Crossed products of crossed modules of Hopf monoids in a braided - PowerPoint PPT Presentation

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections Let H be a Hopf monoid. An object M in C is said to be a left H -module if there is a morphism φ M : H ⊗ M → M in C satisfying that φ M ◦ ( η H ⊗ M ) = id M , φ M ◦ ( H ⊗ φ M ) = φ M ◦ ( µ H ⊗ M ) . Given two left H -modules ( M , φ M ) and ( N , φ N ) , f : M → N is a morphism of left H -modules if φ N ◦ ( H ⊗ f ) = f ◦ φ M . Let ( B , φ B ) be a left H -module. If B is a monoid and η B and µ B are left H -module morphisms, i.e., φ B ◦ ( H ⊗ η B ) = ε H ⊗ η B , φ B ◦ ( H ⊗ µ B ) = µ B ◦ ( φ B ⊗ φ B ) ◦ ( H ⊗ c H , B ⊗ B ) ◦ ( δ H ⊗ B ⊗ B ) , we will say that ( B , φ B ) is a left H -module monoid. If B is a comonoid and ε B and δ B are left H -module morphisms, i.e., ε B ◦ φ B = ε H ⊗ ε B , δ B ◦ φ B = ( φ B ⊗ φ B ) ◦ δ H ⊗ B , where δ H ⊗ B = ( H ⊗ c H , B ⊗ B ) ◦ ( δ H ⊗ δ B ) , ( B , φ B ) is said to be a left H -module comonoid. Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections If H is a Hopf monoid, B a monoid and f : H → B a monoid morphism, the adjoint action of H on B associated to f is defined as ad f , B = µ B ◦ ( µ B ⊗ B ) ◦ ( f ⊗ B ⊗ ( f ◦ λ H )) ◦ ( H ⊗ c H , B ) ◦ ( δ H ⊗ B ) . Then ( B , φ B ) is a left H -module monoid with φ B = ad f , B . Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections If H is a Hopf monoid, B a monoid and f : H → B a monoid morphism, the adjoint action of H on B associated to f is defined as ad f , B = µ B ◦ ( µ B ⊗ B ) ◦ ( f ⊗ B ⊗ ( f ◦ λ H )) ◦ ( H ⊗ c H , B ) ◦ ( δ H ⊗ B ) . Then ( B , φ B ) is a left H -module monoid with φ B = ad f , B . In particular, if B = H and f = id H the action defined above (called the adjoint action of H ) is the following: ad id H , H = µ H ◦ ( µ H ⊗ λ H ) ◦ ( H ⊗ c H , H ) ◦ ( δ H ⊗ H ) . In what follows we will denote this action by ad H . Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections Some definitions of crossed modules of Hopf monoids The setting 1 Some definitions of crossed modules of Hopf monoids 2 A new definition 3 Crossed products of crossed modules of Hopf monoids 4 Projections 5 Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections First definition The notion of crossed module of groups was introduced by Whitehead J.H.C. Whitehead. Combinatorial homotopy II, Bull. Amer. Math. Soc. 55, 453-496, 1949. in his investigation of the monoidal structure of second relative homotopy groups. Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections First definition The notion of crossed module of groups was introduced by Whitehead J.H.C. Whitehead. Combinatorial homotopy II, Bull. Amer. Math. Soc. 55, 453-496, 1949. in his investigation of the monoidal structure of second relative homotopy groups. Let B , H be groups and let β : B → H be a group morphism. Let φ B ( h , b ) = h b φ B : H × B → B , be an action of H over B . The triple B H = ( B , H , β ) is a crossed module of groups if the following identities hold: (i) β ( h b ) = h β ( b ) h − 1 . β ( b ) b ′ = bb ′ b − 1 (Peiffer identity). (ii) Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections First definition The notion of crossed module of groups was introduced by Whitehead J.H.C. Whitehead. Combinatorial homotopy II, Bull. Amer. Math. Soc. 55, 453-496, 1949. in his investigation of the monoidal structure of second relative homotopy groups. Let B , H be groups and let β : B → H be a group morphism. Let φ B ( h , b ) = h b φ B : H × B → B , be an action of H over B . The triple B H = ( B , H , β ) is a crossed module of groups if the following identities hold: (i) β ( h b ) = h β ( b ) h − 1 . β ( b ) b ′ = bb ′ b − 1 (Peiffer identity). (ii) Groups are Hopf monoids in the category Set . Then the previous definition is a definition of crossed module of Hopf monoids. Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections First definition The notion of crossed module of groups was introduced by Whitehead J.H.C. Whitehead. Combinatorial homotopy II, Bull. Amer. Math. Soc. 55, 453-496, 1949. in his investigation of the monoidal structure of second relative homotopy groups. Let B , H be groups and let β : B → H be a group morphism. Let φ B ( h , b ) = h b φ B : H × B → B , be an action of H over B . The triple B H = ( B , H , β ) is a crossed module of groups if the following identities hold: (i) β ( h b ) = h β ( b ) h − 1 . β ( b ) b ′ = bb ′ b − 1 (Peiffer identity). (ii) Groups are Hopf monoids in the category Set . Then the previous definition is a definition of crossed module of Hopf monoids. In this setting, H H = ( H , H , id H ) is an example of is a crossed module of groups with φ H ( h , b ) = hbh − 1 (the adjoint action). Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections Second definition Assume that C is symmetric with isomorphism of symmetry c . Let H be a cocom- mutative Hopf monoid in C and let B be a Hopf monoid in C . Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections Second definition Assume that C is symmetric with isomorphism of symmetry c . Let H be a cocom- mutative Hopf monoid in C and let B be a Hopf monoid in C . In J.M. Fernández Vilaboa, M.P. López López, E. Villanueva Nóvoa. Cat 1 -Hopf algebras and crossed modules, Comm. Algebra 35, 181-191, 2007. we can find a definition of crossed module of Hopf monoids. Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections Second definition Assume that C is symmetric with isomorphism of symmetry c . Let H be a cocom- mutative Hopf monoid in C and let B be a Hopf monoid in C . In J.M. Fernández Vilaboa, M.P. López López, E. Villanueva Nóvoa. Cat 1 -Hopf algebras and crossed modules, Comm. Algebra 35, 181-191, 2007. we can find a definition of crossed module of Hopf monoids. In the previous conditions, if β : B → H is a morphism of Hopf monoids, the triple B H = ( B , H , β ) is a crossed module of Hopf monoids if the following assertions hold: Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections Second definition Assume that C is symmetric with isomorphism of symmetry c . Let H be a cocom- mutative Hopf monoid in C and let B be a Hopf monoid in C . In J.M. Fernández Vilaboa, M.P. López López, E. Villanueva Nóvoa. Cat 1 -Hopf algebras and crossed modules, Comm. Algebra 35, 181-191, 2007. we can find a definition of crossed module of Hopf monoids. In the previous conditions, if β : B → H is a morphism of Hopf monoids, the triple B H = ( B , H , β ) is a crossed module of Hopf monoids if the following assertions hold: (i) ( β ⊗ B ) ◦ δ B = ( β ⊗ B ) ◦ c B , B ◦ δ B . Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections Second definition Assume that C is symmetric with isomorphism of symmetry c . Let H be a cocom- mutative Hopf monoid in C and let B be a Hopf monoid in C . In J.M. Fernández Vilaboa, M.P. López López, E. Villanueva Nóvoa. Cat 1 -Hopf algebras and crossed modules, Comm. Algebra 35, 181-191, 2007. we can find a definition of crossed module of Hopf monoids. In the previous conditions, if β : B → H is a morphism of Hopf monoids, the triple B H = ( B , H , β ) is a crossed module of Hopf monoids if the following assertions hold: (i) ( β ⊗ B ) ◦ δ B = ( β ⊗ B ) ◦ c B , B ◦ δ B . (ii) There exists a morphism φ B : H ⊗ B → B such that ( B , φ B ) is a left H -module monoid and comonoid. Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections Second definition Assume that C is symmetric with isomorphism of symmetry c . Let H be a cocom- mutative Hopf monoid in C and let B be a Hopf monoid in C . In J.M. Fernández Vilaboa, M.P. López López, E. Villanueva Nóvoa. Cat 1 -Hopf algebras and crossed modules, Comm. Algebra 35, 181-191, 2007. we can find a definition of crossed module of Hopf monoids. In the previous conditions, if β : B → H is a morphism of Hopf monoids, the triple B H = ( B , H , β ) is a crossed module of Hopf monoids if the following assertions hold: (i) ( β ⊗ B ) ◦ δ B = ( β ⊗ B ) ◦ c B , B ◦ δ B . (ii) There exists a morphism φ B : H ⊗ B → B such that ( B , φ B ) is a left H -module monoid and comonoid. (iii) The antipode of B is a morphism of left H -modules. Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections Second definition Assume that C is symmetric with isomorphism of symmetry c . Let H be a cocom- mutative Hopf monoid in C and let B be a Hopf monoid in C . In J.M. Fernández Vilaboa, M.P. López López, E. Villanueva Nóvoa. Cat 1 -Hopf algebras and crossed modules, Comm. Algebra 35, 181-191, 2007. we can find a definition of crossed module of Hopf monoids. In the previous conditions, if β : B → H is a morphism of Hopf monoids, the triple B H = ( B , H , β ) is a crossed module of Hopf monoids if the following assertions hold: (i) ( β ⊗ B ) ◦ δ B = ( β ⊗ B ) ◦ c B , B ◦ δ B . (ii) There exists a morphism φ B : H ⊗ B → B such that ( B , φ B ) is a left H -module monoid and comonoid. (iii) The antipode of B is a morphism of left H -modules. (iv) β ◦ φ B = ad H ◦ ( H ⊗ β ) . Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections Second definition Assume that C is symmetric with isomorphism of symmetry c . Let H be a cocom- mutative Hopf monoid in C and let B be a Hopf monoid in C . In J.M. Fernández Vilaboa, M.P. López López, E. Villanueva Nóvoa. Cat 1 -Hopf algebras and crossed modules, Comm. Algebra 35, 181-191, 2007. we can find a definition of crossed module of Hopf monoids. In the previous conditions, if β : B → H is a morphism of Hopf monoids, the triple B H = ( B , H , β ) is a crossed module of Hopf monoids if the following assertions hold: (i) ( β ⊗ B ) ◦ δ B = ( β ⊗ B ) ◦ c B , B ◦ δ B . (ii) There exists a morphism φ B : H ⊗ B → B such that ( B , φ B ) is a left H -module monoid and comonoid. (iii) The antipode of B is a morphism of left H -modules. (iv) β ◦ φ B = ad H ◦ ( H ⊗ β ) . (v) φ B ◦ ( β ⊗ B ) = ad H (Peiffer identity). Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections Second definition Assume that C is symmetric with isomorphism of symmetry c . Let H be a cocom- mutative Hopf monoid in C and let B be a Hopf monoid in C . In J.M. Fernández Vilaboa, M.P. López López, E. Villanueva Nóvoa. Cat 1 -Hopf algebras and crossed modules, Comm. Algebra 35, 181-191, 2007. we can find a definition of crossed module of Hopf monoids. In the previous conditions, if β : B → H is a morphism of Hopf monoids, the triple B H = ( B , H , β ) is a crossed module of Hopf monoids if the following assertions hold: (i) ( β ⊗ B ) ◦ δ B = ( β ⊗ B ) ◦ c B , B ◦ δ B . (ii) There exists a morphism φ B : H ⊗ B → B such that ( B , φ B ) is a left H -module monoid and comonoid. (iii) The antipode of B is a morphism of left H -modules. (iv) β ◦ φ B = ad H ◦ ( H ⊗ β ) . (v) φ B ◦ ( β ⊗ B ) = ad H (Peiffer identity). In this setting, H H = ( H , H , id H ) is an example of is a crossed module of Hopf monoids for φ H = ad H because C is symmetric and H is cocommutative. Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections Third definition Assume that Vect K is a category of vector spaces over a field K . Let H , B be Hopf monoids (algebras) in Vect K . Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections Third definition Assume that Vect K is a category of vector spaces over a field K . Let H , B be Hopf monoids (algebras) in Vect K . In Y. Frégier, F. Wagemann. On Hopf 2-algebras, Int. Math. Res. Notices 2011, 3471- 3501, 2011. we can find a definition of crossed module of Hopf monoids. Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections Third definition Assume that Vect K is a category of vector spaces over a field K . Let H , B be Hopf monoids (algebras) in Vect K . In Y. Frégier, F. Wagemann. On Hopf 2-algebras, Int. Math. Res. Notices 2011, 3471- 3501, 2011. we can find a definition of crossed module of Hopf monoids. In the previous conditions, if β : B → H is a morphism of Hopf monoids, the triple B H = ( B , H , β ) is a crossed module of Hopf monoids if the following assertions hold: Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections Third definition Assume that Vect K is a category of vector spaces over a field K . Let H , B be Hopf monoids (algebras) in Vect K . In Y. Frégier, F. Wagemann. On Hopf 2-algebras, Int. Math. Res. Notices 2011, 3471- 3501, 2011. we can find a definition of crossed module of Hopf monoids. In the previous conditions, if β : B → H is a morphism of Hopf monoids, the triple B H = ( B , H , β ) is a crossed module of Hopf monoids if the following assertions hold: (i) There exists a morphism φ B : H ⊗ B → B such that ( B , φ B ) is a left H -module monoid and comonoid. Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections Third definition Assume that Vect K is a category of vector spaces over a field K . Let H , B be Hopf monoids (algebras) in Vect K . In Y. Frégier, F. Wagemann. On Hopf 2-algebras, Int. Math. Res. Notices 2011, 3471- 3501, 2011. we can find a definition of crossed module of Hopf monoids. In the previous conditions, if β : B → H is a morphism of Hopf monoids, the triple B H = ( B , H , β ) is a crossed module of Hopf monoids if the following assertions hold: (i) There exists a morphism φ B : H ⊗ B → B such that ( B , φ B ) is a left H -module monoid and comonoid. (ii) β ◦ φ B = ad H ◦ ( H ⊗ β ) . Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections Third definition Assume that Vect K is a category of vector spaces over a field K . Let H , B be Hopf monoids (algebras) in Vect K . In Y. Frégier, F. Wagemann. On Hopf 2-algebras, Int. Math. Res. Notices 2011, 3471- 3501, 2011. we can find a definition of crossed module of Hopf monoids. In the previous conditions, if β : B → H is a morphism of Hopf monoids, the triple B H = ( B , H , β ) is a crossed module of Hopf monoids if the following assertions hold: (i) There exists a morphism φ B : H ⊗ B → B such that ( B , φ B ) is a left H -module monoid and comonoid. (ii) β ◦ φ B = ad H ◦ ( H ⊗ β ) . (iii) φ B ◦ ( β ⊗ B ) = ad H (Peiffer identity). Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections Third definition Assume that Vect K is a category of vector spaces over a field K . Let H , B be Hopf monoids (algebras) in Vect K . In Y. Frégier, F. Wagemann. On Hopf 2-algebras, Int. Math. Res. Notices 2011, 3471- 3501, 2011. we can find a definition of crossed module of Hopf monoids. In the previous conditions, if β : B → H is a morphism of Hopf monoids, the triple B H = ( B , H , β ) is a crossed module of Hopf monoids if the following assertions hold: (i) There exists a morphism φ B : H ⊗ B → B such that ( B , φ B ) is a left H -module monoid and comonoid. (ii) β ◦ φ B = ad H ◦ ( H ⊗ β ) . (iii) φ B ◦ ( β ⊗ B ) = ad H (Peiffer identity). In this setting, H H = ( H , H , id H ) is not an example of is a crossed module for φ H = ad H . Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections Fourth definition Assume that Vect K is a category of vector spaces over a field K . Let H , B , be Hopf monoids (algebras) in Vect K . Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections Fourth definition Assume that Vect K is a category of vector spaces over a field K . Let H , B , be Hopf monoids (algebras) in Vect K . In S. Majid. Strict quantum 2-groups, arXiv:1208.6265v1, 2012. we can find a definition of crossed module of Hopf monoids (Hopf algebra crossed module). Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections Fourth definition Assume that Vect K is a category of vector spaces over a field K . Let H , B , be Hopf monoids (algebras) in Vect K . In S. Majid. Strict quantum 2-groups, arXiv:1208.6265v1, 2012. we can find a definition of crossed module of Hopf monoids (Hopf algebra crossed module). In the previous conditions, if β : B → H is a morphism of Hopf monoids, the triple B H = ( B , H , β ) is a crossed module of Hopf monoids if the following assertions hold: Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections Fourth definition Assume that Vect K is a category of vector spaces over a field K . Let H , B , be Hopf monoids (algebras) in Vect K . In S. Majid. Strict quantum 2-groups, arXiv:1208.6265v1, 2012. we can find a definition of crossed module of Hopf monoids (Hopf algebra crossed module). In the previous conditions, if β : B → H is a morphism of Hopf monoids, the triple B H = ( B , H , β ) is a crossed module of Hopf monoids if the following assertions hold: (i) There exists a morphism φ B : H ⊗ B → B such that ( B , φ B ) is a left H -module monoid and comonoid. Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections Fourth definition Assume that Vect K is a category of vector spaces over a field K . Let H , B , be Hopf monoids (algebras) in Vect K . In S. Majid. Strict quantum 2-groups, arXiv:1208.6265v1, 2012. we can find a definition of crossed module of Hopf monoids (Hopf algebra crossed module). In the previous conditions, if β : B → H is a morphism of Hopf monoids, the triple B H = ( B , H , β ) is a crossed module of Hopf monoids if the following assertions hold: (i) There exists a morphism φ B : H ⊗ B → B such that ( B , φ B ) is a left H -module monoid and comonoid. (ii) The antipode of B is a morphism of left H -modules. Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections Fourth definition Assume that Vect K is a category of vector spaces over a field K . Let H , B , be Hopf monoids (algebras) in Vect K . In S. Majid. Strict quantum 2-groups, arXiv:1208.6265v1, 2012. we can find a definition of crossed module of Hopf monoids (Hopf algebra crossed module). In the previous conditions, if β : B → H is a morphism of Hopf monoids, the triple B H = ( B , H , β ) is a crossed module of Hopf monoids if the following assertions hold: (i) There exists a morphism φ B : H ⊗ B → B such that ( B , φ B ) is a left H -module monoid and comonoid. (ii) The antipode of B is a morphism of left H -modules. (iii) The identity ( φ B ⊗ H ) ◦ ( H ⊗ c H , B ) ◦ ( δ H ⊗ B ) = c H , B ◦ ( H ⊗ φ B ) ◦ ( δ H ⊗ B ) holds. Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections Fourth definition Assume that Vect K is a category of vector spaces over a field K . Let H , B , be Hopf monoids (algebras) in Vect K . In S. Majid. Strict quantum 2-groups, arXiv:1208.6265v1, 2012. we can find a definition of crossed module of Hopf monoids (Hopf algebra crossed module). In the previous conditions, if β : B → H is a morphism of Hopf monoids, the triple B H = ( B , H , β ) is a crossed module of Hopf monoids if the following assertions hold: (i) There exists a morphism φ B : H ⊗ B → B such that ( B , φ B ) is a left H -module monoid and comonoid. (ii) The antipode of B is a morphism of left H -modules. (iii) The identity ( φ B ⊗ H ) ◦ ( H ⊗ c H , B ) ◦ ( δ H ⊗ B ) = c H , B ◦ ( H ⊗ φ B ) ◦ ( δ H ⊗ B ) holds. (iv) β ◦ φ B = ad H ◦ ( H ⊗ β ) . Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections Fourth definition Assume that Vect K is a category of vector spaces over a field K . Let H , B , be Hopf monoids (algebras) in Vect K . In S. Majid. Strict quantum 2-groups, arXiv:1208.6265v1, 2012. we can find a definition of crossed module of Hopf monoids (Hopf algebra crossed module). In the previous conditions, if β : B → H is a morphism of Hopf monoids, the triple B H = ( B , H , β ) is a crossed module of Hopf monoids if the following assertions hold: (i) There exists a morphism φ B : H ⊗ B → B such that ( B , φ B ) is a left H -module monoid and comonoid. (ii) The antipode of B is a morphism of left H -modules. (iii) The identity ( φ B ⊗ H ) ◦ ( H ⊗ c H , B ) ◦ ( δ H ⊗ B ) = c H , B ◦ ( H ⊗ φ B ) ◦ ( δ H ⊗ B ) holds. (iv) β ◦ φ B = ad H ◦ ( H ⊗ β ) . (v) φ B ◦ ( β ⊗ B ) = ad H (Peiffer identity). Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections Fourth definition Assume that Vect K is a category of vector spaces over a field K . Let H , B , be Hopf monoids (algebras) in Vect K . In S. Majid. Strict quantum 2-groups, arXiv:1208.6265v1, 2012. we can find a definition of crossed module of Hopf monoids (Hopf algebra crossed module). In the previous conditions, if β : B → H is a morphism of Hopf monoids, the triple B H = ( B , H , β ) is a crossed module of Hopf monoids if the following assertions hold: (i) There exists a morphism φ B : H ⊗ B → B such that ( B , φ B ) is a left H -module monoid and comonoid. (ii) The antipode of B is a morphism of left H -modules. (iii) The identity ( φ B ⊗ H ) ◦ ( H ⊗ c H , B ) ◦ ( δ H ⊗ B ) = c H , B ◦ ( H ⊗ φ B ) ◦ ( δ H ⊗ B ) holds. (iv) β ◦ φ B = ad H ◦ ( H ⊗ β ) . (v) φ B ◦ ( β ⊗ B ) = ad H (Peiffer identity). In this setting, if the antipode of H is an isomorphism, H H = ( H , H , id H ) is an example of is a crossed module for φ H = ad H because (iii) holds and Vect K is symmetric. Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections A new definition The setting 1 Some definitions of crossed modules of Hopf monoids 2 A new definition 3 Crossed products of crossed modules of Hopf monoids 4 Projections 5 Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections In this point the category C is braided with braiding c Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections In this point the category C is braided with braiding c Definition Let H be a Hopf monoid in C . We will say that a left H -module ( X , φ X ) is in the cocommutativity class of H if c H , X is a morphism of left H -modules. This is equivalent to the condition ( φ X ⊗ H ) ◦ ( H ⊗ c H , X ) ◦ ( δ H ⊗ X ) = c − 1 H , X ◦ ( H ⊗ φ X ) ◦ ( δ H ⊗ X ) Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections In this point the category C is braided with braiding c Definition Let H be a Hopf monoid in C . We will say that a left H -module ( X , φ X ) is in the cocommutativity class of H if c H , X is a morphism of left H -modules. This is equivalent to the condition ( φ X ⊗ H ) ◦ ( H ⊗ c H , X ) ◦ ( δ H ⊗ X ) = c − 1 H , X ◦ ( H ⊗ φ X ) ◦ ( δ H ⊗ X ) Proposition Let H and B be Hopf monoids, and let f : H → B be a bimonoid morphism. The following assertions are equivalent. (i) ( ad f , B ⊗ ( f ◦ λ H )) ◦ ( H ⊗ c H , B ) ◦ ( δ H ⊗ B ) = c − 1 B , B ◦ (( f ◦ λ H ) ⊗ ad f , B ) ◦ ( δ H ⊗ B ) . (ii) B is a left H -module comonoid via ad f , B . Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections In this point the category C is braided with braiding c Definition Let H be a Hopf monoid in C . We will say that a left H -module ( X , φ X ) is in the cocommutativity class of H if c H , X is a morphism of left H -modules. This is equivalent to the condition ( φ X ⊗ H ) ◦ ( H ⊗ c H , X ) ◦ ( δ H ⊗ X ) = c − 1 H , X ◦ ( H ⊗ φ X ) ◦ ( δ H ⊗ X ) Proposition Let H and B be Hopf monoids, and let f : H → B be a bimonoid morphism. The following assertions are equivalent. (i) ( ad f , B ⊗ ( f ◦ λ H )) ◦ ( H ⊗ c H , B ) ◦ ( δ H ⊗ B ) = c − 1 B , B ◦ (( f ◦ λ H ) ⊗ ad f , B ) ◦ ( δ H ⊗ B ) . (ii) B is a left H -module comonoid via ad f , B . As a consequence, if λ H is an isomorphism we have that H is a left H -module comonoid via ad H if and only if ( H , ad H ) is in the cocommutativity class of H . Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections Definition A left-left entwining structure on C consists of a triple ( A , D , ψ A , D ) , where A is a monoid, D a comonoid, and ψ A , D : A ⊗ D → D ⊗ A is a morphism satisfying the conditions (a1) ψ A , D ◦ ( η A ⊗ D ) = D ⊗ η A , (a2) ( D ⊗ µ A ) ◦ ( ψ A , D ⊗ A ) ◦ ( A ⊗ ψ A , D ) = ψ A , D ◦ ( µ A ⊗ D ) , (a3) ( δ D ⊗ A ) ◦ ψ A , D = ( D ⊗ ψ A , D ) ◦ ( ψ A , D ⊗ D ) ◦ ( A ⊗ δ D ) , (a4) ( ε D ⊗ A ) ◦ ψ A , D = A ⊗ ε D . Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections Definition A left-left entwining structure on C consists of a triple ( A , D , ψ A , D ) , where A is a monoid, D a comonoid, and ψ A , D : A ⊗ D → D ⊗ A is a morphism satisfying the conditions (a1) ψ A , D ◦ ( η A ⊗ D ) = D ⊗ η A , (a2) ( D ⊗ µ A ) ◦ ( ψ A , D ⊗ A ) ◦ ( A ⊗ ψ A , D ) = ψ A , D ◦ ( µ A ⊗ D ) , (a3) ( δ D ⊗ A ) ◦ ψ A , D = ( D ⊗ ψ A , D ) ◦ ( ψ A , D ⊗ D ) ◦ ( A ⊗ δ D ) , (a4) ( ε D ⊗ A ) ◦ ψ A , D = A ⊗ ε D . If we only have the conditions (a1) and (a2) we will say that ( A , D , ψ A , D ) is a left-left semi-entwining structure. Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections Definition A left-left entwining structure on C consists of a triple ( A , D , ψ A , D ) , where A is a monoid, D a comonoid, and ψ A , D : A ⊗ D → D ⊗ A is a morphism satisfying the conditions (a1) ψ A , D ◦ ( η A ⊗ D ) = D ⊗ η A , (a2) ( D ⊗ µ A ) ◦ ( ψ A , D ⊗ A ) ◦ ( A ⊗ ψ A , D ) = ψ A , D ◦ ( µ A ⊗ D ) , (a3) ( δ D ⊗ A ) ◦ ψ A , D = ( D ⊗ ψ A , D ) ◦ ( ψ A , D ⊗ D ) ◦ ( A ⊗ δ D ) , (a4) ( ε D ⊗ A ) ◦ ψ A , D = A ⊗ ε D . If we only have the conditions (a1) and (a2) we will say that ( A , D , ψ A , D ) is a left-left semi-entwining structure. In a similar way, we can define the notions of right-right, right-left and left-right (se- mi)entwining structure. Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections Definition A left-left entwining structure on C consists of a triple ( A , D , ψ A , D ) , where A is a monoid, D a comonoid, and ψ A , D : A ⊗ D → D ⊗ A is a morphism satisfying the conditions (a1) ψ A , D ◦ ( η A ⊗ D ) = D ⊗ η A , (a2) ( D ⊗ µ A ) ◦ ( ψ A , D ⊗ A ) ◦ ( A ⊗ ψ A , D ) = ψ A , D ◦ ( µ A ⊗ D ) , (a3) ( δ D ⊗ A ) ◦ ψ A , D = ( D ⊗ ψ A , D ) ◦ ( ψ A , D ⊗ D ) ◦ ( A ⊗ δ D ) , (a4) ( ε D ⊗ A ) ◦ ψ A , D = A ⊗ ε D . If we only have the conditions (a1) and (a2) we will say that ( A , D , ψ A , D ) is a left-left semi-entwining structure. In a similar way, we can define the notions of right-right, right-left and left-right (se- mi)entwining structure. For example, ( A , D , ψ D , A : D ⊗ A → A ⊗ D ) will be a right-right semi-entwining structure if conditions (b1) ψ D , A ◦ ( D ⊗ η A ) = η A ⊗ D , (b2) ( µ A ⊗ D ) ◦ ( A ⊗ ψ D , A ) ◦ ( ψ D , A ⊗ A ) = ψ D , A ◦ ( D ⊗ µ A ) , hold. Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections Definition Let X and Y be monoids and comonoids and let ψ Y , X : Y ⊗ X → X ⊗ Y be a morphism. We will say that ψ Y , X is in the cocommutativity class of Y if the following equality ( ψ Y , X ⊗ Y ) ◦ ( Y ⊗ c Y , X ) ◦ ( δ Y ⊗ X ) = ( c − 1 Y , X ⊗ Y ) ◦ ( Y ⊗ ψ Y , X ) ◦ ( δ Y ⊗ X ) , holds. Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections Definition Let X and Y be monoids and comonoids and let ψ Y , X : Y ⊗ X → X ⊗ Y be a morphism. We will say that ψ Y , X is in the cocommutativity class of Y if the following equality ( ψ Y , X ⊗ Y ) ◦ ( Y ⊗ c Y , X ) ◦ ( δ Y ⊗ X ) = ( c − 1 Y , X ⊗ Y ) ◦ ( Y ⊗ ψ Y , X ) ◦ ( δ Y ⊗ X ) , holds. Lemma Let X and Y be monoids and comonoids and let ψ Y , X : Y ⊗ X → X ⊗ Y be a morphism such ( ε X ⊗ Y ) ◦ ψ Y , X = Y ⊗ ε X holds. The following assertions are equivalent. (i) δ X ⊗ Y ◦ ψ Y , X = ( ψ Y , X ⊗ ψ Y , X ) ◦ δ Y ⊗ X . (ii) ψ Y , X is in the cocommutativity class of Y , and satisfy the conditions ( δ X ⊗ Y ) ◦ ψ Y , X = ( X ⊗ ψ Y , X ) ◦ ( ψ Y , X ⊗ X ) ◦ ( Y ⊗ δ X ) , ( X ⊗ δ Y ) ◦ ψ Y , X = ( ψ Y , X ⊗ Y ) ◦ ( Y ⊗ c Y , X ) ◦ ( δ Y ⊗ X ) (1) Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections Proposition Let X and Y be bimonoids. The following assertions are equivalent. (i) There is a morphism ψ Y , X : Y ⊗ X → X ⊗ Y such that ( Y , X , ψ Y , X ) is a left- left entwining structure and ( X , Y , ψ Y , X ) a right-right semi-entwining structure satisfying (1). Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections Proposition Let X and Y be bimonoids. The following assertions are equivalent. (i) There is a morphism ψ Y , X : Y ⊗ X → X ⊗ Y such that ( Y , X , ψ Y , X ) is a left- left entwining structure and ( X , Y , ψ Y , X ) a right-right semi-entwining structure satisfying (1) ( X ⊗ δ Y ) ◦ ψ Y , X = ( ψ Y , X ⊗ Y ) ◦ ( Y ⊗ c Y , X ) ◦ ( δ Y ⊗ X ) Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections Proposition Let X and Y be bimonoids. The following assertions are equivalent. (i) There is a morphism ψ Y , X : Y ⊗ X → X ⊗ Y such that ( Y , X , ψ Y , X ) is a left- left entwining structure and ( X , Y , ψ Y , X ) a right-right semi-entwining structure satisfying (1). (ii) There is a morphism φ X : Y ⊗ X → X such that ( X , φ X ) is a left Y -module monoid and comonoid. Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections Proposition Let X and Y be bimonoids. The following assertions are equivalent. (i) There is a morphism ψ Y , X : Y ⊗ X → X ⊗ Y such that ( Y , X , ψ Y , X ) is a left- left entwining structure and ( X , Y , ψ Y , X ) a right-right semi-entwining structure satisfying (1). (ii) There is a morphism φ X : Y ⊗ X → X such that ( X , φ X ) is a left Y -module monoid and comonoid. Proof (i) ⇒ (ii) Define φ X = ( X ⊗ ε Y ) ◦ ψ Y , X . (ii) ⇒ (i) Define ψ Y , X = ( φ X ⊗ Y ) ◦ ( Y ⊗ c Y , X ) ◦ ( δ Y ⊗ X ) . Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections Proposition Let X and Y be bimonoids. The following assertions are equivalent. (i) There is a morphism ψ Y , X : Y ⊗ X → X ⊗ Y such that ( Y , X , ψ Y , X ) is a left- left entwining structure and ( X , Y , ψ Y , X ) a right-right semi-entwining structure satisfying (1). (ii) There is a morphism φ X : Y ⊗ X → X such that ( X , φ X ) is a left Y -module monoid and comonoid. Proof (i) ⇒ (ii) Define φ X = ( X ⊗ ε Y ) ◦ ψ Y , X . (ii) ⇒ (i) Define ψ Y , X = ( φ X ⊗ Y ) ◦ ( Y ⊗ c Y , X ) ◦ ( δ Y ⊗ X ) . Moreover, ψ Y , X is in the cocommutativity class of Y iff so is ( X , φ X ) . Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections Definition Let β : X → Y be a morphism of Hopf monoids and let ψ Y , X : Y ⊗ X → X ⊗ Y be a morphism. We will say that X Y = ( X , Y , β ) is a crossed module of Hopf monoids if Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections Definition Let β : X → Y be a morphism of Hopf monoids and let ψ Y , X : Y ⊗ X → X ⊗ Y be a morphism. We will say that X Y = ( X , Y , β ) is a crossed module of Hopf monoids if (c1) ( Y , X , ψ Y , X ) is a left-left entwining structure and ( X , Y , ψ Y , X ) a right-right semi- entwining structure. Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections Definition Let β : X → Y be a morphism of Hopf monoids and let ψ Y , X : Y ⊗ X → X ⊗ Y be a morphism. We will say that X Y = ( X , Y , β ) is a crossed module of Hopf monoids if (c1) ( Y , X , ψ Y , X ) is a left-left entwining structure and ( X , Y , ψ Y , X ) a right-right semi- entwining structure. (c2) ψ Y , X is in the cocommutativity class of Y . Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections Definition Let β : X → Y be a morphism of Hopf monoids and let ψ Y , X : Y ⊗ X → X ⊗ Y be a morphism. We will say that X Y = ( X , Y , β ) is a crossed module of Hopf monoids if (c1) ( Y , X , ψ Y , X ) is a left-left entwining structure and ( X , Y , ψ Y , X ) a right-right semi- entwining structure. (c2) ψ Y , X is in the cocommutativity class of Y . (c3) ψ Y , X satisfies (1). Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections Definition Let β : X → Y be a morphism of Hopf monoids and let ψ Y , X : Y ⊗ X → X ⊗ Y be a morphism. We will say that X Y = ( X , Y , β ) is a crossed module of Hopf monoids if (c1) ( Y , X , ψ Y , X ) is a left-left entwining structure and ( X , Y , ψ Y , X ) a right-right semi- entwining structure. (c2) ψ Y , X is in the cocommutativity class of Y . (c3) ψ Y , X satisfies (1). (c4) ( β ⊗ ε Y ) ◦ ψ Y , X = ad Y ◦ ( Y ⊗ β ) . Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections Definition Let β : X → Y be a morphism of Hopf monoids and let ψ Y , X : Y ⊗ X → X ⊗ Y be a morphism. We will say that X Y = ( X , Y , β ) is a crossed module of Hopf monoids if (c1) ( Y , X , ψ Y , X ) is a left-left entwining structure and ( X , Y , ψ Y , X ) a right-right semi- entwining structure. (c2) ψ Y , X is in the cocommutativity class of Y . (c3) ψ Y , X satisfies (1). (c4) ( β ⊗ ε Y ) ◦ ψ Y , X = ad Y ◦ ( Y ⊗ β ) . (c5) ( X ⊗ ε Y ) ◦ ψ Y , X ◦ ( β ⊗ X ) = ad X (Peiffer identity). Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections Definition Let β : X → Y be a morphism of Hopf monoids and let ψ Y , X : Y ⊗ X → X ⊗ Y be a morphism. We will say that X Y = ( X , Y , β ) is a crossed module of Hopf monoids if (c1) ( Y , X , ψ Y , X ) is a left-left entwining structure and ( X , Y , ψ Y , X ) a right-right semi- entwining structure. (c2) ψ Y , X is in the cocommutativity class of Y . (c3) ψ Y , X satisfies (1). (c4) ( β ⊗ ε Y ) ◦ ψ Y , X = ad Y ◦ ( Y ⊗ β ) . (c5) ( X ⊗ ε Y ) ◦ ψ Y , X ◦ ( β ⊗ X ) = ad X (Peiffer identity). Equivalently, there is a morphism φ X : Y ⊗ X → X such that (d1) ( X , φ X ) is a left Y -module monoid and comonoid. (d2) ( X , φ X ) is in the class of cocommutativity of Y . (d2) β ◦ φ X = ad Y ◦ ( Y ⊗ β ) . (d3) φ X ◦ ( β ⊗ X ) = ad X (Peiffer identity). Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections Definition Let β : X → Y be a morphism of Hopf monoids and let ψ Y , X : Y ⊗ X → X ⊗ Y be a morphism. We will say that X Y = ( X , Y , β ) is a crossed module of Hopf monoids if (c1) ( Y , X , ψ Y , X ) is a left-left entwining structure and ( X , Y , ψ Y , X ) a right-right semi- entwining structure. (c2) ψ Y , X is in the cocommutativity class of Y . (c3) ψ Y , X satisfies (1). (c4) ( β ⊗ ε Y ) ◦ ψ Y , X = ad Y ◦ ( Y ⊗ β ) . (c5) ( X ⊗ ε Y ) ◦ ψ Y , X ◦ ( β ⊗ X ) = ad X (Peiffer identity). Equivalently, there is a morphism φ X : Y ⊗ X → X such that (d1) ( X , φ X ) is a left Y -module monoid and comonoid. (d2) ( X , φ X ) is in the class of cocommutativity of Y . (d2) β ◦ φ X = ad Y ◦ ( Y ⊗ β ) . (d3) φ X ◦ ( β ⊗ X ) = ad X (Peiffer identity). If λ X is an isomorphism, X X = ( X , X , id X ) is a crossed module of Hopf monoids φ X = ad X . Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections Definition A morphism between two crossed modules of Hopf monoids X Y = ( X , Y , β ) and T G = ( T , G , ∂ ) is a pair of Hopf monoid morphisms u : X → T , v : Y → G such that v ◦ β = ∂ ◦ u , ( u ⊗ ε Y ) ◦ ψ Y , X = ( T ⊗ ε G ) ◦ ψ G , T ◦ ( v ⊗ u ) . Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections Definition A morphism between two crossed modules of Hopf monoids X Y = ( X , Y , β ) and T G = ( T , G , ∂ ) is a pair of Hopf monoid morphisms u : X → T , v : Y → G such that v ◦ β = ∂ ◦ u , ( u ⊗ ε Y ) ◦ ψ Y , X = ( T ⊗ ε G ) ◦ ψ G , T ◦ ( v ⊗ u ) . Equivalently, Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections Definition A morphism between two crossed modules of Hopf monoids X Y = ( X , Y , β ) and T G = ( T , G , ∂ ) is a pair of Hopf monoid morphisms u : X → T , v : Y → G such that v ◦ β = ∂ ◦ u , ( u ⊗ ε Y ) ◦ ψ Y , X = ( T ⊗ ε G ) ◦ ψ G , T ◦ ( v ⊗ u ) . Equivalently, Definition A morphism between two crossed modules of Hopf monoids X Y = ( X , Y , β ) and T G = ( T , G , ∂ ) is a pair of Hopf monoid morphisms u : X → T and v : Y → G such that v ◦ β = ∂ ◦ u , u ◦ φ X = φ T ◦ ( v ⊗ u ) . Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections Crossed products of crossed modules of Hopf monoids The setting 1 Some definitions of crossed modules of Hopf monoids 2 A new definition 3 Crossed products of crossed modules of Hopf monoids 4 Projections 5 Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections In the following we will to assume that C is symmetric. Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections In the following we will to assume that C is symmetric. Let X and Y be Hopf monoids and let ψ Y , X : Y ⊗ X → X ⊗ Y be a morphism such that ( Y , X , ψ Y , X ) is a left-left entwining structure and ( X , Y , ψ Y , X ) a right-right semi-entwining structure. Then the smash product of X by Y defined as X # Y = ( X ⊗ Y , η X # Y = η X ⊗ η Y , µ X # Y = ( µ X ⊗ µ Y ) ◦ ( X ⊗ ψ Y , X ⊗ Y )) , is a monoid. Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections In the following we will to assume that C is symmetric. Let X and Y be Hopf monoids and let ψ Y , X : Y ⊗ X → X ⊗ Y be a morphism such that ( Y , X , ψ Y , X ) is a left-left entwining structure and ( X , Y , ψ Y , X ) a right-right semi-entwining structure. Then the smash product of X by Y defined as X # Y = ( X ⊗ Y , η X # Y = η X ⊗ η Y , µ X # Y = ( µ X ⊗ µ Y ) ◦ ( X ⊗ ψ Y , X ⊗ Y )) , is a monoid. Proposition Let X and Y be Hopf monoids and let ψ Y , X : Y ⊗ X → X ⊗ Y be a morphism such that ( Y , X , ψ Y , X ) is a left-left entwining structure and ( X , Y , ψ Y , X ) a right-right semi- entwining structure such that ψ Y , X is in the cocommutativity class of Y and (1) holds. Then the tensor product comonoid structure is compatible with the smash product monoid structure, making X ⊲ ⊳ Y = ( X ⊗ Y , η X ⊲ ⊳ Y = η X # Y , µ X ⊲ ⊳ Y = µ X # Y , ε X ⊲ ⊳ Y = ε X ⊗ ε Y , δ X ⊲ ⊳ Y = δ X ⊗ Y ) a Hopf monoid with antipode λ X ⊲ ⊳ Y = ψ Y , X ◦ ( λ Y ⊗ λ X ) ◦ c X , Y . Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections The main goal of this section is to construct the crossed product of two crossed modules of Hopf monoids. In order to do so, in what follows we consider two crossed modules of Hopf monoids X Y = ( X , Y , β ) and T G = ( T , G , ∂ ) and denote the corresponding morphisms by ψ Y , X and ψ G , T , respectively. Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections The main goal of this section is to construct the crossed product of two crossed modules of Hopf monoids. In order to do so, in what follows we consider two crossed modules of Hopf monoids X Y = ( X , Y , β ) and T G = ( T , G , ∂ ) and denote the corresponding morphisms by ψ Y , X and ψ G , T , respectively. Moreover, let t : Y ⊗ T → X be a morphism and assume that ψ G , X : G ⊗ X → X ⊗ G , ψ T , X : T ⊗ X → X ⊗ T , ψ G , Y : G ⊗ Y → Y ⊗ G are three morphisms that induce left-left entwining structures and right-right semi- entwining structures and such that ψ G , X is in the class of cocommutativity of G , ψ T , X is in the class of cocommutativity of T , ψ G , Y is in the class of cocommutativity of G , (1) holds for the previous morphisms and the Yang-Baxter condition ( ψ Y , X ⊗ G ) ◦ ( Y ⊗ ψ G , X ) ◦ ( ψ G , Y ⊗ X ) = ( X ⊗ ψ G , Y ) ◦ ( ψ G , X ⊗ Y ) ◦ ( G ⊗ ψ Y , X ) also holds. Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections Now define the morphism φ X ⊲ ⊳ T : Y ⊲ ⊳ G ⊗ X ⊲ ⊳ T → X ⊲ ⊳ T as φ X ⊲ ⊳ T = ( µ X ⊗ T ) ◦ ( X ⊗ t ⊗ T ) ◦ ( X ⊗ Y ⊗ δ T ⊗ ε G ) ◦ ( ψ Y , X ⊗ ψ G , T ) ◦ ( Y ⊗ ψ G , X ⊗ T ) . Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections Lemma The following assertions are equivalent. (i) ( X ⊲ ⊳ T , φ X ⊲ ⊳ T ) is a left Y ⊲ ⊳ G -module. (ii) The equalities t ◦ ( η Y ⊗ T ) = ε T ⊗ η X , (2) ( t ⊗ ε G ) ◦ ( Y ⊗ ψ G , T ) ◦ ( ψ G , Y ⊗ T ) = ( X ⊗ ε G ) ◦ ψ G , X ◦ ( G ⊗ t ) , (3) and t ◦ ( µ Y ⊗ T ) = µ X ◦ ( X ⊗ t ) ◦ ( ψ Y , X ⊗ T ) ◦ ( Y ⊗ t ⊗ T ) ◦ ( Y ⊗ Y ⊗ δ T ) (4) hold. Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections Lemma The following assertions are equivalent. (i) φ X ⊲ ⊳ T is a monoid morphism. (ii) The equalities t ◦ ( Y ⊗ η T ) = ε Y ⊗ η X , (5) t ◦ ( Y ⊗ µ T ) = (6) ( µ X ⊗ ε T ) ◦ ( t ⊗ ψ T , X ) ◦ ( Y ⊗ δ T ⊗ X ) ◦ ( Y ⊗ T ⊗ t ) ◦ ( Y ⊗ c Y , T ⊗ T ) ◦ ( δ Y ⊗ T ⊗ T ) and µ X ◦ ( X ⊗ t ) ◦ ( ψ Y , X ⊗ T ) ◦ ( Y ⊗ ψ T , X ) (7) = ( µ X ⊗ ε T ) ◦ ( t ⊗ ψ T , X ⊗ ε Y ) ◦ ( Y ⊗ δ T ⊗ ψ Y , X ) ◦ ( Y ⊗ c Y , T ⊗ X ) ◦ ( δ Y ⊗ T ⊗ X ) , hold. Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections Lemma The following assertions are equivalent. (i) φ X ⊲ ⊳ T is a comonoid morphism. (ii) t is a comonoid morphism and the equality c X , T ◦ ( t ⊗ T ) ◦ ( Y ⊗ δ T ) = ( T ⊗ t ) ◦ ( c Y , T ⊗ T ) ◦ ( Y ⊗ δ T ) (8) holds. Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections Lemma If (2) and (5) hold, Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections Lemma If (2) and (5) hold, t ◦ ( η Y ⊗ T ) = ε T ⊗ η X , t ◦ ( Y ⊗ η T ) = ε Y ⊗ η X Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections Lemma If (2) and (5) hold, the following assertions are equivalent. Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections Lemma If (2) and (5) hold, the following assertions are equivalent. (i) ( X ⊲ ⊳ T , φ X ⊲ ⊳ T ) is in the cocommutativity class of Y ⊲ ⊳ G . Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections Lemma If (2) and (5) hold, the following assertions are equivalent. (i) ( X ⊲ ⊳ T , φ X ⊲ ⊳ T ) is in the cocommutativity class of Y ⊲ ⊳ G . (ii) The equality ( t ⊗ Y ) ◦ ( Y ⊗ c Y , T ) ◦ ( δ Y ⊗ T ) = c Y , X ◦ ( Y ⊗ t ) ◦ ( δ Y ⊗ T ) (9) holds. Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections Lemma The following assertions are equivalent. (i) ( β ⊗ ∂ ) ◦ φ X ⊲ ⊳ T = ad Y ⊲ ⊳ G ◦ ( Y ⊗ G ⊗ β ⊗ ∂ ) (ii) The equalities (( β ◦ t ) ⊗ ∂ ) ◦ ( Y ⊗ δ T ) = ( µ Y ⊗ G ) ◦ ( Y ⊗ ( ψ G , Y ◦ c Y , G ◦ ( λ Y ⊗ ∂ ))) ◦ ( δ Y ⊗ T ) (10) and ( β ⊗ G ) ◦ ψ G , X = ψ G , Y ◦ ( G ⊗ β ) (11) hold. Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections Lemma (Peiffer identity) The following assertions are equivalent. (i) φ X ⊲ ⊳ T ◦ ( β ⊗ ∂ ⊗ X ⊗ T ) = ad X ⊲ ⊳ T (ii) The equalities ( t ⊗ T ) ◦ ( β ⊗ δ T ) = ( µ X ⊗ T ) ◦ ( X ⊗ ( ψ T , X ◦ c X , T ◦ ( λ X ⊗ T ))) ◦ ( δ X ⊗ T ) (12) and ψ G , X ◦ ( ∂ ⊗ X ) = ( X ⊗ ∂ ) ◦ ψ T , X (13) hold. Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections Theorem In the conditions of this section, the following assertions are equivalent. (i) X Y ⊲ ⊳ T G = ( X ⊲ ⊳ T , Y ⊲ ⊳ G , β ⊗ ∂ ) is a crossed module of Hopf monoids via φ X ⊲ ⊳ T . (ii) t is a comonoid morphism and the equalities (2), (3), (4), (5), (6), (7), (8), (9), (10), (11), (12) and (13) hold. Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections Projections The setting 1 Some definitions of crossed modules of Hopf monoids 2 A new definition 3 Crossed products of crossed modules of Hopf monoids 4 Projections 5 Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections We assume that every idempotent morphism q : Y → Y in C splits, i.e., there exist an object Z (image of q ) and morphisms i : Z → Y (injection) and p : Y → Z (projection) such that q = i ◦ p and p ◦ i = id Z . Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections We assume that every idempotent morphism q : Y → Y in C splits, i.e., there exist an object Z (image of q ) and morphisms i : Z → Y (injection) and p : Y → Z (projection) such that q = i ◦ p and p ◦ i = id Z . Definition A projection of Hopf monoids is a quartet ( T , B , u , w ) where T , B are Hopf monoids, and u : T → B , w : B → T are Hopf monoid morphisms such that w ◦ u = id T . A morphism between projections of Hopf monoids ( T , B , u , w ) and ( G , H , v , y ) is a pair ( ∂, γ ) , where ∂ : T → G , γ : B → H are Hopf monoid morphisms such that v ◦ ∂ = γ ◦ u , ∂ ◦ w = y ◦ γ. Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections Let ( T , B , u , w ) be a projection of Hopf monoids. The morphism q B = µ B ◦ ( B ⊗ ( u ◦ λ T ◦ w )) ◦ δ B is an idempotent and, as a consequence, there exist an epimorphism p B , a mo- nomorphism i B , and an object B coT (submonoid of coinvariants) such that the diagram q B ✲ B B ❍ ✚ ❃ ❍ ❥ ✚ p B i B B coT commutes and p B ◦ i B = id B coT . Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections Let ( T , B , u , w ) be a projection of Hopf monoids. The morphism q B = µ B ◦ ( B ⊗ ( u ◦ λ T ◦ w )) ◦ δ B is an idempotent and, as a consequence, there exist an epimorphism p B , a mo- nomorphism i B , and an object B coT (submonoid of coinvariants) such that the diagram q B ✲ B B ❍ ❃ ✚ ❥ ✚ ❍ p B i B B coT commutes and p B ◦ i B = id B coT . Also, ( B ⊗ w ) ◦ δ B ✲ i B ✲ B coT ✲ B B ⊗ T B ⊗ η T is an equalizer diagram and µ B ◦ ( B ⊗ u ) p B ✲ ✲ B ⊗ T ✲ B B coT B ⊗ ε T is a coequalizer diagram. Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections The morphism i B ( p B ) is a monoid (comonoid) morphism, where the monoid and comonoid structures in B coT are η B coT = p B ◦ η B , µ B coT = p B ◦ µ B ◦ ( i B ⊗ i B ) , ε B coT = ε B ◦ i B , δ B coT = ( p B ⊗ p B ) ◦ δ B ◦ i B respectively. Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras

The setting Some definitions of crossed modules of Hopf monoids A new definition Crossed products of crossed modules of Hopf monoids Projections The morphism i B ( p B ) is a monoid (comonoid) morphism, where the monoid and comonoid structures in B coT are η B coT = p B ◦ η B , µ B coT = p B ◦ µ B ◦ ( i B ⊗ i B ) , ε B coT = ε B ◦ i B , δ B coT = ( p B ⊗ p B ) ◦ δ B ◦ i B respectively. The morphism ad u , B ◦ ( T ⊗ i B ) factorizes through the equalizer i B , and the facto- rization ϕ B coT = p B ◦ µ B ◦ ( u ⊗ i B ) : T ⊗ B coT → B coT gives a left T -module monoid and comonoid structure for B coT . Ramón González Rodríguez Crossed products of crossed modules of Hopf algebras