Hopf Categories Eliezer Batista, Stefaan Caenepeel, Timmy Fieremans, - PowerPoint PPT Presentation

Hopf Categories Eliezer Batista, Stefaan Caenepeel, Timmy Fieremans, Joost Vercruysse Ponta Delgada, July 9, 2018

Enriched category theory V = ( V , ⊗ , k ) is a strict monoidal category, X is a class. New monoidal category ( V ( X ) , • , J ) ◮ An object is a family of objects M in V indexed by X × X : M = ( M x , y ) x , y ∈ X . ◮ morphism ϕ : M → N : family of morphisms ϕ x , y : M x , y → N x , y ◮ ( M • N ) x , y = M x , y ⊗ N x , y , J x , y = ke x , y functor ( − ) op : V ( X ) → V ( X ): V op y , x = V x , y , ϕ op y , x = ϕ x , y .

Enriched category theory V -category A ◮ class X ◮ multiplication morphisms m = m x , y , z : A x , y ⊗ A y , z → A x , z ◮ unit morphisms η x : J x , x = ke x , x → A x , x with unit and associativity conditions. J is a V -category. ◮ ( V , ⊗ , k ) = ( Sets , × , {∗} ): ordinary categories ◮ ( V , ⊗ , k ) = ( M k , ⊗ , k ): k -linear categories

Enriched category theory ◮ If V is braided: tensor product in V ( X ) of two V -categories is again a V -category. ◮ Fix a class X : V - X -categories; V - X -functor is functor that is the identity on objects.

Semi-Hopf categories Assume that V is braided. C ( V ) is the category of coalgebras in V . We consider C ( V )-categories, aka semi-Hopf V -categories. Description Coalgebra in V ( X ) is a family of coalgebras ( C x , y ). Structure maps: ∆ x , y : C x , y → C x , y ⊗ C x , y and ε x , y : C x , y → J x , y = ke x , y

Semi-Hopf categories Proposition A semi-Hopf V -category with underlying class X consists of A ∈ V ( X ) which is ◮ a V -category ◮ a coalgebra in V ( X ) ◮ the morphisms ∆ x , y and ε x , y define V -X-functors ∆ : A → A • A and ε : A → J. C ( V )-categories with one object correspond to bialgebras in V

op and cop op If A is a V -category, then A op is also a V -category: multiplication morphisms m op x , y , z = m z , y , x ◦ c A y , x , A x , y : A op x , y ⊗ A op y , z = A y , x ⊗ A z , y → A op x , z = A z , x and unit morphisms η op = η x . x If A is a C ( V )-category, then A op is also a C ( V )-category, with coalgebra structure maps ∆ op x , y = ∆ y , x and ε op x , y = ε y , x . cop Let C be a coalgebra in V ( X ). The coopposite coalgebra C cop is equal to C as an object of V ( X ), with comultiplication maps ∆ cop x , y = c C x , y , C x , y ◦ ∆ x , y : C x , y → C x , y ⊗ C x , y , and counit maps ε x , y . If A is a C ( V )-category, then A cop is also a C ( V )-category; the V -category structures on A and A cop coincide.

Hopf categories Definition A Hopf V -category is a semi-Hopf V -category A together with a morphism S : A → A op in V ( X ) ( S x , y : A x , y → A y , x ) such that m x , y , x ◦ ( A x , y ⊗ S x , y ) ◦ ∆ x , y = η x ◦ ε x , y : A x , y → A x , x ; m y , x , y ◦ ( S x , y ⊗ A x , y ) ◦ ∆ x , y = η y ◦ ε x , y : A x , y → A y , y , for all x , y ∈ X . Over M k : for h ∈ A x , y : h (1) S x , y ( h (2) ) = ε x , y ( h )1 x ; S x , y ( h (1) ) h (2) = ε x , y ( h )1 y . A Hopf V -category with one object is a Hopf algebra in V .

Hopf-categories and groupoids V = ( Sets , × , {∗} ). Every set is in a unique way a coalgebra in Sets . C ( Sets ) = Sets . C ( Sets )-categories = categories. Proposition A Hopf Sets -category is the same thing as a groupoid (i.e. a category in which all morphisms are isomorphisms).

Hopf-categories: first properties Theorem Let A be a Hopf V -category. The antipode S is a morphism of C ( V ) -categories H → H opcop . Proposition Let A be a k-linear Hopf category. For x , y ∈ X, the following assertions are equivalent. 1. S x , y ( h (2) ) h (1) = ε x , y ( h )1 y , for all h ∈ A x , y ; 2. h (2) S x , y ( h (1) ) = ε x , y ( h )1 x , for all h ∈ A x , y ; 3. S y , x ◦ S x , y = A x , y .

Hopf-categories: first properties Let A and B be Hopf V -categories. A C ( V )-functor f : A → B is called a Hopf V -functor if S B f ( x ) , f ( y ) ◦ f x , y = f y , x ◦ S A x , y , (1) for all x , y ∈ X . Proposition Let A and B be Hopf V -categories. If f : A → B is a C ( V ) -functor, then it is also a Hopf V -functor.

The representation category Let A be a V -category. A left A -module is an object M in V ( X ) together with a family of morphisms in V ψ = ψ x , y , z : A x , y ⊗ M y , z → M x , z + associativity and unit conditions. A morphism ϕ : M → N in V ( X ) between left A -modules is called left A -linear if ϕ x , z ◦ ψ x , y , z = ψ x , y , z ◦ ( A x , y ⊗ ϕ y , z ) Category: A V ( X )

The representation category Proposition Let A be a C ( V ) -category. Then there is a monoidal structure on A V ( X ) such that the forgetful functor A V ( X ) → V ( X ) is monoidal. Bewijs. (in case V = M k ). We need actions A x , y ⊗ M y , z ⊗ N y , z → M x , z ⊗ N x , z and A x , y ⊗ ke y , z → ke x , z . Take a · ( m ⊗ n ) = a (1) m ⊗ a (2) n and a · 1 = ε ( a ) .

Duality: V -opcategories

Hopf categories and Hopf group (co)algebras

Hopf categories and weak Hopf algebras Proposition Let A be a k-linear Hopf category, with | A | = X a finite set. Then A = ⊕ x , y ∈ X A x , y is a weak Hopf algebra. Example Take a groupoid with finitely many objects; apply the linearization functor to obtain a k -linear Hopf category; in packed form it becomes the groupoid algebra, which is well-known to be a weak Hopf algebra. Proposition Let C be a k-linear Hopf opcategory, with | C | = X a finite set. Then C = ⊕ x , y ∈ X C x , y is a weak Hopf algebra.

Hopf categories and duoidal categories ◮ M. Aguiar, S. Mahajan, “Monoidal functors, species and Hopf algebras”, CRM Monogr. ser. 29 , Amer. Math. Soc. Providence, RI, (2010). ◮ G. B¨ ohm, Y. Chen, L. Zhang, “On Hopf monoids in duoidal categories”, J. Algebra 394 (2013), 139-172.

Hopf categories and duoidal categories Definition A duoidal category is a category M with ◮ monoidal structure ( ⊙ , I ) ◮ monoidal structure ( • , J ) ◮ δ : I → I • I ◮ ̟ : J ⊙ J → J ◮ τ : I → J ◮ ζ A , B , C , D : ( A • B ) ⊙ ( C • D ) → ( A ⊙ C ) • ( B ⊙ D ) ◮ ( J , ̟, τ ) is an algebra in ( M , ⊙ , I ) ◮ ( I , δ, τ ) is a coalgebra in ( M , • , J ) ◮ 6 more commutative diagrams (2 associativity and 4 unit)

Hopf categories and duoidal categories Let X be a set. ( M k ( X ) , • , J ) is a monoidal category. Second monomial structure: ( M ⊙ N ) x , z = ⊕ y ∈ X M x , y ⊗ N y , z . � ke x , x if x = y I x , y = 0 if x � = y ◮ τ : I → J : natural inclusion ◮ δ : I → I • I = I : identity map ◮ ( J ⊙ J ) x , y = ⊕ z ∈ X ke x , z ⊗ ke z , y = ⊕ z ∈ X kze x , y = kXe x , y . ̟ : J ⊙ J → J ̟ x , y : ⊕ z ∈ X kze x , y → ke x , y ̟ x , y ( � z ∈ X α z ze x , y ) = � z ∈ X α z e x , y .

Hopf categories and duoidal categories � (( M • N ) ⊙ ( P • Q )) x , y = M x , z ⊗ N x , z ⊗ P z , y ⊗ Q z , y ; z ∈ X � (( M ⊙ P ) • ( N ⊙ Q )) x , y = M x , u ⊗ P u , y ⊗ N x , v ⊗ Q v , y , u , v ∈ X ζ M , N , P , Q , x , y is the map switching the second and third tensor factor, followed by the natural inclusion. Theorem Let X be a set. ( M k ( X ) , ⊙ , I , • , J , δ, ̟, τ, ζ ) is a duoidal category.

Hopf categories and duoidal categories Definition Let ( M , ⊙ , I , • , J , δ, ̟, τ, ζ ) be a duoidal category. A bimonoid is an object A , together with an algebra structure ( µ, η ) in ( M , ⊙ , I ) and a coalgebra structure (∆ , ε ) in ( M , • , J ) subject to the compatibility conditions ∆ ◦ µ = ( µ • µ ) ◦ ζ ◦ (∆ ⊙ ∆); ̟ ◦ ( ε ⊙ ε ) = ε ◦ µ ; ( η • η ) ◦ δ = ∆ ◦ η ; ε ◦ η = τ.

Hopf categories and duoidal categories Theorem Let X be a set, and let A ∈ M k ( X ) . We have a bijective correspondence between bimonoid structures on A over the duoidal category ( M k ( X ) , ⊙ , I , • , J , δ, ̟, τ, ζ ) from and k-linear semi-Hopf category structures on A.

Hopf modules Definition A is a k -linear semi-Hopf category. A Hopf module over A is M ∈ M k ( X ) such that ◮ M ∈ M k ( X ) A , with structure maps ψ x , y , z ◮ M ∈ M k ( X ) A : M is a right comodule over A as a coalgebra in M k ( X ), with structure maps ρ x , y ◮ ρ x , z ( ma ) = m [0] a (1) ⊗ m [1] a (2) Category of Hopf modules: M k ( X ) A A . New category: D ( X ) consisting of families of k -modules N = ( N x ) x ∈ X indexed by X .

An adjoint pair of functors Proposition We have a pair of adjoint functors ( F , G ) between the categories D ( X ) and M k ( X ) A A . Bewijs. F ( N ) x , y = N x ⊗ A x , y , with ( n ⊗ a ) b = n ⊗ ab ; ρ x , y ( n ⊗ a ) = n ⊗ a (1) ⊗ a (2) , G ( M ) = M co A ∈ D ( X ) is given by the formula = M co A x , x M co A = { m ∈ M x , x | ρ x , x ( m ) = m ⊗ 1 x } . x , x x

The fundamental theorem Canonical maps: can z can z x , y : A z , x ⊗ A x , y → A z , y ⊗ A x , y , x , y ( a ⊗ b ) = ab (1) ⊗ b (2) . Theorem For a k-linear semi-Hopf category A with underlying class X, the following assertions are equivalent. 1. A is a k-linear Hopf category; 2. the pair of adjoint functors ( F , G ) is a pair of inverse equivalences between the categories D ( X ) and M k ( X ) A A ; 3. the functor G is fully faithful; 4. can z x , y is an isomorphism, for all x , y , z ∈ X; x , y and can y 5. can x x , y are isomorphisms, for all x , y ∈ X.