Contents 1. Introduction 2 2. Preliminaries 4 2.1. Some results on ...
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Contents 1. Introduction 2 2. Preliminaries 4 2.1. Some results on ...

Contents 1. Introduction 2 2. Preliminaries 4 2.1. Some results on ... Contents 1. Introduction 2 2. Preliminaries 4 2.1. Some results on ...

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44Definition 3.64. Let R : A → B be a functor. The functor R is called monadic ifit has a left adjoint L : B → A for which the functor K = Υ (Id RL ) : A → RL B is anequivalence of categories.The following is a slightly improved version of Theorem 3.14 p. 101 [BW].Theorem 3.65 (Generalized Beck’s Precise Tripleability Theorem). Let R : A → Bbe a functor and let A = (A, m A , u A ) be a monad on the category B. Then R isψ-monadic if and only if1) R has a left adjoint L : B → A,2) ψ : A → RL is a monads isomorphism where RL = (RL, RɛL, η) with η andɛ unit and counit of (L, R) ,3) for every ( Y, A µ Y)∈ A B, there exist Coequ A(rY, L A µ Y), where r = Θ (ψ) =(ɛL) ◦ (Lψ) , and R preserves the coequalizerCoequ Fun (r A U, L A Uλ A ) ,4) R reflects isomorphisms.In this case in A there exist coequalizers of R-contractible coequalizer pairs and Rpreserves them.Corollary 3.66 (Beck’s Precise Tripleability Theorem). Let R : A → B be afunctor. Then R is monadic if and only if1) R has a left adjoint L : B → A,2) for every ( Y, RL µ Y)∈ RL B, there exist Coequ A(ɛLY, L RL µ Y)and R preservesthe coequalizerCoequ Fun (ɛL RL U, L RL Uλ RL ) ,3) R reflects isomorphisms.In this case in A there exist coequalizers of R-contractible coequalizer pairs and Rpreserves them.Theorem 3.67 (Generalized Beck’s Theorem for Monads). Let (L, R) be an adjunctionwhere L : B → A and R : A → B, let A = (A, m A , u A ) be a monad onthe category B and let ψ : A = (A, m A , u A ) → RL = (RL, RɛL, η) be a monadsmorphism such that ψY is an epimorphism for every Y ∈ B. Let K ψ = Υ (ψ) =(R, (Rɛ) ◦ (ψR)) and A UK ψ (f) = A UΥ (ψ) (f) = R (f) for every morphism f in A.Then K ψ : A → A B is full and faithful if and only if for every X ∈ A we have that(X, ɛX) = Coequ A (LRɛX, ɛLRX) .Corollary 3.68 (Beck’s Theorem for Monads). Let (L, R) be an adjunction whereL : B → A and R : A → B. Then K = Υ (Id RL ) : A → RL B is full and faithful ifand only if for every X ∈ A we have that (X, ɛX) = Coequ A (LRɛX, ɛLRX) .4. ComonadsDefinition 4.ong>1.ong> A comonad on a category A is a triple C = ( C, ∆ C , ε C) , whereC : A → A is a functor, ∆ C : C → CC and ε C : C → A are functorial morphismssatisfying the coassociativity and the counitality conditions(∆ C C ) ◦ ∆ C = ( C∆ C) (◦ ∆ C and) CεC◦ ∆ C = C = ( ε C C ) ◦ ∆ C .

Definition 4.ong>2.ong> A morphism between two comonads C = ( C, ∆ C , ε C) and D =(D, ∆ D , ε D) on a category A is a functorial morphism ϕ : C → D such that∆ C ◦ ϕ = (ϕϕ) ◦ ∆ D and ε C ◦ ϕ = ε D .Example 4.3. Let ( C, ∆ C , ε C) an A-coring where A is a ring. Then• C is an A-A-bimodule• ∆ C : C → C ⊗ A C is a morphism of A-A-bimodules• ε C : C → A is a morphism of A-A-bimodules satisfying the following(∆ C ⊗ A C ) ◦∆ C = ( C ⊗ A ∆ C) ◦∆ C , ( C ⊗ A ε C) (◦∆ C = r −1Cand ε C ⊗ A C ) ◦∆ C = l −1Cwhere r C : C⊗ A A → C and l C : A⊗ A C → C are the right and left constraints.LetC = − ⊗ A C : Mod-A → Mod-A∆ C = − ⊗ A ∆ C : − ⊗ A C → − ⊗ A C ⊗ A Cε C = r − ◦ ( − ⊗ A ε C) : − ⊗ A C → − ⊗ A A → −Then, dually to the case of the R-ring, C = ( C, ∆ C , ε C) is a comonad on thecategory Mod-A.Proposition 4.4 ([H]). Let (L, R) be an adjunction with unit η and counit ɛ whereL : B → A and R : A → B. Then LR = (LR, LηR, ɛ) is a comonad on the categoryA.Proof. Dual to the proof of Proposition 3.4.Definition 4.5. A left comodule functor for a comonad C = ( C, ∆ C , ε C) on acategory A is a pair ( Q, C ρ Q)where Q : B → A is a functor and C ρ Q : Q → CQ isa functorial morphism such that(C C ρ Q)◦ C ρ Q = ( ∆ C Q ) ◦ C ρ Q and Q = ( ε C Q ) ◦ C ρ Q .Definition 4.6. A right comodule functor for a comonad C = ( C, ∆ C , ε C) on acategory A is a pair ( P, ρ C P)where P : A → B is a functor and ρCP : P → P C is afunctorial morphism such that(ρCP C ) ◦ ρ C P = ( P ∆ C) ◦ ρ C P and P = ( P ε C) ◦ ρ C P .45□Definition 4.7. For two comonads C = ( (C, ∆ C , ε C) on a category A and D =D, ∆ D , ε D) on a category B, a C-D-bicomodule functor is a triple ( )Q, C ρ Q , ρ D Q ,where Q : B → A is a functor and ( ) ( )Q, C ρ Q is a left C-comodule, Q, ρDQ is a rightD-comodule such that in addition(Cρ D Q) ◦ C ρ Q = (C ρ Q D ) ◦ ρ D Q.Definition 4.8. A morphism between two left C-comodule functors ( Q, C ρ Q)and(Q ′ , C ρ Q)is a morphism f : Q → Q ′ in A such thatC ρ Q ◦ f = (Cf) ◦ C ρ Q .

Definiti<strong>on</strong> 4.<str<strong>on</strong>g>2.</str<strong>on</strong>g> A morphism between two com<strong>on</strong>ads C = ( C, ∆ C , ε C) and D =(D, ∆ D , ε D) <strong>on</strong> a category A is a functorial morphism ϕ : C → D such that∆ C ◦ ϕ = (ϕϕ) ◦ ∆ D and ε C ◦ ϕ = ε D .Example 4.3. Let ( C, ∆ C , ε C) an A-coring where A is a ring. Then• C is an A-A-bimodule• ∆ C : C → C ⊗ A C is a morphism of A-A-bimodules• ε C : C → A is a morphism of A-A-bimodules satisfying the following(∆ C ⊗ A C ) ◦∆ C = ( C ⊗ A ∆ C) ◦∆ C , ( C ⊗ A ε C) (◦∆ C = r −1Cand ε C ⊗ A C ) ◦∆ C = l −1Cwhere r C : C⊗ A A → C and l C : A⊗ A C → C are the right and left c<strong>on</strong>straints.LetC = − ⊗ A C : Mod-A → Mod-A∆ C = − ⊗ A ∆ C : − ⊗ A C → − ⊗ A C ⊗ A Cε C = r − ◦ ( − ⊗ A ε C) : − ⊗ A C → − ⊗ A A → −Then, dually to the case of the R-ring, C = ( C, ∆ C , ε C) is a com<strong>on</strong>ad <strong>on</strong> thecategory Mod-A.Propositi<strong>on</strong> 4.4 ([H]). Let (L, R) be an adjuncti<strong>on</strong> with unit η and counit ɛ whereL : B → A and R : A → B. Then LR = (LR, LηR, ɛ) is a com<strong>on</strong>ad <strong>on</strong> the categoryA.Proof. Dual to the proof of Propositi<strong>on</strong> 3.4.Definiti<strong>on</strong> 4.5. A left comodule functor for a com<strong>on</strong>ad C = ( C, ∆ C , ε C) <strong>on</strong> acategory A is a pair ( Q, C ρ Q)where Q : B → A is a functor and C ρ Q : Q → CQ isa functorial morphism such that(C C ρ Q)◦ C ρ Q = ( ∆ C Q ) ◦ C ρ Q and Q = ( ε C Q ) ◦ C ρ Q .Definiti<strong>on</strong> 4.6. A right comodule functor for a com<strong>on</strong>ad C = ( C, ∆ C , ε C) <strong>on</strong> acategory A is a pair ( P, ρ C P)where P : A → B is a functor and ρCP : P → P C is afunctorial morphism such that(ρCP C ) ◦ ρ C P = ( P ∆ C) ◦ ρ C P and P = ( P ε C) ◦ ρ C P .45□Definiti<strong>on</strong> 4.7. For two com<strong>on</strong>ads C = ( (C, ∆ C , ε C) <strong>on</strong> a category A and D =D, ∆ D , ε D) <strong>on</strong> a category B, a C-D-bicomodule functor is a triple ( )Q, C ρ Q , ρ D Q ,where Q : B → A is a functor and ( ) ( )Q, C ρ Q is a left C-comodule, Q, ρDQ is a rightD-comodule such that in additi<strong>on</strong>(Cρ D Q) ◦ C ρ Q = (C ρ Q D ) ◦ ρ D Q.Definiti<strong>on</strong> 4.8. A morphism between two left C-comodule functors ( Q, C ρ Q)and(Q ′ , C ρ Q)is a morphism f : Q → Q ′ in A such thatC ρ Q ◦ f = (Cf) ◦ C ρ Q .

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