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

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.<strong>1.</strong> 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 .