Applications of the Kleisli and Eilenberg-Moore 2-adjunctions

We collect some isomorphisms of categories and bijections of structures using the Kleisli and Eilenberg-Moore 2-adjunctions.

In Section 8, we apply the adjunction to the most known case of liftings of functors and commutative diagrams for the forgetful functor, check [1] and [10], just to mention a couple of references.
In Section 9, we relate actions of the category C over its Kleisli category C F with strong monads.
In Section 10, we finalize with left and right functor algebras for a monad and relate this to certain liftings and extensions, respectively, for the underlying functors, cf. [4].
We give some remarks on notation. Suppose that we had an adjunction of the form L ⊣ R, then the unit and counit for this adjunction will be denoted as η RL and ε LR , respectively. We are completely aware that this notation is very cumbersome, nonetheless, it is clear and in the case of the proliferation of several adjunctions it has the property of avoiding the need for extra greek letters to denote the new units and counits. As the article develops, the reader will see the advantage in using this notation.
We will be working with monoidal categories denoted as (C, ⊗, I, a, l, r) and also as (C, ⊗, I) , as a contraction, that leaves understood the natural constrain transformations. We will be working with the constant functor δ I : 1 −→ C, on I, where 1 is the category with only one object 0 and only one arrow 1 0 . That is to say, δ I (0) = I.
On the other hand, it is known that a category with binary products and a terminal object has a canonical (cartesian) monoidal structure. This is the case for the category Cat, of small categories. The natural constraint transformations, taken on components, are functors, for example, for C, D, E, a C,D,E : (C × D) × E −→ C × (D × E) is the obvious functor. In order to compact the notation, we will agree that in the case that the component be the object C, C, C, the asociativity functor will be denoted simply as a C . In turn, the respective constraint functors will be denoted as l C and r C .
The horizontal composition in a general 2-category A will be denoted as · or by juxtaposition, this notation will be used indistinctively. The vertical composition on 2-cells will be given the symbol •.

Formal Kleisli 2-Adjunction
Consider a 2-category such that A op admits the construction of algebras. Due to this property of the 2-category A op , we will be able to construct a 2-adjunction of the form If we describe the adjunction over A and not on the opposite one then the 2-category Mnd(A op ) will be isomorphic to Mnd • (A) and the 2-category Adj R (A op ) will be isomorphic to Adj L (A). Note that in [9], the category A op is denoted as A * .
The description of the 2-category Mnd • (A) is given as follows.
1.-The 0-cells are monads in A, i.e. (A, f, µ f , η f ). The short notation (A, f ) will be used for such a monad. The mate is described by This morphism can be represented as 3.-The 2-cells are made of a pair of 2-cells in A, (α, β) as in such that they fulfill one of the following equivalent conditions Remark 2.1 Note that the previous conditions can be seen as commutative surface diagrams.
This 2-cell can be displayed as follows The n-cell structure described arrange itself to form a 2-category.
Before going into the details on the construction of the 2-functor Ψ K , we develop some calculations. These calculations are dual to those made in [9]. Note that we are going to be switching between the 2-categories A op and Mnd(A op ) to A and Mnd • (A), respectively.
Since the 2-category A op admits the construction of algebras, the functor Inc A op : A op −→ Mnd(A op ) admits a right adjoint, denoted as Alg A op : Mnd(A op ) −→ A op . These 2-functors are going to be short denoted as I • and A • respectively.
The corresponding counit, on the component Following [9], for any monad (A, f op ) in Mnd(A op ), there exists an adjunction in A, such that it generates the monad (A, f ), with unit η f and counit ε gv f . It can be checked that . Take the following composition of morphisms of monads Since the counit is universal from y y r r r r r r r r r r In particular, g h m = m π g f and ι h m • g h π = m π ι f . Note that the associated mate to the first equality is ρ π = v h m π ε gv h • η h mv f and that ρ π g f = π.
this translates, in the non-opposite case, into the following assigment On the other hand, we have an equality of 2-cells Therefore, to the 2-cell g h ϑ there corresponds, through the asignment (3), We change, at this point, the notation as β ϑ = ϑ.
Without any further ado, we provide the description of the 2-functor Ψ K , where ϑ is given as above.
The description of the functor Φ K is given as follows 3.-For the transformation of adjunctions (α, β) : Yet again, following [9], it can be shown that for the adjunction l ⊣ r, there exists a dual comparison 1-cell k rl : A rl −→ B, such that l = k rl g rl , v rl = rk rl and ε rl l = k rl ι rl .
The unit of the 2-adjunction in (1), In turn, the counit ε ΨΦ K : Proof : We prove only one of the triangular identities, i.e.

Formal Eilenberg-Moore 2-Adjunction
Consider a 2-category A which admits the construction of algebras. With this property of A, we will construct a 2-adjunction of the form The 2-category Adj R (A) is described as follows  2.-The 1-cells are pairs, of 1-cells in A, (j, k) such that the first diagram is the 2-cell mate to the second one The mate is described by This morphism can be represented as  such that they fulfill one of the following equivalent conditions Remark 3.1 Note that the previous conditions can be seen as commutative surface diagrams.
This 2-cell can be displayed as follows The described n-cell structure arrange itself to form a 2-category.
The description of 2-functor Φ E is given as follows Before the description of the 2-functor Ψ E , we realize some calculations.
In Theorem 2, at [9], the author proved that if A admits the construction of algebras then for any monad (A, f ) in Mnd(A), there exists an adjunction in A such that it generates the monad (A, f ), with unit η f and counit ε d u f . It can be checked that The previous counit, ε IA , is universal from the functor Inc A , in particular, for the 1- y y r r r r r r r r r r In particular, pu f = u h p ϕ and pχ f • ϕu f = χ h p ϕ . Observe that the associated mate, to the first Consider a 2-cell of monads, θ : Because of the construction of algebras for A, the 2-adjunction provides an isomorphism of categories, for A in A and (X, f ) in Mnd(A), given by the following assignment cf. [9]. On the other hand, we have an equality of 2-cells Therefore, to the 2-cell θu f there corresponds, through the assignment (5), a 2-cell Alg A (θu f )η AI (A f ) := β θ , where β θ : p ϕ −→ q ψ and such that u h β θ = θu f . We change the notation as follows β θ = θ.
With these calculations at hand, we define the 2-functor Ψ E .
The unit and the counit for this 2-adjunction are given as follows. The component of the unit, at In [9], Theorem 3, the author proved the existence of a comparison 1-cell k rl : B −→ A rl , such that u rl k rl = r and d rl = k rl l. Therefore, we can make the following definition In turn, the component of the counit, at We prove only one of the triangular identities and the other one is left to the reader. Using the definition of the unit and counit for this 2-adjunction, the triangular identity

Eilenberg-Moore 2-Adjunction
In this section, we apply the results of the Section 3 to the 2-category 2 Cat, the 2-category of small categories and functors, because the 2-category 2 Cat admits the construction of algebras. The 2adjunction given in that section gives a usual adjunction, Since the complete description, for a general A, has been given above, we only give some remarks on the derived properties for this particular 2-category.
The description of the 2-functor Ψ E , for this particular 2-category, is given by the following entries (ii) On morphisms, p, P ϕ (p) = P p.
(iii) The natural transformation λ ϕ is the mate of the identity U H P ϕ = P U F . Using (4), we get the component of λ ϕ at A, in C, The induced natural tranformationθ : It is clear that this definition is equivalent to the condition θ U F = U Hθ .
Since we have a 2-adjunction, the following isomorphism of categories takes place, natural for all L ⊣ R and (X , H): 5 Monoidal Liftings (Eilenberg-Moore Type)

Colax Monads
In this section, we give the definition of a colax monad.
That is to say, the natural transformations ξ : F · ⊗ −→ ⊗ · (F × F ) and γ : F · δ I −→ δ I fulfills the commutativity on the following diagrams F (I ⊗ A) apart from the fact that they are natural transformations, they fulfill additionally the following commutative diagrams Since the natural transformation γ has only one component, at 0, then this natural transformation and its component will be denoted indistinctly as γ.

2.-Morphisms and natural transformations of monads of the form
: 3.-Monoidal structures for the Eilenberg-Moore category, (C F , ⊗,Î,â,l,r) such that the following diagram of arrows and surfaces commutes (a) Consider a colax monad (F, ξ, γ), µ F , η F , for the monoidal structure (C, ⊗, I). In particular, the multiplication and the unit of the monad are colax natural transformations and the first diagrams in (9) and (10) commute. Therefore, we have a monad morphism (⊗, ξ) : Likewise, the commutativity of the second diagrams in (9) and (10) implies that (δ I , γ) : (1, 1 1 ) −→ (C, F ) is a morphism of monads. Note that the requirement (δ I , γ) is a monad morphism is equivalent to the statement (I, γ) is an Eilenberg-Moore algebra.
Since (⊗, ξ) is a morphism of monads then the following morphisms are also morphisms of monads ⊗·(⊗×C), ⊗(ξ×F )•ξ(⊗×C) and (⊗·(C×⊗)·a C , ⊗(F ×ξ)a C •ξ(C×⊗)a C ) from ((C×C)×C, (F ×F )×F ) to (C, F ) and due to the commutativity of the diagram (7), the following is a 2-cell in Mnd a Likewise, because (⊗, ξ) and (δ I , γ) are monad morphisms, is also a monad morphism. Using the commutativity of the first diagram in (8), we can consider the monad 2-cell In a similar way, the following is a monad transformation, r : Note that the aforementioned claims can be reverted.

⇒ 3)
Take the monad morphism (⊗, ξ) : (C × C, F × F ) −→ (C, F ). In order to use the isomorphism (6), we make L ⊣ R = D F × D F ⊣ U F × U F and (X , H, µ H , η H ) = (C, F, µ F , η F ). Therefore, to this monad morphism corresponds a morphism of adjunctions of the form (⊗, ⊗ ξ ) : (11a) commutes. According to the definition of Ψ E , the functor ⊗ ξ acts as follows The previous action is defined at the beginning of the proof of Theorem 7.1, [8].
If in the isomorphism (6), we make L ⊣ R = 1 1 ⊣ 1 1 and (X , H, µ H , η H ) = (C, F, µ F , η F ). The monad morphism (δ I , γ) has an associated morphism of adjunctions of the form (δ I , δ γ I ) : (1 1 ⊣ 1 1 ) −→ D F ⊣ U F such that a diagram like (11b) commutes. According to the definition of Ψ E , the functor δ γ I acts as follows On morphisms, If we make the following definitionÎ = (I, γ), then δ γ I := δÎ . The algebra (I, γ) is the unit of the monoidal structure on C F . Suppose that we have a natural transformation of the form a : (⊗ ·(⊗ × C), . Therefore, to the previous 2-cell of monads, we can associate a 2-cell of adjunctions of the form In order to reduce expressions, we used and will be using the following notation It can be prove that [⊗ · (⊗ × C)] · ξ 2 = ⊗ · ( ⊗ × C F ). On objects and morphisms In the same way, we can check that [⊗ · (C × ⊗) · a C ] · ξ 2 = ⊗ · (C F × ⊗) · a C F . We change the notation β a for a and we get a natural transformation a :  (6), we make L ⊣ R = D F ⊣ U F and (X , H, µ H , η H ) = (C, F, µ F , η F ), it can be obtained a 2-cell in the 2-category Adj R ( 2 Cat) of the form (l, β l ) : We change the notation from β l tol.
In the same way as before, it can be proved that Since the natural transformations a, l and r fulfill the coherence conditions for a monoidal struture and U F is faithfull thenâ,l andr fulfill the pentagon and the triangle coherence conditions. Therefore, (C F , ⊗,Î,â,l,r) is a monoidal structure over C F .

⇒ 2)
Note that the aforementioned statements can be reverted. For example, take the morphism of adjunctions (a, a) : Everytime we used the isomorphism (6), the monad (C, F, µ F , η F ) was always taken fixed, therefore the implication 2 ⇒ 3 is natural in the monad (C, F, µ F , η F ).
The authors did not check for the naturality of the implication 1 ⇒ 2, but the reader can do it.

Kleisli 2-Adjunction
Based on either [2] or [3], the following 2-adjunction takes place which can also be deduced from the general 2-adjunction given by (1). In this sense, we provide only a few remarks on the structure for the several objects that build this 2-adjunction.
The description of 2-functor, Ψ K , is given completely in order to provide the necessary notation. The structure of such 2-functor goes as follows 1.-On 0-cells, Ψ K (C, F ) = G F ⊣ V F , i.e. the Kleisli adjunction.
(i) On objects, X in C F , P π X = P X.
(ii) On morphisms, where C x ♯ is the notation for the codomain of the morphism x ♯ as in C F , which in this case is Y . (iii) In order to define ρ π we have to prove that the following equality of functors takes place, G H P = P π G F . On objects and morphisms f : where the second equality takes place because of the unitality condition on π and the third one is due to the naturality on π. Using (2), we get the mate for this identity whose component, at X in C F , is ρ π X = µ H P X · HπX · η H P F X = πX.
Since we have a 2-adjunction, the following isomorphism of categories takes place, natural in (X , H) and L ⊣ R 7 Monoidal Extensions (Kleisli Type)

Lax Monads
Dual to colax monads, we give the definition of a lax monad.
We are going to make use of the isomorphism (14). The result we want to obtain using this isomorphism is the following. Theorem 7.3 There is a bijective correspondance between the following structures 1. -Colax monads ((F, ζ, ω), µ F , η F ), for the monoidal structure (C, ⊗, I, a, l, r).
Since (⊗, ζ) and (δ I , ω) are monad morphisms so is (⊗ · (δ I × C) · l −1 C , ζ (δ I × C) l −1 C • ⊗ (ω × F ) l −1 C ) and taking into account the commutativity of the diagram (16a), we can state that the following is a 2-cell in Mnd • ( 2 Cat) In the very same way, the following is a 2-cell of monads, r : The previous assertions can be reverted.

⇒ 3)
Suppose we have a monad morphism (⊗, ζ). Use the isomorphism (14), with (D, H, µ H , η H ) = (C × C, F × F, µ F × µ F , η F × η F ) and L ⊣ R = G F ⊣ V F to get an associated morphism of adjunctions (⊗, ⊗ ζ ) : such that a diagram like (19a) commutes. According to the definition of Ψ K , the functor ⊗ ζ acts as follows. On objects, and on morphisms, where C x ♯ is codomain of the morphism x ♯ for example. We rename ⊗ ζ as ⊗.
For the monad morphism, (δ I , ω) : (1, 1 1 ) −→ (C, F ), use the mentioned isomorphism with (D, H, µ H , η H ) = (1 , 1 1 , 1 1 1 , 1 1 1 ), i.e. the trivial monad on the category 1, and L ⊣ R = G F ⊣ V F . Therefore, there exists an adjunction morphism (δ I , [δ I ] ω ) : According to the 2-functor Ψ K , the functor [δ I ] ω : 1 −→ C F , acts in the following way Suppose that we have the following 2-cell in Mnd, In order to continue with the calculations, we use the following notation, for the sake of simplification According to the isomorphism of categories given by (14), to the previous 2-cell in Mnd • ( 2 Cat) corresponds a 2-cell, (α a , β a ) in Adj L ( 2 Cat), where α a = a and we rename β a =ã and such that Therefore, we have a natural transformation a : ⊗ · ( ⊗ × C F ) −→ ⊗ · (C F × ⊗) · a C F that will be part of a monoidal structure on C F . According to the 2-functor Ψ K , the component of a at ((X, Y ), Z) is Therefore, we obtain a natural transformationl : Using the definition of the functor Ψ K on the 2-cell l, the component ofl, on the object X in C F , is Similarly, for the monad morphism r : F ), we obtain a natural transformationr : The proof of the coherence conditions are left to the reader.

⇒ 2)
Using the isomorphism, given by (14), we get the return of the proof. For example, the image, under Φ K , for the 2-cell of adjunctions (a,ã) : Note that a C is a morphism of adjunctions.

Liftings to the Eilenberg-Moore algebras & Extensions to the Kleisli Categories
This is probably the more explored section of all this article, a few examples of the detailed proofs for the following statements are found in [1] and [10]. In this section, we treated these statements only as direct consequences of the isomorphisms of categories given by (6) and (14).
9 Actions on the Kleisli Category

Categorical Actions
In this section we give the definition of a categorical action.

Strong Monads
In this section we give the definition of a strong monad.
Definition 9.1 A right strong monad ((F, σ r ), µ F , η F ), on the monoidal category (C, ⊗, I), is a usual monad (F, µ F , η F ), on C, with a natural transformation σ r : Definition 9.2 A left strong monad ((F, σ r ), µ F , η F ) on a monoidal category (C, ⊗, I), is a usual monad (F, µ F , η F ) on C, together with a natural transformation σ l A,B : F A ⊗ B −→ F (A ⊗ B) such that fulfills the commutativity of dual diagrams like (21) and (22).
In [6], the author relates actions of a category C over its Kleisli category and strong monads. In this section, with the use of the 2-adjunction of Kleisli, we rediscover this relation between certain actions on the Kleisli categories for a given monad (C, F, µ F , η F ) with strong monads. Theorem 9.3 There exists a bijection between the following structures 1.-Right strong monads ((F, σ r ), µ F , η F ) on the monoidal category (C, ⊗, I, a, r, l).
The commutativity of the diagram (22a) implies that the following is a transformation of monads a By the same reason as before, the following is a morphism of monads (⊗·(δ I ×C)·l −1 C , σ r (δ I ×C) l −1 C ). Furthermore, due to the commutativity of (22b), we have a 2-cell of monads l : (⊗·(δ I ×C)·l −1 The return of the implication is inmediate. For example, a monad transformation a as indicated implies that a diagram like (22a) commutes.

⇒ 3)
In the isomorphism (14), make (X , H) = (C × C, C × F ) and L ⊣ R = G F ⊣ V F . Therefore, exists a bijection between morphisms of monads of the form (⊗, ϕ) and morphisms of adjunctions (⊗, ⊗ ϕ ), where the corresponding induced functor is denoted as ⊠ = ⊗ ϕ . This pair of functors make a diagram like (23) commute. In particular, the action of the second functor, on morphisms, is Yet again, use the isomorphism (14) with (X , H) = ((C × C) × C, (C × C) × F ) and L ⊣ R = G F ⊣ V F . Therefore, there exists a bijection between the tranformations of monads of the form a and transformations of adjunctions (a,ã). Where the second natural tranformation has the form a : Note that we use the following short notation We prove only that [⊗ · (⊗ × C)] ϕ(⊠×C) = ⊠(⊗ × C F ). Since for objects there is nothing to prove, let ((f, g), x ♯ ) : ((A, B), X) −→ ((A ′ , B ′ ), Y ) be a morphim in (C × C) × C F , therefore At this moment, we change the notation to a = ν. Therefore, the refered natural transformation can be written as ν : ⊠(⊗ × C F ) −→ ⊠(C × ⊠)a C * : (C × C) × C F −→ C F , according to the requirement. Note that the notation a C * stands for the object a C,C,C F . The component of the natural transformation ν, on the object ((A, A ′ ), X), is The same procedure can be applied to the natural transformation l, in order to get a natural transformation j : ⊠(δ I × C F )l −1 C F −→ 1 C F : C F −→ C F , whose component, on the object X, is j X = (η F X · l X ) ♯ .
The reader is invited to realize the remain calculations in order to prove that the structure (C F , ⊠, ν, j) is that of an left C-action on C F .

⇒ 2)
Since we have the isomorphism, the return is already given, nonetheless we comment some part of the procedure.
Under the isomorphism, a commutative diagram like (23) give rise to a morphism of monads of the form (⊗, ϕ ⊠ ) : (C × C, C × F ) −→ (C, F ) where the commutative diagrams that fulfill this morphism are (21) and (22).
We state the dual theorem

Conclusions and Future Work
This review has the objective to show how several situations for the theory of monads are connected in a very simple way, through a 2-adjunction. Any person who has tought a course on monads would agree that this structure, of a 2-adjunction, can be used as an educational purpose in the sense that a simple structure can account for several situations and which can spare the, otherwise cumbersome, details of the proofs.
For future work we have a few recommendations. The reader may find interesting to extent the part of strong monads and actions over the Kleisli categories to strong symmetrical monads and use the actions for the Eilenberg-Moore case. It would be interesting also to contextualize the case of the monoidal liftings and monoidal extensions according to the formal theory of monoidal monads, and the standar objects, given in [11].
The reader may want to find more situations in the monad theory that can use the isomorphism provided by this pair of 2-adjunctions, the authors will certainly pursue this issue.