernández Vilaboa, R. González Rodríguez, R., A.B. Rodríguez Raposo, Equi- valences for weak crossed products, arXiv:1505.05532 (2015)

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Ramón González Rodríguez Equivalences and iterations for weak crossed products