Azumaya monads and comonads

The definition of Azumaya algebras over commutative rings $R$ require the tensor product of modules over $R$ and the twist map for the tensor product of any two $R$-modules. Similar constructions are available in braided monoidal categories and Azumaya algebras were defined in these settings. Here we introduce Azumaya monads on any category $\A$ by considering a monad $\bF$ on $\A$ endowed with a distributive law $\lambda: FF\to FF$ satisfying the Yang-Baxter equation (BD-law). This allows to introduce an {\em opposite monad} $\bF^\la$ and a monad structure on $FF^\la$. For an {\em Azumaya monad} we impose the condition that the canonical comparison functor induces an equivalence between the category $\A$ and the category of $\bF\bF^\la$-modules. Properties and characterisations of these monads are studied, in particular for the case when $F$ allows for a right adjoint functor. Dual to Azumaya monads we define {\em Azumaya comonads} and investigate the interplay between these notions. In braided categories $(\V,\ot,I,\tau)$, for any $\V$-algebra $A$, the braiding induces a BD-law $\tau_{A,A}:A\ot A\to A\ot A$ and $A$ is called left (right) Azumaya, provided the monad $A\ot-$ (resp. $-\ot A$) is Azumaya. If $\tau$ is a symmetry, or if the category $\V$ admits equalisers and coequalisers, the notions of left and right Azumaya algebras coincide. The general theory provides the definition of coalgebras in $\V$. Given a cocommutative $\V$-coalgebra $\bD$, coalgebras $\bC$ over $\bD$ are defined as coalgebras in the monoidal category of $\bD$-comodules and we describe when these have the Azumaya property. In particular, over commutative rings $R$, a coalgebra $C$ is Azumaya if and only if the dual $R$-algebra $C^*=\Hom_R(C,R)$ is an Azumaya algebra.

separable algebras, that is, the multiplication A ⊗ R A → A splits as (A, A)-bimodule map.
Braided monoidal categories allow for similar constructions as module categories over commutative rings and so -with some care -Azumaya monoids (algebras) and Brauer groups can be defined for such categories. For finitely bicomplete categories this was worked out by J. Fisher-Palmquist in [8], for symmetric monoidal categories it was investigated by B. Pareigis in [21], and for braided monoidal categories the theory was outlined by F. van Oystaeyen and Y. Zhang in [29] and B. Femić in [7]. It follows from the observations in [21] that -even in symmetric monoidal categories -the category equivalence requested for an Azumaya monoid A does not imply separability of A (defined as for R-algebras).
In our approach to Azumaya (co)monads we focus on properties of monads and comonads on any category A inducing equivalences between certain related categories. Our main tools are distributive laws between monads (and comonads) as used in the investigations of Hopf monads in general categories (see [17], [19]).
We begin by recalling basic facts about the related theory -including Galois functors -in Section 1. Then, in Section 2, we consider monads F = (F, m, e) on any category A endowed with a distributive law λ : F F → F F satisfying the Yang Baxter equation (BD-laws). The latter enables the definition of a monad F λ = (F λ , m λ , e λ ) where F λ = F , m λ = m · λ, and e λ = e. Furthermore, λ can be considered as distributive law λ : F λ F → F F λ and this allows to define a monad structure on F F λ . Then, for any object A ∈ A, F (A) allows for an F F λ -module structure, thus inducing a comparison functor K : A → A F F λ . We call F an Azumaya monad (in 2.3) if this functor is an equivalence of categories. Properties and characterisations of such monads are given, in particular for the case that they allow for a right adjoint functor (Theorem 2.10).
These notions lead to an intrinsic definition of Azumaya comonads as outlined in Section 3 where also the relationship between the Azumaya properties of a monad F and a right adjoint comonad R is investigated (Proposition 3.15). It turns out that for a Cauchy complete category A, F is an Azumaya monad and F F λ is a separable monad if and only if R is an Azumaya comonad and G κ G is a separable comonad (Theorem 3.16).
In Section 4, our theory is applied to study Azumaya algebras in braided monoidal categories (V, ⊗, I, τ ). Then, for any V-algebra A, the braiding induces a distributive law τ A,A : A ⊗ A → A ⊗ A, and A is called left (right) Azumaya if the monad A ⊗ − : V → V (resp. − ⊗ A : V → V) is Azumaya. In [29], V-algebras which are both left and right Azumaya are used to define the Brauer group of V. We will get various characterisations for such algebras but will not pursue their role for the Brauer group. In braided monoidal categories with equalisers and coequalisers, the notions of left and right Azumaya algebras coincide (Theorem 4.19).
The results from Section 3 provide an extensive theory of Azumaya coalgebras in braided categories V and the basics for this are described in Section 5. Besides the formal transfer of results known for algebras, we introduce coalgebras C over cocommutative coalgebras D and for this, Section 3 provides conditions which make them Azumaya. This extends the corresponding notions studied for coalgebras over cocommutative coalgebras in vector space categories by B. Torrecillas, F. van Oystaeyen and Y. Zhang in [28]. Over a commutative ring R, Azumaya coalgebras C turn out to be coseparable and are characterised by the fact that the dual algebra C * = Hom(C, R) is an Azumaya R-algebra. Notice that coalgebras with the latter property were first studied by K. Sugano in [27].

Preliminaries
Throughout this section A will stand for any category.
1.1. Modules and comodules. For a monad T = (T, m, e) on A, we write A T for the Eilenberg-Moore category of T -modules and denote the corresponding forgetful-free adjunction by η T , ε T : φ T ⊣ U T : A T → A. Dually, if G = (G, δ, ε) is a comonad on A, we write A G for the Eilenberg-Moore category of G-comodules and denote the corresponding forgetful-cofree adjunction by For any monad T = (T, m, e) and an adjunction η, ε : T ⊣ R, there is a comonad R = (R, δ, ε), where m ⊣ δ, ε ⊣ e (mates) and there is an isomorphism of categories (e.g. [17]) Note that, for any (A, θ) ∈ A R , Ψ −1 (A, θ) = (A, T (A) Given a distributive law λ : T S → ST , the triple ST = (ST, m ′ m · SλT, e ′ e) is a monad on A (e.g. [1], [32]). Notice that the monad structure on ST depends on λ and if the choice of λ needs to be specified we write (ST ) λ . Furthermore, a distributive law λ corresponds to a monad S λ = ( S, m, e) on A T that is lifting of S to A T in the sense that U T S = SU T , U T m = m ′ U T and U T e = e ′ U T . This defines the Eilenberg-Moore category (A T ) S λ of S λ -modules whose objects are triples ((A, t), s), with (A, t) ∈ A T , (A, s) ∈ A S with a commutative diagram When no confusion can occur, we shall just write S instead of S λ .
1.3. Proposition. In the setting of 1.2, let λ : T S → ST be an invertible monad distributive law.
(1) κ −1 : GH → HG is again a comonad distributive law of H over G; (2) GH allows for a comonad structure (GH) κ −1 and κ : HG → GH is a comonad isomorphism (HG) κ → (GH) κ −1 defining a category equivalence (3) κ −1 induces a lifting G κ −1 : A H → A H of G to A H and an equivalence of categories leading to the commutative diagram 1.6. Mixed distributive laws. Given a monad T = (T, m, e) and a comonad G = (G, δ, ε) on A, a mixed distributive law (or entwining) from T to G is a natural transformation ω : Given a mixed distributive law ω : T G → GT from the monad T to the comonad G, we write G ω = ( G, δ, ε) for a comonad on A T lifting G (e.g. [32,Section 5]).
It is well-known that for any object (A, h) of A T , and the objects of (A T ) G are triples (A, h, ϑ), where (A, h) ∈ A T and (A, ϑ) ∈ A G with commuting diagram Distributive laws and adjoint functors. Let λ : T S → ST be a distributive law of a monad T = (T, m, e) over a monad S = (S, m ′ , e ′ ) on A. If T admits a right adjoint comonad R (with η, ε : T ⊣ R), then the composite is a mixed distributive law from S to R (e.g. [2], [17]) and the assignment is a mixed distributive law from the monad S to the comonad H. Moreover, there is an adjunction α, β : S λ ⊣ H (λ −1 )⋄ : A T → A T , where S λ is the lifting of S to A T considered in 1.2 (e.g. [10,Theorem 4]) and the canonical isomorphism Ψ from (1.1) yields the commutative diagram Note that U T (α) = η and U T (β) = ε.
1.9. Entwinings and adjoint functors. For a monad T = (T, m, e) and a comonad G = (G, δ, ε), consider an entwining ω : T G → GT . If T admits a right adjoint comonad R (with η, ε : T ⊣ R), then the composite is a comonad distributive law of G over R (e.g. [2], [17]) inducing a lifting G ω of G to A R and thus an Eilenberg-Moore category (A R ) Gω of G ω -comodules whose objects are triples ((A, d), g) with commutative diagram The following notions will be of use for our investigations.
1.10. Monadic and comonadic functors. Let η, ε : F ⊣ R : B → A be an adjoint pair of functors. Then the composite RF allows for a monad structure RF on A and the composite F R for a comonad structure F R on B. By definition, R is monadic and F is comonadic provided the respective comparison functors are equivalences, For an endofunctor we have, under some conditions on the category: 1.11. Lemma. Let F : A → A be a functor that allows for a left and a right adjoint functor and assume A to have equalisers and coequalisers. Then the following are equivalent: is a monad, then the above are also equivalent to (d) the free functor φ F : A → A F is comonadic.
Proof. Since F is a left as well as a right adjoint functor, it preserves equalisers and coequalisers. Moreover, since A is assumed to have both equalisers and coequalisers, it follows from Beck's monadicity theorem (see [14]) and its dual that F is monadic or comonadic if and only if it is conservative.
(a)⇔(d) follows from [16,Corollary 3.12]. ⊔ ⊓ 1.12. T -module functors. Given a monad T = (T, m, e) on A, a functor R : B → A is said to be a (left) T -module if there exists a natural transformation α : T R → R with α · eR = 1 and α · mR = α · T α. This structure of a left T -module on R is equivalent to the existence of a functor R : If R is such a functor, then R(B) = (R(B), α B ) for some morphism α B : T R(B) → R(B) and the collection {α B , B ∈ B} forms a natural transformation α : For any T -module (R : B → A, α) admitting an adjunction F ⊣ R : B → A with unit η : 1 → RF , the composite is a monad morphism from T to the monad RF on A generated by the adjunction F ⊣ R. This yields a functor A t R : 1.14. G-comodule functors. Given a comonad G = (G, δ, ε) on a category A, a functor L : B → A is a left G-functor if there exists a natural transformation α : L → GL with εL · α = 1 and δL · α = Gα · α. This structure on L is equivalent to the existence of a functor L : B → A G with commutative diagram (dual to 1.12) If a G-functor (L, α) admits a right adjoint S : A → B, with counit σ : LS → 1, then (see Propositions II.1.1 and II.1.4 in [6]) the composite is a comonad morphism from the comonad generated by the adjunction L ⊣ S to G. L : B → A is said to be a G-Galois comodule functor provided t L : LS → G is an isomorphism.
Dual to Proposition 1.13 we have (see also [18], [19]): 1.15. Proposition. The functor L is an equivalence of categories if and only if the functor L is comonadic and a G-Galois comodule functor.
1.16. Right adjoint for L. If the category B has equalisers of coreflexive pairs and L ⊣ S, the functor L (in 1.14) has a right adjoint S, which can be described as follows (e.g. [6], [15]): With the composite If σ denotes the counit of the adjunction L ⊣ S, then for any (A, ϑ) ∈ A G , where σ : LS → 1 is the counit of the adjunction L ⊣ S.
is a split monomorphism. (a) m has a natural section ω : F → F F such that F m·ωF = ω·m = mF ·F ω; (2) For a comonad G = (G, δ, ε) on A, the following are equivalent: (a) δ has a natural retraction ̺ : GG → G such that ̺G · Gδ = δ · ̺ = G̺ · δG; (b) the forgetful functor U G : A G → A is separable. (2) G is separable if and only ifε : GF → 1 A a split epimorphism. Given a comonad structure G on G with corresponding monad structure F on F (see 1.1), there are pairs of adjoint functors (2) U G is separable if and only if U F is separable and then any object of A G is injective relative to U G and every object of A F is projective relative to U F .
The following generalises criterions for separability given in [

Proof. (i) Inspection shows that
is the identify and hence F U is separable. By 1.17, this implies that U is also separable.

Azumaya monads
An algebra A over a commutative ring R is Azumaya provided A induces an equivalence between M R and the category A M A of (A, A)-bimodules. The construction uses properties of the monad A ⊗ R − on M R and the purpose of this section is to trace this notion back to the categorical essentials to allow the formulation of the basic properties for monads on any category. Throughout again A will denote any category.
2.1. Definitions. Given an endofunctor F : A → A on A, a natural transformation λ : F F → F F is said to satisfy the Yang-Baxter equation provided it induces commutativity of the diagram For a monad F = (F, m, e) on A, a monad distributive law λ : F F → F F satisfying the Yang-Baxter equation is called a (monad) BD-law (see [11,Definition 2.2]).
Here the interest in the YB-condition for distributive laws lies in the fact that it allows to define opposite monads and comonads 2.2. Proposition. Let F = (F, m, e) be a monad on A and λ : F F → F F a BD-law.
(2) can be seen by direct computation (e.g. [3], [11], and [17] Proof. With our previous notation we have the commutative diagram Proof. That (a) and (b) are equivalent follows from Proposition 1.15.
(b)⇔(c) In both cases, F is monadic and thus F allows for an adjunction, say L ⊣ F with unit η : 1 → F L. Write T for the monad on A generated by this adjunction. Since the left F F λ -module structure on the functor F is the composite it follows from 1.12 that the monad morphism t K : F F λ → T induced by the diagram Thus F is F F λ -Galois if and only if t K is an isomorphism. ⊔ ⊓ 2.6. The isomorphism A F F λ ≃ (A F λ ) F . According to 1.2, for any BD-law λ : yields an isomorphism of categories P λ : There is a comparison functor Proof. Direct calculation shows that To apply Proposition 1.13 to the functor K F , we will need a functor left adjoint to φ F λ whose existence is not a consequence of the Azumaya condition. For this the invertibility of λ plays a crucial part.

2.7.
Proposition. Let F = (F, m, e) be a monad on A with an invertible BD-law λ : (2) There is an isomorphism of categories with commutative diagrams is an equivalence of categories. Note that if λ : F F → F F is a BD-law, then λ can be seen as a BD-law λ : F λ F λ → F λ F λ , and it is not hard to see that the corresponding comparison functor . Modulo this identification, the functor K ′ F λ corresponds to the functor K F λ . It now follows from the preceding remark: 2.9. Azumaya monads with right adjoints. Let F = (F, m, e) be a monad with an invertible BD-law λ : F F → F F . Assume F to admit a right adjoint functor R, with η, ε : F ⊣ R, inducing a comonad R = (R, δ, ε) (see 1.1). Since λ : F λ F → F F λ is an invertible distributive law, there is a comonad R = R (λ −1 )⋄ on A F λ lifting the comonad R and is right adjoint to the monad F (see 1.7) yielding a category isomorphism So the A-component α A of the induced R-comodule structure α : φ F λ → Rφ F λ on the functor φ F λ induced by the commutative diagram (2.2) (see 1.14), is the composite

It then follows that for any (
These observations lead to the following characterisations of Azumaya monads. 2.10. Theorem. Let F = (F, m, e) be a monad on A, λ : F F → F F an invertible BD-law, and R a comonad right adjoint to F (with η, ε : F ⊣ R). Then the following are equivalent: Proof. Recall first that the monad F λ is of effective descent type means that φ F λ is comonadic.
By Proposition 1.15, the functor K making the triangle (2.2) commute is an equivalence of categories (i.e., the monad F is Azumaya) if and only if the monad F λ is of effective descent type and the comonad morphism t : φ F λ U F λ → R is an isomorphism. Moreover, according to [19,Theorem 2

.12], t is an isomorphism if and only if for any object
This completes the proof.

⊔ ⊓
The existence of a right adjoint of the comparison functor K can be guaranteed by conditions on the base category.
2.11. Right adjoint for K. With the data given above, assume A to have equalisers of coreflexive pairs. Then (2) for any A ∈ A, R K(A) is the equaliser , , Proof. (1) According to 1.16, R((A, h), ϑ) is the object part of the equaliser of Definition. Write F F for the subfunctor of the functor F determined by the equaliser of the diagram Since R is right adjoint to the functor K, K is fully faithful if and only if R K ≃ 1.

2.13.
Theorem. Assume A to admit equalisers of coreflexive pairs. Let F = (F, m, e) be a monad on A, λ : F F → F F an invertible BD-law, and R a comonad right adjoint to F . Then the comparison functor K :

and faithful if and only if the monad F is central; (ii) an equivalence of categories if and only if the monad F is central and the functor R is conservative.
Proof. (i) follows from the preceding proposition.
(ii) Since F is central, the unit η : 1 → R K of the adjunction K ⊣ R is an isomorphism by (i). If ε is the counit of the adjunction, then it follows from the triangular identity R ε · η R = 1 that R ε is an isomorphism. Since R is assumed to be conservative (reflects isomorphisms), this implies that ε is an isomorphism, too. Thus K is an equivalence of categories.

Azumaya comonads
Following the pattern for monads we introduce the corresponding definitions for comonads. Again A denotes any category. The following results and definitions are dual to those in the preceding section.
3.1. Definition. For a comonad G = (G, δ, ε) on A, a comonad distributive law κ : GG → GG (see 1.4) satisfying the Yang-Baxter equation is called a comonad BD-law or just a BD-law if the context is clear.

Proposition. If G is an Azumaya comonad on A, then the functor G admits a right adjoint.
This leads to a first characterisation of Azumaya comonads.
3.6. Theorem. Consider a comonad G = (G, δ, ε) on A with a comonad BD-law κ : GG → GG. The following are equivalent: (a) G is an Azumaya comonad; Write G for the lifting of the comonad G to A G κ corresponding to the distributive law κ : GG κ → G κ G. Then (see 1.4), the assignment yields an isomorphism of categories There is a comparison functor K κ : (2) There is an isomorphism of categories (3) κ −1 induces a comparison functor with commutative diagrams (with K κ from 3.7) Note that, for κ invertible, it follows from the diagrams in the Sections 3.7, 3.8 that G is an Azumaya comonad if and only if the functor is an equivalence of categories. Dualising Proposition 2.8 gives: 3.9. Proposition. Let G = (G, δ, ε) be a comonad on A with an invertible BD-law κ : GG → GG and assume κ 2 = 1. Then the comonad G is Azumaya if and only if the comonad G κ is so.
3.10. Azumaya comonads with left adjoints. Again let G = (G, δ, ε) be a comonad on A with an invertible BD-law κ : GG → GG. Assume now that the functor G admits a left adjoint functor L, with η, ε : L ⊣ G, inducing a monad L = (L, m, e) on A (see 1.1). Since κ is invertible, κ −1 can be seen as a distributive law G κ G → GG κ . It then follows from the dual of 1.7 that the composite is a mixed distributive law from the monad L to the comonad G κ leading to an isomorphism of categories where L is the lifting of L to A G κ (corresponding to ω). Then the composite thus inducing commutativity of the diagram 3.11. Theorem. Let G = (G, δ, ε) be a comonad on A with an invertible comonad BD-law κ : GG → GG and L a monad left adjoint to G (with η, ε : L ⊣ G). Then the following are equivalent: (a) G is an Azumaya comonad; Proof. This follows by applying the dual of Theorem 2.10 to the last diagram. ⊔ ⊓ 3.12. Proposition. If A has coequalisers of reflexive pairs, then given as the coequaliser LG(A) 3.13. Definition. Write G G for the quotient functor of the functor G determined by the coequaliser of the diagram We call the comonad G cocentral if G G is (isomorphic to) the identity functor.
3.14. Theorem. Assume A to admit coequalisers of reflexive pairs. Let G = (G, δ, ε) be a comonad on A, κ : GG → GG an invertible comonad BD-law, and L a monad left adjoint to G. Then the comparison functor The next observation shows the transfer of the Galois property to an adjoint functor.
Proof. By Theorems 2.10 and 3.11, we have to show that, for any (A, h) ∈ A F λ , the composite is an isomorphism if and only if, for any (A, θ) ∈ A R κ , this is so for the composite By symmetry, it suffices to prove one implication. So suppose that the functor φ F λ is R-Galois. Since m ⊣ δ, δ is the composite in which the top left triangle commutes by one of the triangular identities for F ⊣ R and the other partial diagrams commute by naturality, one sees that 1) -is an object of the category A F λ . It then follows that t (A,θ) = t (A,ε A ·F (θ)) . Since the functor φ F λ is assumed to be R-Galois, the morphism t (A,ε A ·F (θ)) , and hence also t (A,θ) , is an isomorphism, as desired.
⊔ ⊓ In view of the properties of separable functors (see 1.19) and Definition 2.3, for an Azumaya monad F , F F λ is a separable monad if and only if F is a separable functor. In this case φ F λ is also a separable functor, that is, the unit e : 1 → F splits.
Proof. (a)⇒(b)⇒(c) follow by the preceding remarks. (c)⇒(a) Since A is assumed to be Cauchy complete, by [16,Corollary 3.17], the splitting of e implies that the functor φ F λ is comonadic. Now the assertion follows by Theorem 2.10.
Since ε is the mate of e, ε is a split epimorphism if and only if e is a split monomorphism (e.g. [17, 7.4]) and the splitting of ε implies that the functor φ R κ is monadic.
Applying now Theorems 2.10, 3.11 and Proposition 3.15 gives the desired result. ⊔ ⊓ 4. Azumaya algebras in braided monoidal categories 4.1. Algebras and modules in monoidal categories. Let (V, ⊗, I, τ ) be a strict monoidal category ( [14]). An algebra A = (A, m, e) in V (or V-algebra) consists of an object A of V endowed with multiplication m : A ⊗ A → A and unit morphism e : I → A subject to the usual identity and associative conditions. For a V-algebra A, a left A-module is a pair (V, ρ V ), where V is an object of V and Left A-modules are objects of a category A V whose morphisms between objects f : Similarly, one has the category V A of right A-modules.
The forgetful functor A U : A V → V, taking a left A-module (V, ρ V ) to the object V , has a left adjoint, the free A-module functor There is another way of representing the category of left A-modules involving modules over the monad associated to the V-algebra A.
Any V-algebra A = (A, m, e) defines a monad A l = (T, η, µ) on V by putting The corresponding Eilenberg-Moore category V A l of A l -modules is exactly the category A V of left A-modules, and A U ⊣ F is the familiar forgetful-free adjunction between V A l and V. This gives in particular that the forgetful functor A U : A V → V is monadic. Hence the functor A U creates those limits that exist in V.
Symmetrically, writing A r for the monad on V whose functor part is − ⊗ A, the category V A is isomorphic to the Eilenberg-Moore category V Ar of A r -modules, and the forgetful functor U A : V A → V is monadic and creates those limits that exist in V.
If V admits coequalisers, A is a V-algebra, (V, ̺ V ) ∈ V A a right A-module, and (W, ρ W ) ∈ A V a left A-module, then their tensor product (over A) is the object part of the coequaliser • if D is another V-algebra and P ∈ C V D , then the canonical morphism induced by the associativity of the tensor product, is an isomorphism in A V D ; Note that (co)algebras in this monoidal category are called A-(co)rings. where V ⊗ r W := W ⊗ V , and (V op ) r = (V op , ⊗ r , I) (see, for example, [24]). Note that (V op ) r = (V r ) op .
Coalgebras and comodules in a monoidal category V = (V, ⊗, I) are respectively algebras and modules in V op = (V op , ⊗, I). If C = (C, δ, ε) is a V-coalgebra, we write V C (resp. C V) for the category of right (resp. left) C-comodules. Thus, V C = (V op ) C and C V = C (V op ). Moreover, if C ′ is another V-coalgebra, then the category C V C ′ of (C, C ′ )-bicomodules is C (V op ) C ′ . Writing C l (resp. C r ) for the comonad on V with functor-part C ⊗ − (resp. − ⊗ C), one has that V C (resp. C V) is just the category of C l -comodules (resp. C r -comodules).
4.5. Duality in monoidal categories. One says that an object V of V admits a left dual, or left adjoint, if there exist an object V * and morphisms db : yield the identity morphisms. db is called the unit and ev the counit of the adjunction. We use the notation (db, ev : V * ⊣ V ) to indicate that V * is left adjoint to V with unit db and counit ev. This terminology is justified by the fact that such an adjunction induces an adjunction of functors as well as an adjunction of functors i.e., for any X, Y ∈ V, there are bijections Any adjunction (db, ev : V * ⊣ V ), induces a V-algebra and a V-coalgebra, Dually, one says that an object V of V admits a right adjoint if there exist an object V ♯ and morphisms db ′ : I → V ♯ ⊗ V and ev ′ : V ⊗ V ♯ → I such that the composites yield the identity morphisms.
) is a morphism from the Vcoalgebra C V,V * to the V-coalgebra C.

4.7.
Definition. Let V ∈ V be an object with a left dual (V * , db, ev).
subject to two hexagon coherence identities (e.g. [14]). A symmetric monoidal category is a monoidal category with a braiding τ such that τ V,W · τ W,V = 1 for all V, W ∈ V. If V is a braided category with braiding τ , then the monoidal category V r becomes a braided category with braiding given by τ V,W := τ W,V . Furthermore, given V-algebras A and B, the triple is also a V-algebra, called the braided tensor product of A and B.
The braiding also has the following consequence (e.g [26]): If an object V in V admits a left dual (V * , db : I → V ⊗ V * , ev : V * ⊗ V → I), then (V * , db ′ , ev ′ ) is right adjoint to V with unit and counit Thus there are isomorphisms (V * ) ♯ ≃ V and (V ♯ ) * ≃ V , and we have the following 4.9.
Definition. An object V of a braided monoidal category V is said to be finite if V admits a left (and hence also a right) dual.
For the rest of this section, V = (V, ⊗, I, τ ) will denote a braided monoidal category.
Finite objects in a braided monoidal category have the following relationship between the related functors to be (co)monadic or conservative. Recall that a morphism f : V → W in V called a copure epimorphism (monomorphism) if for any X ∈ V, the morphism f ⊗ X : V ⊗ X → W ⊗ X (and hence also the morphism X ⊗ f : X ⊗ V → X ⊗ W ) is a regular epimorphism (monomorphism). Proof. First observe that since V is assumed to admit a left dual, it admits also a right dual (see 4.8). Hence the equivalence of the properties listed in (a) (and in (c)) follows from 1.11. It only remains to show the equivalence of (a) and (b), since the equivalence of (c) and (d) will then follow by duality.
(a)⇒(b) If V ⊗ − : V → V is monadic, then it follows from [12, Theorem 2.4] that each component of the counit of the adjunction V * ⊗ − ⊣ V ⊗ −, which is the natural transformation ev ⊗ −, is a regular epimorphism. Thus, ev : V * ⊗ V → I is a copure epimorphism.
(b)⇒(a) To say that ev : V * ⊗ V → I is a copure epimorphism is to say that each component of the counit ev ⊗ − of the adjunction V * ⊗ − ⊣ V ⊗ − is a regular epimorphism, which implies (see, for example, [12]) that V ⊗ − : V → V is conservative. ⊔ ⊓ 4.11. Remark. In Proposition 4.10, if the tensor product preserves regular epimorphisms, then (b) is equivalent to require ev : V * ⊗V → I to be a regular epimorphism.
If the tensor product in V preserves regular monomorphisms, then (d) is equivalent to require db : I → V ⊗ V * to be a regular monomorphism.

4.12.
Opposite algebras. For a V-algebra A = (A, m, e), define the opposite algebra A τ = (A, m τ , e τ ) in V with multiplication m τ = m · τ A,A and unit e τ = e. Denote by A e = A ⊗ A τ and by e A = A τ ⊗ A the braided tensor products.
Then A is a left A e -module as well as a right e A-module by the structure maps By properties of the braiding, the morphism τ A,A : A ⊗ A → A ⊗ A induces a distributive law from the monad (A τ ) l to the monad A l satisfying the Yang-Baxter equation and the monad A l (A τ ) l is just the monad (A e ) l . Thus the category of A l (A τ ) l -modules is the category A e V of left A e -modules. Symmetrically, the category of A r (A τ ) r -modules is the category Ve A of right e A-modules.
given by the assignments The V-algebra A is called left (right) Azumaya provided A l (A r ) is an Azumaya monad.

Remark. It follows from Remark 2.8 that if τ 2
A,A = 1, the monad A l (resp. A r ) is Azumaya if and only if (A τ ) l (resp. (A τ ) l ) is. Thus, in a symmetric monoidal category, a V-algebra is left (right) Azumaya if ond only if its opposite is so.
A basic property of these algebras is the following.

Proposition. Let V be a braided monoidal category and
Proof. It is easy to see that when V and A eV are viewed as right V-categories (in the sense of [22]), K l is a V-functor. Hence, when K l is an equivalence of categories (that is, when A is left Azumaya), its inverse equivalence R is also a V-functor. Moreover, since R is left adjoint to K l , it preserves all colimits that exist in A e V. Obviously, the functor φ (A e ) l : V → A e V is also a V-functor and, moreover, being a left adjoint, it preserves all colimits that exist in V. Consequently, the composite R · φ (A e ) l : V → V is a V-functor and preserves all colimits that exist in V. It then follows from [22,Theorem 4

.2] that there exists an object
is an isomorphism (between the V-algebras A e and S A,A * ); Proof.
is an isomorphism (between the V-algebras e A and S A ♯ ,A ).
is an isomorphism.

Proposition. In any braided monoidal category, an algebra is left (right) Azumaya if and only if its opposite algebra is right (left) Azumaya.
Proof. We just note that if (V, ⊗, I, τ ) is a braided monoidal category and A is a V-algebra, then (τ −,A ) −1 : Under some conditions on V, left Azumaya algebras are also right Azumaya and vice versa: = (A, m, e) be a V-algebra in a braided monoidal category (V, ⊗, I, τ ) with equalisers and coequalisers. Then the following are equivalent: (a) A is a left Azumaya algebra; (c)⇔(d) The composite χ is the upper path and χ 1 is the lower path in the diagram where τ = τ A,A and · = ⊗. The left square is commutative by naturality, the pentagon is commutative since τ is a braiding, and the parallelogram commutes by the associativity of m. So the diagram is commutative and hence χ = In the setting of 4.12, by Proposition 2.2, the assignment  (b) the functor φ (A τ ) l : V → A τV is comonadic and, for any V ∈ V, the composite: is an isomorphism; (c) A is finite, the functor φ (A τ ) l : V → A τV is comonadic, and the composite is an isomorphism. Proof.
is an isomorphism; (c) A is finite, the functor φ (A τ )r : V → V A τ is comonadic, and the composite Given an adjunction (db, ε : V * ⊣ V ) in V, we know from 4.5 that S V,V * = V ⊗ V * is a V-algebra. Moreover, it is easy to see that the morphism Recall from [29] that an object V ∈ V with a left dual (V * , db, ev) is right faithfully projective if the morphism ev : V * ⊗ S V,V * V → I induced by ev : V * ⊗ V → I is an isomorphism. Dually, an object V ∈ V with a right dual (V ♯ , db ′ , ev ′ ) is left faithfully Since in a braided monoidal category an object is left faithfully projective if and only if it is right faithfully projective (e.g. [7, Proposition 4.14.]), we do not have to distinguish between left and right faithfully projective objects and we shall call them just faithfully projective.
where η A is the unit of the adjunction A ⊗ − ⊣ [A, −], is an isomorphism.

(d) A is faithfully projective and the composite
V being braided left closed implies that V is also right closed. So assume V to be closed. If A is a V-algebra, and (V, ρ V ) ∈ A V, then for any X ∈ V, and the assignment X → (V ⊗ X, ρ V ⊗ X) defines a functor V ⊗ − : V → A V. When V admits equalisers, this functor has a right adjoint where one of the morphisms is [̺ V , W ], and the other one is the composition Symmetrically, for V, W ∈ V A , one defines {V, W } A .
The functor K = ΨK : V → ( A τ V) [A,−] (in diagram (4.2)) has as right adjoint R : ( A τ V) [A,−] → V (see 1.16), and since Ψ is an isomorphism of categories, the composition R Ψ is right adjoint to the functor K : V → ( A τ V) A l . Using now that P (see 2.6) is an isomorphism of categories, we conclude that R ΨP is right adjoint to the functor P −1 K : V → A e V. For any (V, h) ∈ A e V, we put Taking into account the description of the functors P, Ψ and R, one gets that A V can be obtained as the equaliser of the diagram Proof. The V-algebra A is left Azumaya provided the functor K l : V → A eV is an equivalence of categories. It follows from equation (1.6) that the composite For the other direction we note that, under the conditions (i) and (ii), the counit of the adjunction P −1 K l ⊣ R ΨP (and hence also of the adjunction K l = ΨK ⊣ R) is an isomorphism and the functor φ (A τ ) l (and hence also K l ) is conservative (again [16, Theorem 2.1(2.(i))]), implying (as in the proof of Theorem 2.13 (ii)) that K l is an equivalence of categories.
⊔ ⊓ Symmetrically, for any (V, h) ∈ Ve V defining V A as the equaliser of the diagram A is called central if it is both left and right central.

Proposition. Let V be a braided closed monoidal category with equalisers. Then (i) any left (resp. right) Azumaya algebra is left (resp. right) central;
(ii) if, in addition, V admits also coequalisers, then any V algebra that is Azumaya on either side is central.
Proof. (i) follows by the Theorems 4.27 and 4.28, while (ii) follows from (i) and Theorem 4.19.

⊔ ⊓
Recall that for any Valgebra A, an A e -module M is U A e -projective provided for morphisms g : N → L and f : M → L in A e V with U A e (g) a split epimorphism, there exists an h : M → N in A e V with gh = f . This is the case if and only if M is a retract of a (free) A e -module A e ⊗ X with some X ∈ V (e.g. [25]). This is applied in the characterisation of separable algebras. 4.31. Proposition. The following are equivalent for a V-algebra A = (A, m, e): (a) A is a separable algebra; Proof. Since I is a retract of B in V, A is a retract of A⊗B in A e V. Since A⊗B is assumed to be separable in V, A⊗B is a retract of (A⊗B) e in (A⊗B) e V, and hence also in A e V. Thus A is a retract of A e ⊗ B e ≃ (A ⊗ B) e in A e V. Since A e ⊗ B e = φ A e (B e ), it follows that A e ⊗ B e is A e U-projective, and since retracts of a A e U-projectives are A e U-projective, A is A e U-projective and A is separable by Proposition 4.31.
⊔ ⊓ Following [21], a finite object V in V is said to be a progenerator if the counit morphism ev : V * ⊗ V → I is a split epimorphism. The following list describes some of its properties. 4.33. Proposition. Assume V to admit equalisers and coequalisers. For an algebra A = (A, m, e) in V with A admitting a left adjoint (V * , db, ev), consider the following statements: (1) A is a progenerator; (2) the morphism db : I → A ⊗ A * is a split monomorphism; (3) the functor A ⊗ − : V → V is separable; (4) the unit morphism e : I → A is a split monomorphism; Proof. Since A is assumed to be admit a left adjoint (V * , db, ev), the functor A * ⊗ − : V → V is left as well as right adjoint to the functor A ⊗ − : V → V. For any V ∈ V, the composite To say that db : I → A ⊗ A * (resp. ev : A * ⊗ A → I is a split monomorphism (resp. epimorphism) is to say that the unit (resp. counit) of the adjunction A ⊗ − ⊣ A * ⊗ − (resp. A * ⊗ − ⊣ A ⊗ − ) is a split monomorphism (resp. epimorphism). From the observations in 1.17, one gets (1) ⇔ (2) ⇔ (3).
If e : I → A is a split monomorphism, then the natural transformation is a split monomorphism and applying Proposition 1.20 to the pair of functors (A ⊗ − , 1 V ) gives that the functor A ⊗ − : V → V is separable, proving (4) ⇒ (3). If A is a progenerator, then ev : A * ⊗ A → I has a splitting ζ : We claim that φ · e = 1. Indeed, we have ev · A * ⊗ m · ζ ⊗ A · e = ev · A * ⊗ m · A * ⊗ A ⊗ e · ζ = ev · ζ = 1.
The first equality holds by naturality, the second one since e is the unit for the Valgebra A, and the third one since ζ is a splitting for ev : A * ⊗ A → I. Thus (2) implies (4). Now, if A is again a progenerator, then the morphism ev : A * ⊗ A → I has a splitting ζ : I → A * ⊗ A, and direct inspection shows that the morphism is a splitting for the multiplication A ⊗ ev ⊗ A * of the V-algebra A ⊗ A * satisfying condition (b) of Proposition 4.31. Thus A ⊗ A * is a separable V-algebra, proving the implication (2) ⇒ (6).
Finally, suppose that I is projective (w.r.t. regular epimorphisms) in V and that the functor A ⊗ − : V → V is monadic. Then, by [12,Theorem 2.4], each component of the counit of the adjunction A * ⊗ − ⊣ A ⊗ − is a regular epimorphism. Since ev : A * ⊗ A → I is the I-component of the counit, ev is a regular epimorphism, and hence splits, since I is assumed to be projective w.r.t. regular epimorphisms. Thus A is a progenerator. This proves the implication (5) is an equivalence of categories. Obviously this holds if and only if A is an Azumaya R-algebra in the usual sense. We have the commutative diagram where (e ⊗ R A τ ) * is the restriction of scalars functor induced by the ring morphism e ⊗ R A τ : It is not hard to see that, for any (M, h) ∈ A τ M, the (M, h)-component t (M,h) : corresponding to the functor K = ΨK, takes any element a ⊗ R m to the map b → h((ba) ⊗ R m). Thus, writing a · m for h(a ⊗ R m), one has for a, b ∈ A and m ∈ M, In particular, for any Since the canonical morphism i : R → A factorises through the center of A, it follows from of [16,Theorem 8.11] that the functor A ⊗ R − : is comonadic if and only if i is a pure morphism of R-modules. Applying Theorem 4.21 and using that K is an equivalence of categories if and only if K = ΨK is so, we get several characterisations of Azumaya R-algebra.

4.36.
Theorem. An R-algebra A is an Azumaya R-algebra if and only if the canonical morphism i : R → A is a pure morphism of R-modules, and one of the following holds: (a) for any M ∈ A τ M, there is an isomorphism (c) A R is finitely generated projective and there is an isomorphism (d) for any (A, A)-bimodule M, the evaluation map is an isomorphism Proof. For a (von Neumann) regular ring R, i : R → A is always a pure R-module morphism, and hence over such rings the (equivalent) properties (a) to (d) are sufficient to characterise Azumaya algebras.

Azumaya coalgebras in braided monoidal categories
Throughout (V, ⊗, I, τ ) will denote a strict monoidal braided category. The definition of coalgebras C = (C, ∆, ε) in V was recalled in 4.4.
5.1. The coalgebra C e . Let C be a V-coalgebra. The braiding τ C,C : C ⊗ C → C ⊗ C provides a BD-law allowing for the definition of the opposite coalgebra C τ = (C τ , ∆ τ = τ C,C · ∆, ε τ = ε) and a coalgebra Writing τ : C l (C τ ) l → (C τ ) l C l for the induced distributive law of the comonad C l over the comonad (C τ ) l , we have an isomorphism of categories V (C τ ) l C l ≃ V (C e ) l = C e V.

Definition. (see 3.4)
A V-coalgebra C is said to be left Azumaya provided the comonad C l = C ⊗ − : V → V is Azumaya, i.e. the comparison functor is an equivalence of categories. It fits into the commutative diagram C is said to be right Azumaya if the corresponding conditions for C r = − ⊗ C are satisfied. Similar to 4.15 we have: Proof. Suppose that a V-coalgebra C is left Azumaya. Then the functor C ⊗ − : V → V admits a right adjoint [C, − ] : V → V by Proposition 3.5. Write ϑ for the composite (C ⊗ ∆ τ ) · ∆ : C → C ⊗ C ⊗ C. Then for any V ∈ V, K τ (V ) = (C ⊗ V, ϑ ⊗ V ) and thus the V -component of the left C e -comodule structure on the functor C ⊗ − : V → V, induced by the commutative diagram (5.1), is the morphism ϑ ⊗ V : C ⊗ V → C ⊗ C ⊗ C ⊗ V . From 1.14 we then see that the V -component t V of the comonad morphism induced by the above diagram is the composite In this diagram the rectangle is commutative by naturality of composition. Since σ V is the transpose of the morphism (ev C ) I ⊗ V , the transpose of σ V -which is the Hence the triangle in the diagram is also commutative. Now, since it follows from commutativity of the diagram that t I ⊗ V = t V · (C ⊗ σ V ), and since C is assumed to be left Azumaya, both t I and t V are isomorphisms, one concludes that C ⊗ σ V is an isomorphism. Moreover, the functor C ⊗ − : V → V is comonadic, hence conservative. It follows that σ V : [C, I] ⊗ V → [C, V ] is an isomorphism for all V ∈ V. Thus the functor [C, I] ⊗ − : V → V is also right adjoint to the functor C ⊗ − : V → V. It is now easy to see that [C, I] is right adjoint to C. ⊔ ⊓ Theorem 3.6 provides a first characterisation of left Azumaya coalgebras.

5.4.
Theorem. For a V-coalgebra C = (C, ∆, ε), the following are equivalent: (a) C is a left Azumaya V-coalgebra; (b) the functor C ⊗ − : V → V is comonadic and the left (C e ) l -comodule structure on it, induced by the commutative diagram (5.1), is Galois; (c) (i) C is finite with right dual (C ♯ , db ′ : I → C ♯ ⊗ C, ev ′ : C ⊗ C ♯ → I), the functor C ⊗ − : V → V is comonadic and (ii) the composite χ 0 : is an isomorphism (between the V-coalgebras S C,C ♯ and C e ); (d) (i) C is finite with left dual (C * , db : Proof. Under suitable assumptions, the base category V may be replaced by a comodule category over a cocommutative coalgebra. For this we consider the 5.6. Cotensor product. Suppose now that V = (V, ⊗, I, τ ) is a braided monoidal category with equalisers and D = (D, ∆ D , ε D ) is a coalgebra in V. If ((V, ρ V ) ∈ D V and (W, ̺ W ) ∈ V D , then their cotensor product (over D) is the object part of the equaliser Suppose in addition that either -for any V ∈ V, V ⊗ − : V → V and − ⊗ V : V → V preserve equalisers, or -V is Cauchy complete and D is coseparable.
Each of these condition guarantee that for V, W, X ∈ D V D , • the canonical morphism (induced by the associativity of the tensor product) where τ is the restriction of τ , is a braided monoidal category. When D is cocommutative (i.e. τ D,D · ∆ = ∆), then for any ( These two constructions establish an isomorphism between D V and V D , and thus we do not have to distinguish between left and right D-comodules. In this case, the tensor product of two D-comodules is another Dcomodule, and cotensoring over D makes D V (as well as V D ) a braided monoidal category with unit D.

D-coalgebras.
Consider V-coalgebras C = (C, ∆ C , ε C ) and D = (D, ∆ D , ε D ) with D cocommutative. A coalgebra morphism γ : C → D is called cocentral provided the diagram is commutative. When this is the case, (C, γ) is called a D-coalgebra.
To specify a D V-coalgebra structure on an object C ∈ V is to give C a D-coalgebra structure (C = (C, ∆ C , ε C ), γ). Indeed, if γ : C → D is a cocentral morphism, C can be viewed as an object of D V (and V D ) via and ∆ C factors through the i C,C : C ⊗ D C → C ⊗ C by some (unique) morphism is a coalgebra in the braided monoidal category D V. Conversely, any D V-coalgebra, (C, ∆ ′ C : C → C ⊗ D C, ε C : C → D) induces a V-coalgebra Considering C as a (D, C)-bicomodule by C ∆ −→ C ⊗ R C γ⊗C −→ D ⊗ R C, the corestriction functor is isomorphic to C ⊗ C − : C V → D V.
If (C, γ) is a D-coalgebra, then the category C D ( D V) can be identified with the category C V and, modulo this identification, the functor corresponds to the coinduction functor C ⊗ D − : D V → C V.

5.8.
Azumaya D-coalgebras. Let D be a cocommutative V-coalgebra. Then a D-coalgebra C = (C, ∆ C , ε C ) is said to be left Azumaya provided the comonad is Azumaya, i.e. (see 3.4), the comparison functor K τ : D V → C⊗ D C τ V defined by is an equivalence of categories.
In this setting, the results from Section 3 -and also specializing Theorem 5.4 -yield various characterisations of Azumaya D-coalgebras. Now let R be again a commutative ring with identity and M R the category of Rmodules. As an additional notion of interest the dual algebra of a coalgebra comes in.
5.9. Coalgebras in M R . An R-coalgebra C = (C, ∆, ε) consists of an R-module C with the R-linear maps multiplication ∆ : C → C ⊗ R C and counit ε : C → R subject to coassociativity and counitality conditions. C ⊗ R − : M R → M R is a comonad and it is customary to write C M := M C⊗− R for the category of left C-comodules. We write Hom C (M, N) for the comodule morphisms between M, N ∈ C M. In general, C M need not be a Grothendieck category unless C R is a flat R-module (e.g. [4, 3.14]).
The dual module C * = Hom R (C, R) has an R-algebra structure by defining for f, g ∈ C * , f * g = (g ⊗ f ) · ∆ (definition opposite to [4, 1.3]) and there is a faithful functor where ev denotes the evaluation map. The functor Φ is full if and only if for any N ∈ M R , α N : C ⊗ R N → Hom R (C * , N), c ⊗ n → [f → f (c)n], is injective and this is equivalent to C R being locally projective (α-condition, e.g. [4, 4.2]). In this case C M can be identified with the full subcategory σ[ C * C] ⊂ C * M subgenerated by C as C * -module.
The R-module structure of C is of considerable relevance for the related constructions and for convenience we recall: 5.10. Remark. For C R the following are equivalent: (a) C R is finitely generated and projective; (b) C ⊗ R − : M R → M R has a left adjoint; (c) Hom R (C, −) : M R → M R has a right adjoint; (d) C * ⊗ R − → Hom R (C, −), f ⊗ R − → (c → f (c) · −), is a (monad) isomorphism; (e) C ⊗ R − → Hom R (C * , −), c⊗ R − → (f → f (c)·−), is a (comonad) isomorphism; (f) Φ : C M → C * M is a category isomorphism. If this holds, there is an algebra anti-isomorphism End R (C) ≃ End R (C * ) and we denote the canonical adjunction by η C , ε C : C ⊗ R − ⊣ C * ⊗ R −.

5.11
. The coalgebra C e . As in 5.1, the twist map τ C,C : C ⊗ R C → C ⊗ R C provides an (involutive) BD-law allowing for the definition of the opposite coalgebra C τ = (C τ , ∆ τ , ε τ ) and a coalgebra The category C e M of left C e -comodules is just the category of (C, C)-bicomodules (e.g. [13], [4, 3.26]). A direct verification shows that the endomorphism algebra of C as C e -comodule is just the center of C * , that is, Z(C * ) = Hom C e (C, C) ⊂ C Hom(C, C) ≃ C * .
If C R is locally projective, an easy argument shows that C ⊗ R C is also locally projective as R-module and then C e M is a full subcategory of (C e ) * M.

5.12.
Definition. An R-coalgebra C is said to be an Azumaya coalgebra provided the comonad G = C ⊗ R − : M R → M R is Azumaya, i.e. (see 3.4), the comparison functor K : M R → C e M defined by is an equivalence of categories. We have the commutative diagram By Proposition 1.15, the functor K is an equivalence provided (i) the functor C ⊗ R − : R M → R M is comonadic, and (ii) the induced comonad morphism C ⊗ R Hom R (C, −) → C e ⊗ R − is an isomorphism. If R ≃ End C e (C) ≃ Z(C * ), the morphism in (ii) characterises C as a C e -Galois comodule as defined in [31, 4.1] and if C R is finitely generated and projective, the condition reduces to an R-coalgebra isomorphism C ⊗ R C * ≃ C e . In module categories, separable coalgebras are well studied and we recall some of their characterisations (e.g. Section 1.19, [13], [9], [4, 3.29], [2, 2.10]). 5.13. Coseparable coalgebras. An R-coalgebra C = (C, ∆, ε) is called coseparable if any of the following equivalent conditions is satisfied: (a) C ⊗ R − : M R → M R is a separable comonad; (b) ∆ : C → C ⊗ R C splits in C e M; (c) C is (C e , R)-injective; (d) the forgetful functor C M → M R is separable; (e) the forgetful functor C e M → M R is separable; (f) Hom R (C, −) : M R → M R is a separable monad.
For any coseparable coalgebra C, Z(C * ) is a direct summand of C * . Proof.
⊔ ⊓ For an Azumaya coalgebra C, the free functor φ (C τ ) l : M R → C τ M is monadic (see Theorem 3.11), and hence, in particular, it is conservative. It then follows that, for each X ∈ M R , the morphism ε ⊗ R X : C ⊗ R X → X is surjective. For X = R this yields that ε : C → R is surjective (hence splitting). By Theorem 3.16 this means that C is also a coseparable coalgebra.
It follows from the general Hom-tensor relations that the functor K : M R → C e M has a right adjoint C e Hom(C, −) : C e M → M R (e.g. [4, 3.9]) and we denote the unit and counit of this adjunction by η and ε, respectively.
Besides the characterisations derived from Theorem 5.4 we have: 5.14. Characterisation of Azumaya coalgebras. For an R-coalgebra C the following are equivalent: (a) C is an Azumaya coalgebra; (b) (i) ε X : C ⊗ R C e Hom(C, X) → X is an isomorphism for any X ∈ C e M, (ii) η M : M → C e Hom(C, C ⊗ R M) is an isomorphism for any M ∈ M R . (c) C is a C e -Galois comodule, C * is a central R-algebra, and the functor C ⊗ R − : R M → R M is comonadic; (d) C * is an Azumaya algebra.
Proof. This is essentially Theorem 3.16.
⊔ ⊓ As shown in Proposition 5.3, an Azumaya coalgebra C is finite in M R , that is, C R is finitely generated and projective (see Remark 5.10).
Coalgebras C with C R finitely generated and projective for which C * is an Azumaya R-algebra were investigated by K. Sugano in [27]. As an easy consequence he also observed that an R-algebra A with A R finitely generated and projective is Azumaya if and only if A * is an Azumaya coalgebra.
For vector space categories, Azumaya D-coalgebras C over a cocommutative coalgebra D (over a field) were defined and characterised in [28,Theorem 3.14].