Entwining Structures in Monoidal Catrgories
ENTWINING STRUCTURES IN MONOIDAL CATEGORIES
B. MESABLISHVILI
Abstract. Interpreting entwining structures as special instances of J. Beck’s distribu-
tive law, the concept of entwining module can be generalized for the setting of arbitrary
monoidal category. In this paper, we use the distributive law formalism to extend in
this setting basic properties of entwining modules.
1. Introduction
The important notion of entwining structures has been introduced by T. Brzeziński and S.
Majid in [4]. An entwining structure (over a commutative ring K) consists of a K-algebra
A, a K-coalgebra C and a certain K-homomorphism λ : C ⊗K A → A ⊗K C satisfying
some axioms. Associated to λ there is the category MCA(λ) of entwining modules whose
objects are at the same time A-modules and C-comodules, with compatibility relation
given by λ.
The algebra A can be identified with the monad T = −⊗K A : ModK → ModK whose
Eilenberg-Moore category of algebras, (ModK)
T , is (isomorphic to) the category of right A-
modules. Similarly, C can be identified with the comonad G = −⊗K C : ModK → ModK ,
and the corresponding Eilenberg-Moore category of coalgebras with the category of C-
comodules. It turns out that to give an entwining structure C ⊗K A → A⊗K C is to give
a mixed distributive law TG → GT from the monad T to the comonad G in the sense
of J. Beck [2], which are in bijective corresondence with liftings (or extensions) G of the
comonad G to the category (ModK)
T ; or, equivalently, liftings T of the monad T to the
category (ModK)G. Moreover, the categories M
A(λ) , ((ModK)
T )G and ((ModK)G)
T are
isomorphic. Thus, the (mixed) distributive law formalism can be used to study entwining
structures and the corresponding category of modules. In this article -based on this
formalism- we extend in the context of monoidal categories some of basic results on
entwining structures that appear in the literature (see, for example, [5], [6], [11]).
The paper is organized as follows. After recalling the notion of Beck’s mixed distribu-
tive law and the basic facts about it, we define in Section 3 an entwining structure in
any monoidal category. In Section 4, we prove some categorical results that are needed
in the next section, but may also be of independent interest. Finally, in the last section
we present our main results.
We refer to M. Barr and C. Wells [1], S. MacLane [9] and F. Borceux [3] for terminology
Supported by the research project ”Algebraic and Topological Structures in Homotopical and Cate-
gorical Algebra, K-theory and Cyclic Homology“, with financial support of the grant GNSF/ST06/3-004.
2000 Mathematics Subject Classification: 16W30, 18D10, 18 D35.
Key words and phrases: Entwining module, (braided) monoidal category, Hopf algebra.
c© B. Mesablishvili, . Permission to copy for private use granted.
http://arxiv.org/abs/0704.1231v1
and general results on (co)monads, and to T. Brzeziński and R. Wisbauer [5] for coring
and comodule theory.
2. Mixed distributive laws
Let T = (T, η, µ) be a monad and G = (G, ε, δ) a comonad on a category A. A mixed
distributive law from T = (T, η, µ) to G = (G, ε, δ) is a natural transformation
λ : TG → GT
for which the diagrams
|| Gη
yy Tε
// GT , GT
// T,
Tδ // TG2
λG // GTG
and T 2G
Tλ // TGT
λT // GTT
// GGT TG
// GT
commute.
Given a monad T = (T, η, µ) on A, write AT for the Eilenberg-Moore category of T-
algebras, and write FT ⊣ UT : AT → A for the corresponding forgetful-free adjunction.
Dually, if G = (G, ε, δ) is comonad on A, then write AG for the category of G-coalgebras,
and write FG ⊣ UG : AG → A for the corresponding forgetful-cofree adjunction.
2.1. Theorem. ( see [12] ) Let T = (T, η, µ) be a monad and G = (G, ε, δ) a comonad
on a category A. Then the following structures are in bijective correspondences:
• mixed distributive laws λ : TG → GT;
• comonads Ḡ = (Ḡ, ε̄, δ̄) on AT that extend G in the sense that UT Ḡ = GUT , UT ε̄ =
εUT and UT δ̄ = δUT ;
• monads T̄ = (T̄ , η̄, µ̄) on AG that extend T in the sense that UGT̄ = TUG, UGη̄ =
ηUG and UGµ̄ = µUG.
These correspondences are constructed as follows:
• Given a mixed distributive law
λ : TG → GT,
then Ḡ(a, ξa) = (G(a), G(ξa) ·λa), ε̄(a,ξa) = εa, δ̄(a,ξa) = δa, for any (a, ξa) ∈ A
T; and
T̄ (a, νa) = (T (a), λa · T (νa)), η̄(a,νa) = ηa, µ̄(a,νa) = µa for any (a, νa) ∈ AG.
• If Ḡ = (Ḡ, ε̄, δ̄) is a comonad on AT extending the comonad G = (G, ε, δ), then the
corresponding distributive law
λ : TG → GT
is given by
TGη // TGT = UTF TGUTF T = UTF TUT ḠF T
UT εT ḠFT // UT ḠF T = GUTF T = GT,
where εT : F TUT → 1 is the counit of the adjunction F T ⊣ UT .
• If T̄ = (T̄ , η̄, µ̄) is a monad on AG extending T = (T, η, µ), then the corresponding
mixed distributive law is given by
TG = TUGFG = UGT̄FG
UGηGT̄ FG // UGFGUGT̄ FG = UGFGTUGFG = GTG
GTε // GT ,
where ηG : 1 → FGUG is the unit of the adjunction UG ⊣ FG.
It follows from this theorem that if
λ : TG → GT
is a mixed distributive law, then (AG)
T̄ = (AT)Ḡ. We write (A
)(λ) for this category.
An object of this category is a three-tuple (a, ξa, νa), where (a, ξa) ∈ A
T, (a, νa) ∈ AG,
for which G(ξa) · λa · T (νa) = νa · ξa. A morphism f : (a, ξa, νa) → (a
′, ξ′a, ν
a) in (A
is a morphism f : a → a′ in A such that ξ′a · T (f) = f · ξa and ν
a · f = G(f) · νa.
3. Entwining structures in monoidal categories
Let V = (V,⊗, I) be a monoidal category with coequalizers such that the tensor product
preserves the coequalizer in both variables. Then for all algebras A = (A, eA, mA) and
B = (B, eB, mB) and allM ∈ VA, N ∈ AVB and P ∈ BV, the tensor productM⊗AN exists
and the canonical morphism (M⊗AN)⊗BP → M⊗A (M⊗BP ) is an isomorphism. Using
MacLane’s coherence theorem (see, [9], XI.5), we may assume without loss of generality
that V is strict.
It is well known that every algebra A = (A, eA, mA) in V defines a monad TA on V by
• TA(X) = X ⊗ A,
• (ηTA)X = X ⊗ eA : X → X ⊗A,
• (µTA)X = X ⊗mA : X ⊗A⊗A → X ⊗ A,
and that VTA is (isomorphic to) the category VA of right A-modules.
Dually, if C = (C, εC, δC , ) is a coalgebra (=comonoid) in V, then one defines a
comonad GC on V by
• GC(X) = X ⊗ C,
• (εGC)X = X ⊗ εC : X ⊗ C → X,
• (δGC)X = X ⊗ δC : X ⊗ C → X ⊗ C ⊗ C,
and VGC is (isomorphic to) the category V
C of right C-comodules.
Quite obviously, if λ is a mixed distributive law from TA to GC, then the morphism
λ′ = λI : C ⊗ A → A⊗ C
makes the following diagrams commutative:
C ⊗ A
// A⊗ C , A⊗ C
// A,
C ⊗ A
δC⊗A // C ⊗ C ⊗A
C⊗λ′ // C ⊗ A⊗ C
C ⊗A⊗A
λ′⊗A // A⊗ C ⊗A
A⊗λ′ // A⊗A⊗ C
// A⊗ C ⊗ C, C ⊗ A
// A⊗ C .
Conversely, if λ′ : C ⊗A → A⊗C is a morphism for which the above diagrams commute,
then the natural transformation
−⊗ λ′ : TAGC(−) = −⊗ C ⊗A → −⊗ A⊗ C = GCTA(−)
is a mixed distributive law from the monad TA to the comonad GC. It is easy to see that
λ′ = (−⊗ λ′)I . When I is a regular generator in V and the tensor product preserves all
colimits in both variables, it is not hard to show that λ ≃ −⊗ λI . When this is the case,
then the correspondences λ → λI and λ
′ → −⊗ λ′ are inverses of each other.
3.1. Definition. An entwining structure (C,A, λ) consists of an algebra A = (A, eA, mA)
and a coalgebra C = (C, εC, δC) in V and a morphism λ : C ⊗ A → A⊗ C such that the
natural transformation
−⊗ λ : TAGC(−) = −⊗ C ⊗ A → −⊗ A⊗ C = GCTA(−)
is a mixed distributive law from the monad TA to the comonad GC.
Let be (C,A, λ) be an entwining structure and let Ḡ = (Ḡ, ε̄, δ̄) be the comonad on
VA that extends G = GC. Then we know that, for any (V, ξV ) ∈ VA,
Ḡ(V, ξV ) = (V ⊗ C, V ⊗ C ⊗A
V⊗λ // V ⊗ A⊗ C
ξV ⊗C // V ⊗ C).
In particular, since (A,mA) ∈ VA, A⊗ C is a right A-module with right action
ξA⊗C : A⊗ C ⊗ A
A⊗λ // A⊗ A⊗ C
ma⊗C// A⊗ C.
3.2. Lemma. View A ⊗ C as a left A-module through ξ̄A⊗C = mA ⊗ C. Then (A ⊗
C, ξ̄A⊗C, ξA⊗C) is an A-A-bimodule.
Proof. Clearly (A ⊗ C, ξ̄A⊗C) ∈ AV. Moreover, since (A ⊗ λ) · (mA ⊗ C ⊗ A) = (mA ⊗
A⊗ C) · (A⊗ A⊗ λ), it follows from the associativity of mA that the diagram
A⊗ A⊗ C ⊗A
A⊗A⊗λ//
mA⊗C⊗A
A⊗A⊗A⊗ C
A⊗mA⊗C
A⊗ C ⊗ A
A⊗ A⊗ C
A⊗A⊗ C
// A⊗ C
is commutative, which just means that (A⊗ C, ξ̄A⊗C, ξA⊗C) is an A-A-bimodule.
Since ε̄(A,mA) : Ḡ(A,mA) → (A,mA) and δ̄(A,mA) : Ḡ(A,mA) → Ḡ
2(A,mA) are
morphisms of right A-modules, and since UA(ε̄(A,mA)) = εA = (A ⊗ C
−→ A) and
UA(δ̄(A,mA)) = δC =(A ⊗ C
−→ A ⊗ C ⊗ C), it follows that A ⊗ C
−→ A and
−→ A⊗C⊗C are both morphisms of right A-modules. Clearly they are also mor-
phisms of left A-modules with the obvious left A-module structures arising from the multi-
plication mA : A⊗A → A, and hence morphisms of A-A-bimodules. Since C = (C, εC, δC)
is a coalgebra in V, it follows that the triple (A⊗ C)λ = (A⊗ C, ε(A⊗C)λ , δ(A⊗C)λ), where
ε(A⊗C)λ = A ⊗ C
−→ A and δ(A⊗C)λ = A ⊗ C
−→ A ⊗ C ⊗ C, is an A-coring. Since,
for any V ∈ VA, V ⊗A (A⊗ C) ≃ V ⊗ C, the comonad Ḡ is isomorphic to the comonad
G(A⊗C)λ . Thus, any entwining structure (C,A, λ) defines a right A-module structure ξA⊗C
on A ⊗ C such that (A ⊗ C, ξ̄A⊗C = mA ⊗ C, ξA⊗C) is an A-A-bimodule and the triple
(A⊗ C)λ = (A⊗C, ε(A⊗C)λ , δ(A⊗C)λ) is an A-coring. Moreover, when this is the case, the
comonad G(A⊗C)λ on VA extends the comonad GC. It follows that V
(A⊗C)λ
Conversely, let A = (A, eA, mA) be an algebra and C = (C, εC, δC) a coalgebra in V,
and suppose that A⊗ C has the structure ξA⊗C of a right A-module such that the triple
A⊗ C = ((A⊗ C,mA ⊗ C, ξA⊗C), A⊗ C
A⊗εC // A, A⊗ C
A⊗δC // A⊗ C ⊗ C) (1)
is an A-coring. Then it is easy to see that the comonad GA⊗C on VA extends the comonad
GC on V, and thus defines an entwining structure λA⊗C : C ⊗A → A⊗ C.
Summarising, we have
3.3. Theorem. Let A = (A, eA, mA) be an algebra and C = (C, εC, δC) a coalgebra in V.
Then there exists a bijection between right A-module structures ξA⊗C making (A⊗C,mA⊗
C, ξA⊗C) an A-bimodule for which the triple (1) is an A-coring and entwining structures
(C,A, λ), given by:
ξA⊗C // (λA⊗C : C ⊗A
eA⊗C⊗A // A⊗ C ⊗A
ξA⊗C // A⊗ C)
with inverse given by
λ // (ξA⊗C : A⊗ C ⊗A
A⊗λ // A⊗A⊗ C
mA⊗C // A⊗ C)
Under this equivalence V
(A⊗C)λ
4. Some categorical results
Let G = (G, ε, δ) be a comonad on a category A, and let UG : AG → A be the forgetful
functor. Fix a functor F : B → A, and consider a functor F̄ : B → AG making the
diagram
F̄ //
F ��?
UG}}||
commutative. Then F̄ (b) = (F (b), αF (b)) for some αF (b) : F (b) → GF (b). Consider the
natural transformation
ᾱF : F → GF, (3)
whose b-component is αF (b).
It is proved in [7] that:
4.1. Theorem. Suppose that F has a right adjoint R : A → B with unit η : 1 → FU and
counit ε : FU → 1. Then the composite
tF̄ : FU
// GFU
Gε // G.
is a morphism from the comonad G′ = (FU, ε, FηU) generated by the adjunction η, ε :
F ⊣ U : B → A to the comonad G. Moreover, the assignment
F̄ −→ tF̄
yields a one to one correspondence between functors F̄ : B → AG making the diagram (2)
commutative and morphisms of comonads tF̄ : G
′ → G.
Write βU for the composite U
ηU // UFU
UtF̄ // UG .
4.2. Proposition. The equalizer Ū , if it exists, of the following diagram
UUGηG //
// UGUG = UUGFGUG,
where ηG : 1 → FGUG is the unit of the adjunction UG ⊣ FG, is right adjoint to F .
Proof. See [3] or [7].
Let F̄ : B → AG be a functor making (2) commutative and let tF̄ : G
′ → G be the
corresponding morphism of comonads. Consider the following composition
G′ // AG′
F̄ // AG,
where
• KG′ : B → AG′, KG′(b) = (F (b), F (ηb)) is the Eilenberg-Moore comparison functor
for the comonad G′.
• AtF̄ is the functor
((a, θa) ∈ A
) −→ ((a, (tF̄ )a · θa) ∈ AG)
induced by the morphism of comonads tF̄ : G
′ → G.
4.3. Lemma. The diagram
KG′ //
F̄   B
B AG′
is commutative.
Proof. Let b ∈ B. Then KG′(b) = (F (b), F (ηb)) and AtF̄ (F (b), F (ηb)) = (F (b), (tF̄ )F (b) ·
F (ηb)). Since (tF̄ )F (b) is the composite
FUF (b)
(ᾱF )UF (b) // GFUF (b)
GεF (b)// GF (b),
and since by naturality of ᾱF , the diagram
F (b)
(ᾱ)b //
F (ηb)
GF (b)
GF (ηb)
FUF (b)
(ᾱ)UF (b)
// GFUF (b)
commutes, we have
(tF̄ )F (b) · F (ηb) = G(εF (b)) · (ᾱF )UF (b) · F (ηb) = G(εF (b)) ·GF (ηb) · (ᾱF )b = (ᾱF )b = αF (b).
(AtF̄ ·KG′)(b) = AtF̄ (KG′(b)) = AtF̄ (F (b), F (ηb)) = (F (b), (tF̄ )F (b) ·F (ηb)) = (F (b), αF (b)),
which just means that AtF̄ ·KG′ = F̄ .
We are now ready to prove the following
4.4. Theorem. Let G be a comonad on a category A, η, ε : F ⊣ U : B → A an adjunction
and F̄ : B → AG a functor with UG · F̄ = F . Then the following are equivalent:
(i) The functor F̄ is an equivalence.
(ii) The functor F is comonadic and the morphism of comonads
tF̄ : G
′ = (FU, ε, FηU) → G
is an isomorphism.
Proof. Suppose that F̄ is an equivalence of categories. Then F is isomorphic to the
comonadic functor UG and thus is comonadic. Hence the comparison functor KG′ : B →
AG′ is an equivalence and it follows from the commutative diagram (4) that AtF̄ is also
an equivalence, and since the diagram
F̄ //
UG′ !!C
UG~~||
is commutative, tF̄ is an isomorphism of comonads. So (i) =⇒ (ii).
Suppose now that tF̄ : G
′ → G is an isomorphism of comonads and F is comonadic.
• KG′ is an equivalence, since F is comonadic.
• AtF̄ is an equivalence, since tF̄ is an isomorphism.
And it now follows from the commutative diagram (4) that F̄ is also an equivalence. Thus
(ii) =⇒ (i). This completes the proof of the theorem.
4.5. Remark. In [8], J. Gómez-Torrecillas has proved that F̄ is an equivalence of cate-
gories iff tF̄ is an isomorphism of comonads, F is conservative, and for any (X, x) ∈ AG,
F preserves the equalizer of the pair of parallel morphisms
ηU(X)
//UG′(X)
U((tF̄ )X)
//UG(X) . (5)
When tF̄ is an isomorphism of comonads, to say that F preserves the equalizer of the pair
of morphisms (5) is to say that F preserves the equalizer of the pair of morphisms
ηU(X) //
)X )·U(x)
// UG′(X),
which we can rewrite as
ηU(X) //
)X ·x)
// UG′(X) = UFU(X). (6)
Since tF̄ is an isomorphism of comonds, AtF̄ is an equivalence of categories, and thus each
object (X, x′) ∈ AG′ is isomorphic to theG
′-coalgebra (X, (t−1
)X · x), where (X, x) ∈ AG.
It follows that when tF̄ is an isomorphism of comonds, to say that F preserves the equalizer
of (5) for each (X, x) ∈ AG is to say that F preserves the equalizer of (6) for each
(X, x′) ∈ AG′ . Thus, when tF̄ is an isomorphism of comonds, F̄ is an equivalence of
categories iff F is conservative and preserves the equalizer of (6) for each (X, x′) ∈ AG′ ,
which according to (the dual of) Beck’s theorem (see [9]), is to say that the functor F is
comonadic. Hence our theorem 4.4 is equivalent to Theorem 1.7 of [8].
5. Some applications
Let (C,A, λ) be an entwining structure in a monoidal category V = (V,⊗, I), and let
g : I → C be a group-like element of C. (Recall that a morphism g : I → C is said to be
a group-like element of C if the following diagrams
g⊗g ##F
I C ⊗ C
are commutative.)
5.1. Proposition. If C has a group-like element g : I → C, then A is a right C-comodule
through the morphism
gA : A
g⊗A // C ⊗ A
λ // A⊗ C.
Proof. Consider the diagram
g⊗A // C ⊗ A
A A .
The triangle is commutative by (1) of the definition of g and the square is commutative
by the definition of λ (see the second commutative diagram in the definition of entwining
structures).
Now, we have to show that the following diagram
g⊗A // C ⊗ A
λ // A⊗ C
C ⊗ A
g⊗A⊗C
// C ⊗A⊗ C
// A⊗ C ⊗ C
is also commutative, which it is since
(A⊗ δC)λ = (λ⊗ C)(C ⊗ λ)(δC ⊗ A)
by the definition of λ and since the diagram (2) of definition of group-like elements is
commutative.
Suppose now that V admits equalizers. For any (M,αM) ∈ V
C, write ((M,αM)
C, iM)
for the equalizer of the morphisms
(M,αM)
iM // M
αM //
// M ⊗ C.
5.2. Proposition. AC = (A, gA)
C is an algebra in V and iA : A
C → A is an algebra
morphism.
Proof. Consider the diagram
iA //A
//C ⊗ A
//A⊗ C
���������
Since
g ⊗− : 1V = I ⊗− → C ⊗−
is a natural transformation, the diagram
g // C
// C ⊗ A
is commutative. Similarly, since eA⊗− : 1V = I⊗− → C⊗− is a natural transformation,
the following diagram is also commutative:
eA //
// A⊗ C .
Now we have:
λ(g ⊗ A)eA = λ(C ⊗ eA)g = by the definition of λ
= (eA ⊗ C)g = (A⊗ g)eA.
Thus there exists a unique morphism eA : I → A
C for which iA · eAC = eA.
Since
• the diagram
g⊗A⊗A// C ⊗ A⊗A
// C ⊗ A
is commutative by naturality of g ⊗−;
• λ(C ⊗mA) = (mA ⊗ C)(A⊗ λ)(λ⊗ A) by the definition of λ;
• λ(g ⊗ A)iA = (A⊗ g)iA, since iA is an equalizer of λ(g ⊗ A) and A⊗ g;
• the diagram
A⊗A⊗g// A⊗ A⊗ C
// A⊗ C
is commutative by naturality of mA ⊗−,
we have
λ(g ⊗A)mA(iA ⊗ iA) = λ(C ⊗mA)(g ⊗ A⊗ A)(iA ⊗ iA) =
= (mA ⊗ C)(A⊗ λ)(λ⊗ A)(g ⊗ A⊗ A)(iA ⊗ iA) =
= (mA ⊗ C)(A⊗ λ)(A⊗ g ⊗ A)(iA ⊗ iA) = (mA ⊗ C)(A⊗A⊗ g)(iA ⊗ iA) =
= (A⊗ g)mA(iA ⊗ iA).
Thus the morphism mA · (iA ⊗ iA) equalizes the morphisms λ · (g ⊗ A) and A ⊗ g, and
hence there is a unique morphism
mAC : A
C ⊗AC → AC
such that the diagram
AC ⊗ AC
iA⊗iA // A⊗ A
commutes. It is now straightforward to show that the triple (AC, eAC , mAC) is an algebra
in V; moreover, the triangle of the diagram (7) and the diagram (8) show that iA is an
algebra morphism.
5.3. Proposition. (A,mA, gA) ∈ V
Proof. Since (A,mA) ∈ VA and (A, gA) ∈ V
C, it only remains to show that the following
diagram is commutative:
gA⊗A //
A⊗ C ⊗ A
A⊗λ // A⊗ A⊗ C
// A⊗ C.
By the definition of gA, we can rewrite it as
g⊗A⊗A// C ⊗ A⊗ A
λ⊗A //
A⊗ C ⊗ A
A⊗λ // A⊗ A⊗ C
// C ⊗A
// A⊗ C.
But this diagram is commutative, since
• the middle square commutes because of naturality of g ⊗−;
• the right square commutes because of the definition of λ.
The algebra morphism iA : A
C → A makes A an AC-AC-bimodule and thus induces
the extension-of-scalars functor
FiA : VAC → VA
(X, ρX) −→ (X ⊗AC A,X ⊗AC mA),
and the forgetful functor
UiA : VA → VAC
(Y, ̺Y ) −→ (Y, ̺Y · (Y ⊗ iA)),
which is right adjoint to FiA. The corresponding comonad on VA makes A⊗AC A into an
A-coring with the following counit and comultiplication:
ε : A⊗AC A
q // A⊗A
mA // A,
(where q is the canonical morphism) and
δ : A⊗AC A = A⊗AC A
C ⊗AC A
iA⊗ACA// A⊗AC A⊗AC A = (A⊗AC A)A ⊗ (A⊗AC A).
We write A⊗AC A for this A-coring.
5.4. Lemma. For any X ∈ VAC, the triple
(X ⊗AC A,X ⊗AC mA, X ⊗AC gA)
is an object of the category VC
Proof. Clearly (X ⊗AC A,X ⊗AC mA) ∈ VA and ((X ⊗AC A,X ⊗AC gA) ∈ V
C. Moreover,
by (9), the following diagram
X ⊗AC X ⊗AC A⊗ A
X ⊗AC A⊗ C ⊗A
// X ⊗AC A⊗A⊗ C
X ⊗AC A X⊗
// X ⊗AC A⊗ C
is commutative. Thus, (X ⊗AC A,X ⊗AC mA, X ⊗AC gA) ∈ V
The lemma shows that the assignment
X −→ (X ⊗AC A,X ⊗AC mA, X ⊗AC gA)
yields a functor
F̄ : VA → V
(λ) = V
(A⊗C)λ
It is clear that U(A⊗C)λ ·F̄ = FiA, where U(A⊗C)λ : V
(A⊗C)λ
→ VA is the underlying functor.
It now follows from Theorem 3.1 that the composite
A⊗AC A
A⊗gA // A⊗A⊗ C
mA⊗C // A⊗ C
is a morphism of A-corings A⊗AC A → (A⊗ C)λ. We write can for this morphism. We
say that A is (C, g)-Galois if can is an isomorphism of A-corings.
Applying Theorem 4.4 the commutative diagram
F̄ //
FiA=−⊗ACA $$J
(A⊗C)λ
U(A⊗C)λ
we get:
5.5. Theorem. Let (C,A, λ) be an entwining structure, and let g : I → C be a group-like
element of C. Then the functor
F̄ : VAC → V
is an equivalence if and only if A is (C, g)-Galois and the functor F is comonadic.
Let A = (A, eA, mA) and B = (B, eB, mB) be algebras in V and let M ∈ AVB. We call
AM (resp. MB)
• flat, if the functor − ⊗AM : VA → VB (resp. M ⊗B − : BV → AV) preserves
equalizers;
• faithfully flat, if the functor − ⊗AM : VA → VB (resp. M⊗B − : BV → AV) is
conservative and flat (equivalently, preserves and reflects equalizers);
5.6. Theorem. Let (C,A, λ) be an entwining structure, and let g : I → C be a group-like
element of C. If C is flat, then the following are equivalent
(i) The functor
F̄ : VAC → V
(λ) = VA
(A⊗C)λ
is an equivalence of categories.
(ii) A is (C, g)-Galois and ACA is faithfully flat.
Proof. Since any left adjoint functor that is conservative and preserves equalizers is
comonadic by a simple and well-known application (of the dual of) Beck’s theorem, one
direction is clear from Therem 5.5; so suppose that F̄ is an equivalence of categories.
Then, by Theorem 4.5, A is (C, g)-Galois and the functor FiA is comonadic. Since any
comonadic functor is conservative, FiA is also conservative. Thus, it only remains to show
that ACA is flat.
Since C is flat by our assumption, A(A⊗ C) is also flat. It follows that the underlying
functor of the comonad G(A⊗C)λ on VA preserves equalizers. We recall (for example, from
[3]) that if G = (G, εG, δG) is a comonad on a category A, and if A has some type of limits
preserved by G, then the category AG has the same type of limits and these are preserved
by the underlying functor UG : AG → A. Thus the functor U(A⊗C)λ : VA
(A⊗C)λ → VA
preserves equalizers, and since F̄ is an equivalence of categories, the functor FiA = −⊗ACA
also preserves equalizers, which just means that ACA is flat. This completes the proof.
From now on we suppose at all times that our V is a strict braided monoidal category
with braiding σX,Y : X ⊗ Y → Y ⊗X . Then the tensor product of two (co)algebras in V
is again a (co)algebra; the multiplication mA⊗B and the unit eA⊗B of the tensor product
of two algebras A = (A, eA, mA) and B = (B, eB, mB) are given through
mA⊗B = (mA ⊗mB)(A⊗ σA,B ⊗B)
eA⊗B = eA ⊗ eB.
A bialgebra H = (H̄ = (H, eH , mH), H = (H, εH , δH)) in V is an algebra H̄ =
(H, eH , mH) and a coalgebra H = (H, εH , δH), where εH and δH are algebra morphisms,
or, equivalently, eH and mH are coalgebra morphisms.
A Hopf algebra H = (H̄ = (H, eH , mH), H = (H, εH, δH), S) in V is a bialgebra H
with a morphism S : H → H , called the antipode of H, such that
mH(H ⊗ S)δH = mH(S ⊗H)δH .
Recall that for any bialgebra H, the category VH is monoidal: The tensor product
(X, δX) ⊗ (Y, δY ) of two right H-comodules (X, δX) and (Y, δY ) is their tensor product
X ⊗ Y in V with the coaction
δX⊗Y : X ⊗ Y
δX⊗δY // X ⊗H ⊗ Y ⊗H
X⊗σX,Y ⊗Y // X ⊗ Y ⊗H ⊗H
X⊗Y⊗mH // X ⊗ Y ⊗H .
The unit object for this tensor product is I with trivial H-comodule structure eH : I → H.
5.7. Proposition. Let H = (H̄ = (H, eH , mH), H = (H, εH , δH)) be a bialgebra in V.
For any algebra A = (A, eA, mA) in V, the following conditions are equivalent:
• A = (A, eA, mA) is an algebra in the monoidal category V
• A = (A, eA, mA) is an H-comodule algebra; that is, A is a right H-comodule and
the H-comodule coaction αA : A → A⊗H is a morphism of algebras in V from the
algebra A = (A, eA, mA) to the algebra A⊗ H̄ = (A⊗ H̄, eA ⊗ eH , mA⊗H̄).
Suppose now that A = (A, eA, mA) is a right H-comodule algebra with H-coaction
αA : A → A⊗H . By the previous proposition, A is an algebra in the monoidal category
VH , and thus defines a monad TAH = (T
H , η
H , µ
H) on V
H as follows:
• TAH (X, δX) = (X, δX)⊗ (A, αA);
• (ηAH)(X,δX ) = X ⊗ eA;
• (µAH)(X,δX) = X ⊗mA.
It is easy to see that the monad TAH extends the monad T
A; and it follows from Theorem
2.1 that there exists a distributive law λα : T
A ·GH → GH ·T
A from the monad TA to the
comonad GH , and hence an entwining structure (H,A, λ(A,αA)), where λ(A,αA) = (λα)I .
Therefore we have:
5.8. Theorem. Every right H-comodule algebra A = ((A, αA), mA, eA) defines an entwin-
ing structure (H,A, λ(A,αA) : C ⊗A → A⊗ C).
5.9. Proposition. Let A = ((A, αA), mA, eA) be a right H-comodule algebra. Then the
entwining structure λA,αA : H ⊗ A → A⊗H is given by the composite:
H ⊗ A
H⊗αA// H ⊗ A⊗H
σH,A⊗H// A⊗H ⊗H
A⊗mH// A⊗H .
Proof. Since (A⊗ αA) , (H, δH) ∈ V
H , the pair (A⊗H, δA⊗H), where δA⊗H is the com-
posite
δH⊗αA// H ⊗H ⊗H ⊗A
H⊗σH,A⊗H // H ⊗ A⊗H ⊗H
H⊗A⊗mH // H ⊗ A⊗H ,
is also an object of VH , and it follows from Theorem 1.1 that λ(A,αA) is the composite
H ⊗ A
δA⊗H // H ⊗ A⊗H
εH⊗A⊗H // A⊗H.
Consider now the following diagram
H ⊗A⊗H
δH⊗A⊗H //
H ⊗H⊗A⊗H
H⊗σH,A⊗H //
εH⊗H⊗A⊗H
H ⊗ A⊗H ⊗H
H⊗A⊗mH //
εH⊗A⊗H⊗H
H ⊗ A⊗H
εH⊗A⊗H
H ⊗ A⊗H
σH,A⊗H
// A⊗H ⊗H
// A⊗H .
Since in this diagram
• the triangle commutes because εH is the counit for δH ;
• the left square commutes by naturality of σ;
• the right square commutes because −⊗− is a bifunctor,
it follows that
λ(A,αA) = (A⊗mH)(σH,A ⊗H)(H ⊗ αA).
Note that the morphism eH : I → H is a group-like element for the coalgebra H =
(H, εH , δH).
5.10. Proposition. Let H = (H̄ = (H, eH , mH), H = (H, εH, δH)) be a bialgebra in V,
and let A = ((A, αA), eA, mA) be a right H-comodule algebra. Then the right H-comodule
structure on A corresponding to the group-like element eH : I → H as in Proposition 4.1
coincides with αA.
Proof. We have to show that
(A⊗mH)(σH,A ⊗H)(H ⊗ αA)(eH ⊗ A) = αA.
But since
• clearly (H ⊗ αA)(eH ⊗ A) = (eH ⊗A⊗H) · αA;
• (σH,A ⊗H) · (eH ⊗A⊗H) = A⊗ eH ⊗H by naturality of σ;
• (A⊗mH) · (A⊗ eH ⊗H) = 1A⊗H since eH is the identity for mH ,
we have that
(A⊗mH)(σH,A ⊗H)(H ⊗ αA)(eH ⊗ A) =
= (A⊗mH)(σH,A ⊗H)(eH ⊗A⊗H)αA =
= (A⊗mH)(A⊗ eH ⊗H)αA =
= 1A⊗H · αA = αA.
It now follows from Proposition 5.3 that
5.11. Proposition. A = (A, eA, mA) ∈ V
(λA,αA).
Recall that for any (X,αX) ∈ V
H , the algebra XH = (X,αX)
H is the equalizer of the
morphisms
αX //
// X ⊗H.
Applying Theorem 5.5 we get
5.12. Theorem. Let H = (H̄ = (H, eH , mH), H = (H, εH , δH)) be a bialgebra in V, let
A = ((A, αA), eA, mA) be a right H-comodule algebra, and let λ(A,αA) : H ⊗A → A⊗H be
the corresponding entwining structure. Then the functor
F̄ : VAH → V
(λ(A,αA))
(X, νX) −→ (X ⊗AH A,X ⊗AH mA, X ⊗AH αA)
is an equivalence of categories iff the extension-of-scalars functor
FiA : VAH → VA
(X, νX) −→ (X ⊗AH A,X ⊗AH mA)
is comonadic and A is H-Galois (in the sense that the canonical morphism
can : A⊗AH A → A⊗H
is an isomorphism).
Now applying Theorem 5.6 we get
5.13. Theorem. Let H = (H̄ = (H, eH , mH), H = (H, εH , δH)) be a bialgebra in V, let
A = ((A, αA), eA, mA) be a right H-comodule algebra, and let λ(A,αA) : H ⊗ A → A ⊗ H
be the corresponding entwining structure. Suppose that H is flat. Then the following are
equivalent:
(i) The functor
F̄ : VAH → V
(λ(A,αA))
(X, νX) −→ (X ⊗AH A,X ⊗AH mA, X ⊗AH αA)
is an equivalence of categories.
(ii) A is H-Galois and AHA is faithfully flat.
Let H = (H̄ = (H, eH , mH), H = (H, εH , δH)) be a bialgebra in V, and let A =
((A, αA), eA, mA) be a right H-comodule algebra. A right (A,H)-module is a right A-
module which is a right H-comodule such that the H-comodule structure morphism is
a morphism of right A-modules. Morphisms of right (A,H)-modules are right A-module
right H-comodule morphisms. We write VH
for this category. Note that the category VH
is the category (VH)A of right A-modules in the monoidal category V
H , and it follows
from Theorem 2.1 that
5.14. Proposition. VH
(λ(A,αA)).
The following is an immediate consequence of Theorem 5.12.
5.15. Theorem. Let H = (H̄ = (H, eH, mH), H = (H, εH, δH)) be a bialgebra in V, and
let A = ((A, αA), eA, mA) be a right H-comodule algebra. Then the functor
F̄ : VAH → V
is an equivalence of categories iff the extension-of-scalars functor
FiA : VAH → VA
is comonadic and A is H-Galois.
Let H = (H̄ = (H, eH , mH), H = (H, εH , δH), S) be an Hopf algebra in V. Then clearly
H̄ = (H, eH , mH) is a right H-comodule algebra.
5.16. Proposition. The composite
x : H ⊗H
H⊗δH // H ⊗H ⊗H
mH⊗H// H ⊗H
is an isomorphism.
Proof. We will show that the composite
y : H ⊗H
H⊗δH // H ⊗H ⊗H
H⊗S⊗H// H ⊗H ⊗H
mH⊗H// H ⊗H
is the inverse for x. Indeed, consider the diagram
H⊗δH // H ⊗H ⊗H
mH⊗H //
H⊗H⊗δH
H ⊗H ⊗H
H⊗δH⊗H // H ⊗H ⊗H ⊗H
H⊗H⊗S⊗H
mH⊗H⊗H // H ⊗H ⊗H
H⊗S⊗H
H ⊗H ⊗H ⊗H
H⊗mH⊗H
mH⊗H⊗H // H ⊗H ⊗H
H ⊗H ⊗H
// H ⊗H .
We have:
• Square (1) commutes because of coassociativity of δH ;
• Square (2) commutes because of naturality of mH ⊗−;
• Square (3) commutes because −⊗− is a bifunctor;
• Square (4) commutes because of associativity of mH .
yx = (mH ⊗H)(H ⊗ S ⊗H)(H ⊗ δH)(mH ⊗H)(H ⊗ δH) =
= (mH ⊗H)(H ⊗mH ⊗H)(H ⊗H ⊗ S ⊗H)(H ⊗ δH ⊗H)(H ⊗ δH),
but since
mH(H ⊗ S)δH = eH · εH ,
yx = (mH ⊗H)(H ⊗ eHεH ⊗H)(H ⊗ δH) =
= (mH ⊗H)(H ⊗ eH ⊗H)(H ⊗ εH ⊗H)(H ⊗ δH) =
= 1H⊗H ⊗ 1H⊗H = 1H⊗H .
Thus yx = 1. The equality xy = 1 can be shown in a similar way.
5.17. Proposition. (H, δH)
H ≃ (I, eH).
Proof. We will first show that the diagram
H⊗eH //
// H ⊗H
δH //
// H ⊗H
is serially commutative. Indeed, we have:
x(H ⊗ eH) = (mH ⊗H)(H ⊗ δH)(H ⊗ eH) = since δH is an algebra morphism
= (mH ⊗H)(H ⊗ eH ⊗ eH) = since eH is the unit for mH
= H ⊗ eH ;
x(eH ⊗H) = (mH ⊗H)(H ⊗ δH)(eH ⊗H) = since eH is a coalgebra morphism
= (mH ⊗H)(eH ⊗H)δH = 1HδH = δH .
Thus, (H, δH , eH)
is isomorphic to the equalizer of the pair (H⊗ eH , eH ⊗H). But since
eH : I → H is a split monomorphism in V, the diagram
eH // H
H⊗eH //
// H ⊗H
is an equalizer diagram. Hence (H, δH , eH)
H ≃ (I, eH).
5.18. Theorem. Let H = (H̄ = (H, eH , mH), H = (H, εH , δH), S) be a Hopf algebra in
V. Then the functor
V → VH
V → V ⊗H
is an equivalence of categories.
Proof. It follows from Propositions 5.16 and 5.17 that H is H-Galois, and according to
Theorem 5.12, the functor V → VH
is an equivalence iff the functor − ⊗H : V → VH̄ is
comonadic. But since the morphism eH : I → H is a split monomorphism in V, the unit
of the adjunction FeH ⊣ UeH is a split monomorphism, and it follows from 3.16 of [10]
that FeH is comonadic. This completes the proof.
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	Introduction
	Mixed distributive laws
	Entwining structures in monoidal categories
	Some categorical results
	Some applications
