A CATEGORICAL APPROACH TO ALGEBRAS AND COALGEBRAS

Algebraic and coalgebraic structures are often handled independently. In this survey we want to show that they both show up naturally when introducing them from a categorical point of you. Azumaya, Frobenius, separable, and Hopf algebras are obtained when both notions are combined. The starting point and guiding lines for this approach are given by adjoint pairs of functors and their elementary properties.


Introduction
In the last decades categorical techniques turned out to be very effective in algebra and representation theory. Hereby, it was a key observation that module theory of an algebra A over a field K is essentially the theory of the functor an endofunctor of the category of K-vector spaces. An algebra A is defined by K-linear maps multiplication A ⊗ K A → A and unit e : K → A, subject to associativity and unitality conditions. Left A-modules are given by a K-vector space V with K-linear maps : A ⊗ K V → V , also subject to associativity and unitality conditions. Together with A-linear maps, this yields the category A M of left A-modules, The tensor product with product 1 The final version of this paper has been submitted for publication in Intern. Electr.
J. Algebra that is, the functor U A is right adjoint to A ⊗ K −. These basic structures can be defined for arbitrary categories A, replacing M K , and any functor F : A → A, replacing A ⊗ K − : M K → M K . Multiplication and unit are replaced by natural transformations, m : F F → F and η : 1 → F , satisfying the respective associativity and unitality conditions. This gives F a monad structure.
An F -module is an object V ∈ A with a morphism : F (V ) → V and for any W ∈ A, (F (W ), m W ) is an example for this. Morphisms of F -modules are morphisms from A respecting the module structures and they yield the category A F of F -modules with free and forgetful functors

and the bijection
shows that the functor U F is right adjoint to φ F . This shows that structures from module theory can be formulated in great generality. As we will see, if the functor F : A → A has a right adjoint G : A → A, then the monad structure on F provides G with the structure of a comonad. Thus this approach leads naturally to comonads (coalgebras, bocses) and comodules and we will highlight this interplay. Notice that the categorical tools developped are also successfully applied in theoretical computer science and logic (e.g. [25], [37]).

Category theory
The idea that the role of elements in algebraic structures should be taken over by homomorphisms came up from the beginning of the last century. It has finally been poured into a solid frame 1945 by Samual Eilenberg and Saunders MacLane in the seminal paper [6]. For convenience and to fix notation we recall the basic notions and refer to [11] for more details.
which is associative in an obvious sense, (iii) for any object A, Mor A (A, A) contains an identity morphism 1 A , leaving any composition with it unchanged.

Functors.
A covariant functor F : A → B between categories sends (i) an object A from A to an object F (A) in B, It follows from this definition that F induces a set map F defines an equivalence of categories provided there exists a functor G : B → A such that F G and GF both yield the respective identities.
A fully faithful functor F : A → B induces an equivalence between A and a full subcategory of B, the image of F .

Natural transformations.
Given two functors F, G : A → B between categories, a natural transformation ψ : F → G is given by a family of morphisms ψ A : F (A) → G(A), A ∈ A, with commutative diagrams, for any morphism h : ψ is called a (natural) isomorphism if all ψ A are isomorphisms in B.

Separable functors.
A functor F : A → B is said to be separable if, for any A, A ∈ A, the canonical map Φ F (from 2.2) is a split monomorphism, that is, there is a map Clearly, for a separable functor F , Φ F is always injective, and every fully faithful functor is separable. A survey on separability in algebra and category theory is given in [36].
It turned out that the following notion introduced and described by S. Eilenberg and J.C. Moore in [5], is a milestone in category theory.
2.5. Adjoint pair of functors. A pair of (covariant) functors F : A → B, G : B → A between any categories A, B, is said to be adjoint, we write F G, provided there is a bijection natural in A ∈ A and B ∈ B. Such a bijection can be described by natural transformations, called unit and counit, They are obtained as images of the identities of F (A) and G(B), respectively, in the defining bijection.
Adjointness of contravariant functors A → B is defined by considering relating them with covariant functors between opposite categories.
2.6. Remark. The notion of adjointness can be weakened in various ways. For example, instead of being invertible one may require α to be regular, that is, there exists β A,B : Mor A (A, G(B)) → Mor B (F (A), B), natural in A ∈ A and B ∈ B, such that αβα = α (and βαβ = β). This yields a weaker form of the triangular identities (see [34], [18]).
A short argument shows that for adjoint functors F G, -F preserves epimorphisms and coproducts, -G preserves monomorphisms and products.

2.7.
Properties of units and counits. Let F G : B → A be an adjoint pair of functors (notation from 2.5).
(1) ε is an isomorphism if and only if G is a fully faithful functor: G yields an equivalence between B and the image of G.
(2) η is an isomorphism if and only if F is a fully faithful functor: F yields an equivalence between A and the image of F .
(3) ε and η are isomorphisms if F and G both are fully faithful: F and G determine an equivalence between A and B.
(4) ε is a split epimorphism if and only if G is a separable functor.
(5) η is a split monomorphism if and only if F is a separable functor.
2.8. Rings and modules. We follow the notation from [26]. For associative rings R and S, denote by R M and M S the category of left and right modules, respectively. Then any bimodule R P S induces an adjoint pair of functors with bijection, counit, and unit Hom R (P ⊗ S Y, X) Hom S (Y, Hom R (P, X)), . Denote by Gen(P ) (Pres(P )) the category of P -generated (P -presented) R-modules. Clearly the image of P ⊗ S − is contained in Pres(P ).
Choose an (injective) cogenerator Q in R M and put U = Hom R (P, Q). For the S-module U , denote by Cog(U ) (Cop(U )) the category of U -cogenerated (U -copresented) S-modules. The image of Hom R (P, −) is contained in Cog(U ). If R is a cogenerator in R M we can choose U = P * .

Monads and comonads
Monads and comonads on categories are modelled after the algebras and coalgebras on vector spaces.

Monads and their modules.
A monad on any category A is an endofunctor T : A → A with natural transformations, product and unit, subject to associativity and unitality conditions (as for algebras).
T -modules are objects A ∈ A with a morphism : T (A) → A subject to associativity and unitality conditions (as for modules over rings).

Morphisms between T -modules (A, ) and (
The category determined by T -modules and their morphisms is called the Eilenberg-Moore category -or just the module category -of the monad (T, m, η) and we denote it by A T .
For any A ∈ A, T (A) has a T -module structure by m A : T T (A) → T (A) and this leads to the free functor φ T which allows for a right adjoint, the forgetful functor, The adjunction φ T U T is given by the bijection, for A ∈ A, B ∈ A T , Define a new category A T with the objects of A but choosing This is known as the Kleisli category of the monad T and the bijection (3.1) shows that A T is isomorphic to the full subcategory of the Eilenberg-Moore category A T determined by the objects T (A), A ∈ A (free T -modules).
Reversing the arrows in the definitions around monads yields 3.2. Comonads and their comodules. A comonad on a category A is an endofunctor S : A → A with natural transformations, coproduct and counit, δ : S → SS, ε : S → 1 A , subject to coassociativity and counitality conditions (dual to monad case).
S-comodules are objects A ∈ A with a morphism ω : A → S(A) subject to coassociativity and counitality conditions.
Morphisms between comodules (A, ω) and (A , ω ) (or S-morphisms), are morphisms g : The category formed by the S-comodules and their morphisms is called the Eilenberg-Moore category -or just the comodule category -of the comonad (S, δ, ε) and we denote it by M S . For any A ∈ A, the structure map δ A : S(A) → SS(A) makes S(A) an S-comodule. This yields the free functor φ S which is right adjoint to the forgetful functor U S , The adjunction U S φ S is given by the bijection, B ∈ A S , A ∈ A, Define a new category A S with the objects of A but choosing This is known as the Kleisli category of the comonad S. The bijection (3.4) shows that A S is isomorphic to the full subcategory of the Eilenberg-Moore category A S determined by the objects S(A), A ∈ A ((co)free S-comodules).

Rings and modules. For a ring R, any (R, R)-bimodule A defines an endofunctor
A is called an R-ring if this functor allows for a monad structure. If R is commutative and ra = ar for a ∈ A, r ∈ R, an R-ring is called an R-algebra.
An (R, R)-bimodule C is called an R-coring provided the functor C ⊗ R − allows for a comonad structure. If R is commutative and rc = cr for c ∈ C, r ∈ R, an R-coring is called an R-coalgebra.
Notice that not every monad or comonad on R M can be represented by a tensor functor. For an extensive treatment of corings refer to [3].

Adjoints and (co)monads
The notion of adjoints and (co)monads are intimately related. Using naturality of the transformations involved it is straightforward to show: 4.1. From adjoints to (co)monads. Let F G : B → A be an adjoint pair of functors with unit η : 1 A → GF and counit ε : F G → 1 B .
(i) T := GF : A → A is an endofunctor and the natural transformations, product and unit, (ii) S := F G : B → B is an endofunctor and the natural transformations coproduct and counit, Recall that the construction of module and comodule categories in 3.1 and 3.2 show the inverse direction. In fact, these structures were introduced to show that monads as well as comonads can be written as a composition of adjoint functors ( [10], [5]).

4.2.
From (co)monads to adjoints. Let A be any category.
(i) For a monad (T, m, η) on A, the category A T allows for an adjoint pair of functors φ T U T : (ii) For a comonad (S, δ, ε) on A, the category A S allows for an adjoint pair of functors U S φ S : A → A S with S = U S φ S . The same assertions hold replacing the Eilenberg-Moore categories by the corresponding Kleisli categories.
The structure of the (co)monads related to an adjunction are strongly influenced by the properties of the unit and counit. If R P is finitely generated and projective, Hom R (P, −) P * ⊗ R − and the comonad P ⊗ S Hom R (P, −) P ⊗ S P * ⊗ R −, that is, P ⊗ S P * is an R-coring (and P * ⊗ R P S).
If η is an isomorphism, P ⊗ S − is (left) exact on the image of Hom R (P, −) and hence the comonad is a left exact functor commuting with products in R M. Therefore, by the dual of Watts' theorem (e.g. [8]), it is determined by the image of a cogenerator Q ∈ R M, that is, Hom R (P ⊗ S Hom R (P, Q), −) Hom S (Hom R (P, Q), Hom R (P, −)).
In case R is a cogenerator in R M, we put Q = R to obtain This is a comonad, thus P ⊗ S P * ⊗ R − is a monad (by 5.1) which means that P ⊗ S P * is an R-ring (see 3.3).
If unit η and counit ε are isomorphisms, the comonad P * ⊗ R Hom S (P * , −) P * ⊗ R P ⊗ S −, that is, P * ⊗ R P is an S-coring.
As a special case, we look at a situation studied in representation theory.

Adjoint monads and comonads
In 3.2, comonads (S, δ, ε) were defined by reversing arrows in the definition of monads (T, m, η). This does not mean that the two notions are strictly dual in a categorical way: in our definition we do not have anything like the "dual" of a functor T (or S). In this section we will outline that adjointness provides a bijective correspondence between monads and comonads. 5.1. Monads versus comonads Let F G : A → A be an adjoint pair of endofunctors with unit η, counit ε and bijection, for X, Y ∈ A, (i) A monad structure on F induces a comonad structure on G, and vice versa, such that the associated (Eilenberg-Moore) categories A F and A G are isomorphic. (ii) A comonad structure on F induces a monad structure on G, and vice versa, such that the associated Kleisli categories A F and A G are equivalent Proof. (Sketch) (i) Let (F, m, η) be a monad. The adjunction F G induces the diagram Mor(X,?) where the dotted arrow exists by composition of the other maps (α is invertible) and determines a natural transformation δ : G → GG. A similar argument shows the existence of a counit ε : G → 1 A . Explicitly we get The symmetric construction shows that a comonad structure on G leads to a monad structure on F . The equivalence of categories is given by (ii) By arguments symmetric to those in (i), a comodule structure on F induces a module structure on G, and vice versa. The equivalence of Kleisli categories A F and A G follows from the isomorphisms where (1)

Composition of monads and comonads
Over a commutative ring R, the tensorproduct of two R-algebras (A, m, e), (B, m , e ) can be made an algebra defining a product (writing ⊗ := ⊗ R ) with the twist map tw : This map is no longer available if R is not commutative. On the other hand, the product is formally defined replacing tw by any morphism τ : B ⊗ A → A ⊗ B, however, it is not clear which properties it has. So one may ask which conditions are to be satisfied by τ to make the product m AB associative. These lead to commutativity of diagrams of the form and should induce commutative diagrams of functors, with the forgetful functor U B : B M → R M, The questions considered above for the functors A ⊗ − and B ⊗ − can be asked for endofunctors in any category.
6.1. Liftings of endofunctors. Let A be any category with endofunctors F, G : A → A. If (F, m, e) is a monad or (G, δ, ε) is a comonad, we get the two diagrams, respectively, If G (F ) exists it is called a lifting of G (F ) from A to A F (A G ). The following questions come up: (i) when does a lifting G or F exist ? (ii) if F and G are monads, when is G a monad ? (iii) if F and G are comonads, when is F a monad ? (iv) if F is a monad and G is a comonad, when is G a comonad, when is F a monad ?
All these problems can successfully be handled applying distributive laws as introduced and investigated in the 1970's by J. Beck [1] and others (see [31] for an overview and [2], [13], [15], et al. for more details).
Question (ii) above will lead to conditions which make the composition F G a monad on A (as considered for algebras above) and (iii) describes the corresponding properties of comonads. Of particular interest are the questions in (iv) since they reveal an interesting interplay between monads and comonads. 6.2. Mixed distributive laws. Let (F, m, η) be a monad and (G, δ, ε) a comonad on a category A. A natural transformations λ : F G → GF is called a mixed distributive law or (mixed) entwining if it induces commutativity of the diagrams The following are equivalent (with notation from 6.1): (a) G can be lifted from A to G : Given a monad and a comonad on A, objects can have a module and a comodule structure. An entwining allows to require a compatibility of these structures.
6.3. Mixed modules. Let (F, G, λ) be a mixed entwining and assume A ∈ A to be an F -module : F (A) → A and a G-comodule ω : A → G(A). Then (A, , ω) is called a mixed (F, G)-module if we get commutativity of the diagram These objects with morphisms that are module as well as comodule morphisms, form a category which we denote by A G F . In 6.1, the lifting to Eilenberg-Moore categories was considered. A corresponding construction for Kleisli categories is the following. 6.4. Extending of endofunctors. Let A be any category with endofunctors F, G : A → A. If (F, m, e) is a monad or (G, δ, ε) is a comonad, we get the two diagrams (with φ the free functors), respectively, G and F are called the extensions of F and G, respectively. Here again distributive laws apply for further investigation but with the role of monad and comonad interchanged, that is, one needs natural transformations σ : GF → F G inducing commutativity of the corresponding diagrams. Notice that the role of mixed modules (as in 6.3) does not transfer to this situation.

Bimonad and Hopf monads
Of special interest are endofunctors B on A which carry a monad structure (B, m, η) and a comonad structure (B, δ, ε) at the same time. To make these data a bimonad we first require the existence of a mixed distributive law λ : BB → BB.
The relevant commutative diagrams are either for the monad structure or else for the comonad structure -they do not relate product with coproduct, for example. To connect these we need further compatibility conditions, namely commutativity of the diagrams where the bottom diagrams mean that η is comonad morphism and ε a monad morphism. Diagram  Let (B, m, δ, λ) be a bimonad on A (as above). Assume B allows for a right adjoint functor C : A → A. Then by 5.1, C can be endowed with a comonad as well as a monad structure. Furthermore, there is a mixed distributive law λ : CC → CC (derived from λ, see [14, 7.4]) which makes C a bimonad.
The bimonad B has an antipode if and only if the associated bimonad C has an antipode. Thus, given an adjoint pair B C of functors on A, B allows for a Hopf monad structure if and only if so does C.

7.2.
Bimonad on Set. ( [31, 5.19], [14, 7.9]) For any set G, the cartesian product defines an endofunctor which is a comonad with coproduct δ : G → G × G, g → (g, g), and is a monad provided G is a monoid. Then it is a bimonad with entwining Now G × − has an antipode, i.e., is a Hopf monad, if and only if the monoid G is in fact a group. By 7.1, this is also equivalent to Map(G, −) : Set → Set, a right adjoint of G × −, being a Hopf monad. 7.3. Bialgebras. Let R be a commutative ring. An R-module B with an algebra structure (B, m, η) and a coalgebra structure (B, δ, ε) is called a bialgebra if δ and ε are algebra morphisms, or, equivalently, m and η are coalgebra morphisms. These conditions require commutativity of the outer path in the diagram Defining an R-linear map yields a similar rectangle (sides interchanged). It gives a mixed distributive law between the related monad − ⊗ B and the comonad − ⊗ B. It may also be considered as a mixed distributive law between the comonad B ⊗ − and the monad B ⊗ − (see 6.4).
An antipode for the bialgebra B is an R-linear map S : B → B leading to commutativity of the diagram (7.3) for the monad B ⊗ −.
Bialgebras B with an antipode are called Hopf algebras and they are characterised by the fact that φ B B : M R → M B B is an equivalence, that is, every Hopf module is of the form B ⊗ X, for some X ∈ M R (Fundamental Theorem).
In case B is a finite dimensional algebra over a field K, B ⊗ K − is left adjoint to the endofunctor Hom K (B, −) B * ⊗ K −, where B * = Hom K (B, K). Hence B is a Hopf algebra if and only if B * is a Hopf algebra (see 7.1). This is known as the duality principle of Hopf algebras.
Hopf algebras were brought to light by Heinz Hopf in his seminal paper [9] in topology (1941). The algebraic essentials of this notion were extracted by Milnor-Moore in [19] (1965). As a result, interest was also directed to the more elementary notions of coalgebras and corings as building blocks for the theory. It took several years until their value for representation theory was unveiled (1980) by A.V. Roiter in [21], there a coring is called bocs, and the interest is focused on Kleisli categories. In the meantime more attention is paid to coalgebraic aspects of finite dimensional algebras. For this we refer to the recent paper [12] by R. Marczinzik and the references given there. 7.4. Remark. As pointed out in subsection 7.3, the mixed distributive law for a bialgebra was derived from the canonical twist of the tensor product of modules over a commutative ring. This setting was extended to monoidal braided categories, where a general version of such a twist map is required. Concentrating on the properties needed, for endofunctors B on any category it is sufficient to define a local braiding τ : BB → BB to derive the corresponding theory. For details we refer to [14,Section 6] and [16]. 7.5. Rings and modules. The preceding sections show that the coproduct plays an important role in the structure theory of algebras. Summarising we consider an algebra (A, m, η) over a commutative ring R. Assume A allows for a coproduct δ : A → A ⊗ A. Then A becomes (1) a separable algebra if (A, m, δ) satisfies the Frobenius condition and m · δ = 1 A ; (2) an Azumaya algebra if it is separable and R C(A) (center of A); the categroy of C(A)-modules is equivalent to the category A M A of (A, A)-bimodules; (3) a Frobenius algebra if (A, m, δ) satisfies the Frobenius condition and (A, δ) has a counit ε : A → R; every A-module has an A-comodule struture (Frobenius (bi)module, see [35]); (4) a Hopf algebra if (A, m, δ) induce commutativity of (7.1) and (A, δ) has a counit ε : A → R; the category of R-modules is equivalent to the category M A A of mixed (A, A)-bimodules (e.g. [3]).