Hopf monoids in duoidal categories Gabriella B ohm Wigner Research - PowerPoint PPT Presentation

Hopf monoids in duoidal categories Gabriella B¨ ohm Wigner Research Centre for Physics Category Theory 2015, Aveiro 16th of June

Plan 1. ¿ What distinguishes groups among monoids ? 2. A universal approach: bimonoids in duoidal categories 3. Hopf-like conditions 4. Relations between them Based on the works GB, Y.Y. Chen, L.Y. Zhang, On Hopf monoids in duoidal categories , GB, S. Lack, Hopf comonads on naturally Frobenius map-monoidales .

Plan . 1. ¿ What distinguishes groups among monoids ? . . . and groupoids categories (weak) Hopf algebras (weak) bialgebras Hopf algebroids bialgebroids Hopf monads bimonads . . . and so on . . . (weak) bialgebras ? Based on the works GB, Y.Y. Chen, L.Y. Zhang, On Hopf monoids in duoidal categories , GB, S. Lack, Hopf comonads on naturally Frobenius map-monoidales .

Plan 1. ¿ What distinguishes groups among monoids ? . . . and groupoids categories (weak) Hopf algebras (weak) bialgebras Hopf algebroids bialgebroids Hopf monads bimonads . . . and so on . . . (weak) bialgebras ? 2. A universal approach: bimonoids in duoidal categories 3. Hopf-like conditions 4. Relations between them Based on the works GB, Y.Y. Chen, L.Y. Zhang, On Hopf monoids in duoidal categories , GB, S. Lack, Hopf comonads on naturally Frobenius map-monoidales .

Plan 1. ¿ What distinguishes groups among monoids ? . . . and groupoids categories (weak) Hopf algebras (weak) bialgebras Hopf algebroids bialgebroids Hopf monads bimonads . . . and so on . . . (weak) bialgebras ? 2. A universal approach: bimonoids in duoidal categories 3. Hopf-like conditions 4. Relations between them Based on the works GB, Y.Y. Chen, L.Y. Zhang, On Hopf monoids in duoidal categories , GB, S. Lack, Hopf comonads on naturally Frobenius map-monoidales .

1. ¿What distinguishes groups among monoids? — and questions of similar flavour

� � � ¿ What distinguishes groups among monoids ? ⇔ every element of C is invertible ⇔ ⇔ the Hopf map the C × C → C × C , ( a , b ) �→ ( a , ab ) is invertible ⇔ the dual Hopf map the C × C → C × C , ( a , b ) �→ ( ab , b ) is invertible ⇔ for any C -set M (i.e. module over the monoid C in set), the Galois morphism the M × C → M × C , ( m , b ) �→ ( mb , b ) is invertible ⇔ for any map (of sets) f : N → C , the dual Galois morphism the C × N → C × N , ( a , n ) �→ ( af ( n ) , n ) is invertible C -set / C ⇔ the comparison functor ( − ) × C the ( − ) × C : set → C -set / C is an equivalence forgetful set C -set ( − ) × C

� � � � � � � � � � � � � ¿ What distinguishes groupoids among categories ? ⇔ every arrow of C is invertible ⇔ ⇔ the Hopf map a b a a b the C s × t C → C t × t C , ( z ) �→ ( z y , y x y , z y x ) is invertible ⇔ the dual Hopf map a b a b b the C s × t C → C s × s C , ( z y , y x ) �→ ( z y x , y x ) is invertible ⇔ for any C -span M (i.e. module over the monoid C in Span( C 0 , C 0 )), the Galois morphism the M s × t C → M s × s C , ( m , b ) �→ ( mb , b ) is invertible ⇔ for any morphism f : N → C of spans, the dual Galois morphism C -span / C the C s × t N → C s × s N , ( a , n ) �→ ( af ( n ) , n ) is invertible ⇔ the comparison functor forgetful the ( − ) • × t C : set / C 0 → C -span / C is an equivalence C -span span ( − ) • × t C

� � � � � � � � � ¿ . . . and bialgebras among Hopf algebras ? bialgebra over a field k (more precisely a bit later) = µ � A η δ ε � k ) in vec compatible monoid ( A 2 ) and comonoid ( A 2 k A ⇔ ∃ an antipode A σ � A – a ‘convolution inverse’ of A 1 � A : A 2 � A 2 A δ σ 1 ε µ δ � k η A 2 � A 2 � A µ 1 σ 1 µ � A 2 δ 1 � A 3 ⇔ the Hopf map A 2 is invertible µ 1 � A 2 1 δ � A 3 ⇔ the dual Hopf map A 2 is invertible ξ � X , the Galois map XA 1 δ � XA 2 ξ 1 � XA ⇔ for any module XA is invertible ζ � AZ , the dual Galois map AZ 1 ζ � A 2 Z µ 1 � AZ ⇔ for any comodule Z is invertible ⇔ the Fundamental thm of Hopf modules holds: hopf( A ) ( − ) A : vec → hopf( A ) is an equivalence ( − ) A forgetful (where the objects of hopf( A ) are compatible vec mod( A ) ( − ) A A modules and comodules)

. . . and so on . . .

2. A universal approach: bimonoids in duoidal categories

� Duoidal category Definition [Aguiar-Mahajan]. A duoidal category consists of ◮ monoidal categories ( C , • , j ), ( C , ◦ , i ), ξ 0 � j ◦ j , ξ 0 ξ 0 � i 0 ◮ morphisms i • i j ξ � ( w • y ) ◦ ( x • z ) natural in w , x , y , z , ◮ morphisms ( w ◦ x ) • ( y ◦ z ) subject to coherence axioms: ◮ ( ◦ , ξ, ξ 0 ) is a • -monoidal functor and ∼ ∼ ∼ = = = ◮ ( x ◦ y ) ◦ z → x ◦ ( y ◦ z ), x ◦ i → x ← i ◦ x are • -monoidal equivalently, ◮ ( • , ξ, ξ 0 ) is a ◦ -opmonoidal functor and ∼ ∼ ∼ = = = ◮ ( x • y ) • z → x • ( y • z ), x • j → x ← j • x are ◦ -opmonoidal.

� � � � � � � � � Bimonoid In a duoidal category, . the monoidal structure ( ◦ , i ) lifts to the category of ( • , j )-monoids . the monoidal structure ( • , j ) lifts to the category of ( ◦ , i )-comonoids. Definition [Aguiar-Mahajan]. A bimonoid is . a comonoid in the category of ( • , j )-monoids, ⇔ a monoid in the category of ( ◦ , i )-comonoids. . Explicitly, µ η ◮ a monoid ( a • a → a ← j ) δ ε ◮ a comonoid ( a ◦ a ← a → i ) for which ξ 0 � µ µ � δ � a ◦ a a • a . a • a j ◦ j a a j j ξ 0 µ ◦ µ δ • δ ε • ε ε 0 η η ◦ η η � ( a • a ) ◦ 2 � i δ � a ◦ a ε � i ( a ◦ a ) • 2 i • i a a ξ ξ 0

0th example: braided monoidal categories any braided monoidal category ( C , ⊗ , k , τ ) is duoidal: A ◦ B := A ⊗ B , i = k A • B := A ⊗ B , j = k 1 ⊗ τ ⊗ 1 � A ⊗ C ⊗ B ⊗ D A ⊗ B ⊗ C ⊗ D . ξ : ( A ◦ B ) • ( C ◦ D ) ( A • C ) ◦ ( B • D ) bimonoid = usual bimonoid in a braided monoidal category .

� � � � � 1st example: span( X ) [Aguiar-Mahajan] t s � X objects: maps (of sets) X A A s � t morphisms: . X X f t ′ A ′ s ′ duoidal: A ◦ B := A s , t × s , t B , i = X × X (the categorical product) A • B := A s × t B , j = X � a � � c � a c ← ← �→ ( ← ← ) ξ : b d b d ← ← ( ← ← ) . (comonoids are trivial) bimonoid = monoid = small category with object set X

2nd example: vec X × X [Batista – Caenepeel – Vercruysse] the category of X × X -graded vector spaces over a field k , for a set X duoidal: ( V • W ) x , y := V x , y ⊗ W x , y j x , y = k ( V ◦ W ) x , y := � z ∈ X V x , z ⊗ W z , y i x , y = δ x , y k ξ : ( v • w ) ◦ ( v ′ • w ′ ) �→ ( v ◦ v ′ ) • ( w ◦ w ′ ) bimonoid = category enriched in the category of comonoids in vec — when X is a group, this includes semi-Hopf group coalgebras [Turaev]

3rd example: bim( R ) [Aguiar-Mahajan] the category of bimodules over a commutative algebra R duoidal: M ◦ N = M ⊗ R N ≡ M ⊗ N / { m · r ⊗ n − m ⊗ r · n } i = R M • N = M ⊗ R ⊗ R N ≡ M ⊗ N / { r · m · r ′ ⊗ n − m ⊗ r · n · r ′ } j = R ⊗ R ξ : ( m • n ) ◦ ( m ′ • n ′ ) �→ ( m ◦ m ′ ) • ( n ◦ n ′ ) bimonoid = R -bialgebroid (with 1 · r and r · 1 central elements, ∀ r ∈ R ) . [Takeuchi, Lu, Ravenel]

4th example: bim( R op ⊗ R ) [GB – G´ omez-Torrecillas – L´ opez-Centella] the category of R op ⊗ R -bimodules i e i ⊗ f i ∈ R ⊗ R , ψ : R → k ) – for a separable Frobenius k -algebra ( R , � duoidal: M • N = M ⊗ R op ⊗ R N ≡ M ⊗ N / { m · ( r ⊗ s ) ⊗ n − m ⊗ ( r ⊗ s ) · n } duoidal: j = R op ⊗ R (with the regular actions) duoidal: M ◦ N = M ⊗ R op ⊗ R N — wrt some twisted actions duoidal: i = R op ⊗ R (with some twisted actions) duoidal: ξ : ( m • n ) ◦ ( m ′ • n ′ ) �→ � i ( m · ( e i ⊗ 1) ◦ m ′ ) • ( n ◦ (1 ⊗ f i ) · n ′ ) bimonoid= weak bialgebra with base algebra R

4th example: bim( R op ⊗ R ) Definition . A separable Frobenius structure on a k -algebra R consists of ◮ a linear map ψ : R → k i e i ⊗ f i of R ⊗ R (that is, a linear map k → R ⊗ R ) ◮ an element � such that for all r ∈ R , i ψ ( re i ) f i = r = e i ψ ( f i r ) i e i f i = 1 � � and (triangle identities of a duality R ⊣ R in vec). ◮ it can be formulated in any monoidal category instead of vec ◮ it has several equivalent reformulations [Street] ◮ it has a number of nice properties (cf. ‘twisted actions’)

5th example: prof( M ) the category of functors M op × M → set – for a monoidal category ( M , ⊗ , I ) duoidal via the coends � p ∈ Ob( M ) F ( − , p ) × G ( p , − ) F ◦ G = (composition if writing prof( M ) ∼ = CoCont([ M op , set] , [ M op , set])) F ◦ G i = M ( − , − ) � p , q , r , s ∈ Ob( M ) M ( − , p ⊗ q ) × F ( p , r ) × G ( q , s ) × M ( r ⊗ s , − ) F • G = F ◦ G (a convolution formula) j = M ( − , I ) × M ( I , − ). T bimonoids are induced e.g. by monoidal comonads M → M as M ( − , T ( − )).

3. Hopf like conditions

For a bimonoid in a (good enough) duoidal category: Hopf map antipode dual Galois maps dual FTHM dual Hopf map Galois maps FTHM