SLIDE 1 Hopf monoids in duoidal categories Gabriella B¨
Wigner Research Centre for Physics
Category Theory 2015, Aveiro 16th of June
SLIDE 2 Plan
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.
SLIDE 3 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.
SLIDE 4 Plan
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.
SLIDE 5 Plan
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.
SLIDE 6
- 1. ¿What distinguishes groups among monoids?
— and questions of similar flavour
SLIDE 7 ¿ 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 ⇔ the comparison functor the (−) × C : set → C-set/C is an equivalence C-set/C
forgetful
(−)×C
SLIDE 8 ¿ What distinguishes groupoids among categories ?
⇔ every arrow of C is invertible ⇔ ⇔ the Hopf map the Cs ×t C → Ct ×t C, ( z y, y
a
b
y, z
a
a
b
⇔ the dual Hopf map the Cs ×t C → Cs ×s C, ( z y, y
a
b
y
a
b
b
⇔ for any C-span M (i.e. module over the monoid C in Span(C0, C0)), the Galois morphism the Ms ×t C → Ms ×s C, (m, b) → (mb, b) is invertible ⇔ for any morphism f : N → C of spans, the dual Galois morphism the Cs ×t N → Cs ×s N, (a, n) → (af (n), n) is invertible ⇔ the comparison functor the (−)•×t C : set/C0 → C-span/C is an equivalence C-span/C
forgetful
(−)•×tC
SLIDE 9 ¿ . . . and bialgebras among Hopf algebras ?
bialgebra over a field k (more precisely a bit later) = compatible monoid ( A2
µ A
k
η
A
δ
⇔ ∃ an antipode A σ A – a ‘convolution inverse’ of A
1 A :
A
δ
ε
σ1
A2
µ
η
1σ
A2
µ
A ⇔ the Hopf map A2
δ1 A3 1µ A2
is invertible ⇔ the dual Hopf map A2
1δ A3 µ1 A2
is invertible ⇔ for any module XA
ξ X, the Galois map XA 1δ XA2 ξ1 XA
is invertible ⇔ for any comodule Z
ζ AZ , the dual Galois map AZ 1ζ A2Z µ1 AZ
is invertible ⇔ the Fundamental thm of Hopf modules holds: (−)A : vec → hopf(A) is an equivalence (where the objects of hopf(A) are compatible A modules and comodules) hopf(A)
forgetful
(−)A
SLIDE 10 . . . and so on . . .
SLIDE 11
bimonoids in duoidal categories
SLIDE 12 Duoidal category
Definition [Aguiar-Mahajan]. A duoidal category consists of ◮ monoidal categories (C, •, j), (C, ◦, i), ◮ morphisms i • i
ξ0 i
j
ξ0
◮ morphisms (w ◦ x) • (y ◦ z)
ξ (w • y) ◦ (x • z) natural in w, x, y, z,
subject to coherence axioms: ◮ (◦, ξ, ξ0) is a •-monoidal functor and ◮ (x ◦ y) ◦ z
∼ =
→ x ◦ (y ◦ z), x ◦ i
∼ =
→ x
∼ =
← i ◦ x are •-monoidal equivalently, ◮ (•, ξ, ξ0) is a ◦-opmonoidal functor and ◮ (x • y) • z
∼ =
→ x • (y • z), x • j
∼ =
→ x
∼ =
← j • x are ◦-opmonoidal.
SLIDE 13 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, ◮ a monoid (a • a
µ
→ a
η
← j) ◮ a comonoid (a ◦ a
δ
← a
ε
→ i) for which . a • a
µ
δ
a ◦ a a • a
µ ε•ε
ε
ξ0 η
η◦η
η
ξ
(a • a)◦2
µ◦µ
ξ0
i a
δ a ◦ a
a
ε i
SLIDE 14 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 . ξ : A ⊗ B ⊗ C ⊗ D
1⊗τ⊗1
A ⊗ C ⊗ B ⊗ D (A ◦ B) • (C ◦ D) (A • C) ◦ (B • D) bimonoid = usual bimonoid in a braided monoidal category .
SLIDE 15 1st example: span(X)
[Aguiar-Mahajan]
A
t
morphisms: A
t
f
X A′
t′
duoidal: A ◦ B := As,t ×s,t B, i = X × X (the categorical product) A • B := As×t B, j = X ξ :
←
b
←
c
←
d
←
a
←
c
←) (
b
←
d
←) . (comonoids are trivial) bimonoid = monoid = small category with object set X
SLIDE 16 2nd example: vecX×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 := Vx,y ⊗ Wx,y jx,y = k (V ◦ W )x,y :=
z∈X Vx,z ⊗ Wz,y
ix,y = δx,yk 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]
SLIDE 17 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]
SLIDE 18 4th example: bim(Rop ⊗ R)
[GB – G´
- mez-Torrecillas – L´
- pez-Centella]
the category of Rop ⊗ R-bimodules – for a separable Frobenius k-algebra (R,
i ei ⊗ f i ∈ R ⊗ R, ψ : R → k)
duoidal: M • N = M ⊗Rop⊗R N ≡ M ⊗ N/{m · (r ⊗ s) ⊗ n − m ⊗ (r ⊗ s) · n} duoidal: j = Rop ⊗ R (with the regular actions) duoidal: M ◦ N = M ⊗Rop⊗R N — wrt some twisted actions duoidal: i = Rop ⊗ R (with some twisted actions) duoidal: ξ : (m • n) ◦ (m′ • n′) →
i(m · (ei ⊗ 1) ◦ m′) • (n ◦ (1 ⊗ f i) · n′)
bimonoid= weak bialgebra with base algebra R
SLIDE 19 4th example: bim(Rop ⊗ R)
- Definition. A separable Frobenius structure on a k-algebra R consists of
◮ a linear map ψ : R → k ◮ an element
i ei ⊗ f i of R ⊗ R (that is, a linear map k → R ⊗ R)
such that for all r ∈ R,
- i ψ(rei)f i = r = eiψ(f ir)
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’)
SLIDE 20 5th example: prof(M)
the category of functors Mop × M → set – for a monoidal category (M, ⊗, I) duoidal via the coends F ◦ G = p∈Ob(M) F(−, p) × G(p, −) F ◦ G (composition if writing prof(M) ∼ = CoCont([Mop, set], [Mop, set])) i = M(−, −) F • G = p,q,r,s∈Ob(M) M(−, p ⊗ q) × F(p, r) × G(q, s) × M(r ⊗ s, −) F ◦ G (a convolution formula) j = M(−, I) × M(I, −). bimonoids are induced e.g. by monoidal comonads M
T
→ M as M(−, T(−)).
SLIDE 22 For a bimonoid in a (good enough) duoidal category:
Galois maps FTHM dual FTHM Hopf map dual antipode dual Galois maps Hopf map
SLIDE 23 Fundamental theorem of Hopf modules (FTHM)
Let a be a bimonoid in a duoidal category (C, ◦, •) — then . mod(a) is monoidal via ◦ and a is a comonoid therein, . comod(a) is monoidal via • and a is a monoid therein.
- Definition. A Hopf module is a module over the monoid a in comod(a)
. ⇔ a comodule over the comonoid a in mod(a). Explicitly, an object n equipped with ◮ an associative action n • a
ν0 n
◮ a coassociative coaction n ν0 a ◦ n n • a
ν0
ν0
a ◦ n (a ◦ n) • (a ◦ a)
ξ (a • a) ◦ (n • a) µ◦ν0
- category hopf(a) of Hopf modules.
SLIDE 24 Fundamental theorem of Hopf modules (FTHM)
Comparison functor: hopf(a)
forgetful
(−)•a
(−)•a
mod(a) [Dubuc-Beck]
- a morphism of comonads on mod(a)
. — the so called Galois morphism: (j ◦ −) • a 1•δ (j ◦ −) • (a ◦ a)
ξ (j • a) ◦ (− • a) = a ◦ (− • a) 1◦action a ◦ −
SLIDE 25 Fundamental theorem of Hopf modules (FTHM)
- Theorem. For a duoidal category C in which idempotent morphisms split and
comod(j)
forgetful C (−)•i mod(i) is fully faithful, and a bimonoid a in C, tfae.
◮ the comparison functor comod(j)
(−)•a hopf(a) is an equivalence;
◮ the Galois morphism (j ◦ −) • a → a ◦ − is an invertible nat transformation.
- Proof. Applying comonadicity criteria.
SLIDE 26 The dual fundamental theorem of Hopf modules
- Theorem. For a duoidal category C in which idempotent morphisms split and
mod(i)
forgetful C j◦(−) comod(j) is fully faithful, and a bimonoid a in C, tfae.
◮ the comparison functor mod(i)
a◦(−) hopf(a) is an equivalence;
◮ the dual Galois morphism − • a → a ◦ (− • i) is invertible.
SLIDE 27 ¡ At this level of generality, we can not go any further!
⇓
We restrict the class of duoidal categories.
SLIDE 28 Duoidal categories from map-monoidales
[Street] Let B be a monoidal bicategory (monoidal product = juxtaposition, unit = I), any object B a monoidal category B(B, B) Definition. A monoidale (aka pseudo-monoid) in B is given by 1-cells MM
m M
I
u
- which are associative and unital up-to coherent iso 2-cells.
- Definition. A map-monoidale is a monoidale s.t. m ⊣ m∗ and u ⊣ u∗.
For a map-monoidale M, B(M, M) is duoidal: x ◦ y = M
y M x M
i = M
1 M
composition x • y = M m∗ MM
xy MM m M
j = M u∗ I
u M
convolution . ξ = M m∗ MM wz MM
m∗
xy MM m M
M
m
- unit
- bimonoid in B(M, M) = monoidal comonad (aka bicomonad) on M
SLIDE 29 Duoidal categories from map-monoidales
- Examples. One can regard as a map-monoidale
0th the single object if a braided monoidal category is regarded as a 0th monoidal bicategory 1st any set X in Spanv op 2nd any set X in Vec(−×−) v op 3rd any commutative algebra R in Bim 4th for any separable Frobenius algebra R, Rop ⊗ R in Bim 5th any monoidal category M with left and right duals in Prof yielding the duoidal structures of the examples.
SLIDE 30 The fundmental theorem of Hopf modules
Proposition. For a map-monoidale M in a monoidal bicategory B, comod(j)
forgetful B(M, M) (−)•i mod(i) is fully faithful.
Corollary (FTHM). For a map-monoidale M in a monoidal bicategory B in which idempotent 2-cells split, and a bicomonad a on M, tfae. ◮ the comparison functor comod(j)
(−)•a hopf(a) is an equivalence;
◮ the Galois morphism (j ◦ −) • a → a ◦ − is an invertible nat transformation.
SLIDE 31 For a bicomonad on a map-monoidale in a monoidal bicategory:
Galois maps FTHM antipode dual FTHM dual Hopf map Hopf map dual Galois maps if idempotents split
SLIDE 32 Hopf- and Galois conditions
a: bimonoid in B(M, M) = monoidal comonad (aka bicomonad) on M Definition [Brugui` eres-Lack-Virelizier]. a is a Hopf comonad if the Hopf map MM
aa
MM
m
M MM
a1
MM
m
a
(Cf. lifting of a coclosed structure on M to Ma [Chikhladze, Lack, Street].)
SLIDE 33 Hopf- and Galois conditions
- Definition. M is well-pointed if ∃ I
v
→ M s.t. B(MM, M)
B(v1,M)
B(M, M) is conservative.
- Examples. The map monoidales
0th the single object if a braided monoidal category is regarded as a 0th monoidal bicategory 1st any set X in Spanv op 2nd any set X in Vec(−×−)v op 3rd any commutative algebra R in Bim 4th for any separable Frobenius algebra R, Rop ⊗ R in Bim 5th any monoidal category M with left and right duals in Prof — yielding the duoidal structures of the examples — all are well-pointed.
- Theorem. For a bicomonad a on a well-pointed map-monoidale M, tfae.
◮ the Hopf map m.aa → a.m.a1 is iso, ◮ the dual Galois morphism − • a → a ◦ (− • i) is invertible.
SLIDE 34 For a bicomonad on a map-monoidale in a monoidal bicategory:
antipode dual FTHM Hopf map dual Galois maps Galois maps FTHM dual Hopf map if well−pointed if idempotents split
SLIDE 35 To treat the dual conditions, further restriction is needed.
SLIDE 36 Naturally Frobenius map-monoidale
Definition [L´
- pez Franco]. A map-monoidale (M, m, u) is naturally Frobenius if
MMM
m1 1m
m
MMM
1m m1
m
MM
1m∗
m M m∗
m∗1
m M m∗
= unit
= unit
- are iso 2-cells.
- Examples. The map monoidales
0th the single object if a braided monoidal category is regarded as a 0th monoidal bicategory 1st any set X in Spanv op 2nd any set X in Vec(−×−)v op 3rd any commutative algebra R in Bim 4th for any separable Frobenius algebra R, Rop ⊗ R in Bim 5th any monoidal category M with left and right duals in Prof — yielding the duoidal structures of the examples — all are naturally Frobenius.
SLIDE 37 Dual fundamental theorem of Hopf modules
Proposition. For a naturally Frobenius map-monoidale M in a monoidal bi- category B, mod(i)
forgetful B(M, M) j◦(−) comod(j) is fully faithful (thus an
equivalence). Corollary (Dual FTHM). For a naturally Frobenius map-monoidale M in a monoid- al bicategory B in which idempotent 2-cells split, and a bicomonad a on M, tfae. ◮ the comparison functor mod(i)
a◦(−) hopf(a) is an equivalence;
◮ the dual Galois morphism − • a → a ◦ (− • i) is invertible.
SLIDE 38 For a bicomonad on a naturally Frobenius map-monoidale:
Galois maps FTHM antipode Hopf map dual Galois maps dual Hopf map dual FTHM if idempotents split if well−pointed if idempotents split
SLIDE 39 Dual Hopf- and Galois conditions
Theorem. For a bicomonad a on a well-pointed naturally Frobenius map- monoidale M, tfae. ◮ the dual Hopf map aa.m∗ → 1a.m∗.a is iso, ◮ the Galois morphism (j ◦ −) • a → a ◦ − is an invertible nat transformation.
SLIDE 40 For a bicomonad on a naturally Frobenius map-monoidale:
Galois maps FTHM antipode dual FTHM Hopf map dual dual Galois maps Hopf map if idempotents split if well−pointed if idempotents split if well−pointed
SLIDE 41 Antipode
For a naturally Frobenius map-monoidale MM
m M
I
u
the unit I
u M m∗ MM and counit MM m M u∗ I.
The mate of M
a
→ M under this duality is a− : M
1u MM 1m∗ MMM 1a1 MMM m1 MM u∗1 M.
(−)− is a monoidal biequivalence ⇒ a− is a bimonoid whenever a is so.
SLIDE 42 Antipode
- Theorem. For a bicomonad a on a naturally Frobenius map-monoidale M, tfae.
◮ the Hopf map m.aa → a.m.a1 is iso ◮ ∃ σ : a → a− (called an antipode) rendering commutative a δ
ε
ϕa,a (j ◦ (a • a)) • i (j◦µ)•i
ε
- a ◦ a 1◦σ a ◦ a− ψa,a i • ((a • a) ◦ j)
i•(µ◦j)
j • i
ξ0•i
(j ◦ j) • i
(j◦η)•i
(j ◦ a) • i i i • j
i•ξ0
i • (j ◦ j)
i•(η◦j)
i • (a ◦ j) (where ϕf ,g : g − ◦f → (j ◦(f •g))•i and ψf ,g : g ◦f − → i •((f •g)◦j) are built up canonically from the (co)unitality and (co)associativity isomorphisms and the unit of the adjunction m ⊣ m∗). ¡ σ is a generalized convolution inverse of 1 : a → a !
SLIDE 43 For a bicomonad on a naturally Frobenius map-monoidale:
Galois maps FTHM dual FTHM Hopf map dual antipode dual Galois maps Hopf map if idempotents split if well−pointed if idempotents split if well−pointed
SLIDE 44 For a bicomonad on a naturally Frobenius map-monoidale:
Galois maps FTHM dual FTHM Hopf map dual antipode dual Galois maps Hopf map if idempotents split if well−pointed if idempotents split if well−pointed
SLIDE 45 Antipode
about the proof. ◮ Associated to a, there is a monad T and a comonad G on B(MM, M) and a mixed distributive law between them. ◮ With an appropriate choice of x, y : MM → M, the Hopf map is a morphism
- f mixed modules GTx → GTy.
◮ Thanks to the various adjunctions present, the full subcategory of two objects {GTx, GTy} in the category of mixed modules is isomorphic to a category
- f two objects X and Y whose morphisms
X → X are 2-cells a → i • (a ◦ j) X → Y are 2-cells a → a Y → X are 2-cells a → a− Y → Y are 2-cells a → (j ◦ a) • i and whose composition is given by the generalized convolution product. ◮ The image of the Hopf map under this isomorphism is a
1 a .
Hence the Hopf map is invertible ⇔ a
1 a is convolution invertible.
SLIDE 46 Antipode
- Theorem. Whenever the antipode a → a− exists, it is unique and a morphism
- f monoids and comonoids.
- Examples. Applying this notion of antipode in our examples, we re-obtain
0th the antipode of a Hopf monoid in a braided monoidal category 1st the inverse operation in a groupoid 2nd the antipode of ‘Hopf categories’ by Batista-Caenepeel-Vercruysse; 2nd in particular of Turaev’s Hopf group coalgebras 3rd the antipode of a Hopf algebroid 4th the antipode of a weak Hopf algebra 5th the antipode of a Brugui` eres-Virelizier Hopf monad.
