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

Corollary 3.58. Let (L, R) be an adjunction where L : B → A and R : A → B.Let r = Θ (Id RL ) = ɛL. Assume that, for every ( )( )Y, RL µ Y ∈ RL B, there existsCoequ A ɛLY, L RL µ Y . Then we can consider the functor K = Υ (IdRL ) : A → RL B.Its left adjoint D : RL B → A is full and faithful if and only if R preserves thecoequalizer(D, d) = Coequ Fun (ɛL RL U, L RL Uλ RL ) .Theorem 3.59. Let (L, R) be an adjunction where L : B → A and R : A → B,let A = (A, m A , u A ) be a monad on the category B and let ψ : A = (A, m A , u A ) →RL = (RL, RɛL, η) be a monad morphism. Let r = Θ (ψ) = (ɛL) ◦ (Lψ) andl = Γ (ψ) = (Rɛ) ◦ (ψR). Assume that, for every ( )( )Y, A µ Y ∈ A B, there existsCoequ A rY, L A µ Y . Then we can consider the functor Kψ = Υ (ψ) : A → A B andits left adjoint D ψ : A B → A. The functor K ψ is an equivalence of categories if andonly if1) R preserves the coequalizer(D ψ , d ψ ) = Coequ Fun (r A U, L A Uλ A )2) R reflects isomorphisms and3) ψ : A → RL is a monad isomorphism.Definition 3.60. Let A = (A, m A , u A ) be a monad on the category B and let(R, A µ R)be a left A-module functor. We say that(R, A µ R)is a left A-coGaloisfunctor if R has a left adjoint L and if the canonical morphismcocan := (A µ R L ) ◦ (Aη) : A → RLis a monad isomorphism, where η denotes the unit of the adjunction (L, R).Corollary 3.6<strong>1.</strong> Let ( R, A µ R)be a left A-coGalois functor where R : A → Bpreserves coequalizers, R reflects isomorphisms and A = (A, m A , u A ) is a monad onB. Assume that, for every ( Y, A µ Y)∈ A B, there exists Coequ A(rY, L A µ Y)wherer = (ɛL) ◦ (Lcocan) where L is the left adjoint of R and ɛ is the counit of theadjunction (L, R). Then we can consider the functor K cocan : A → A B and its leftadjoint D cocan : A B → A. Then the functor K cocan is an equivalence of categories.Theorem 3.62 ( Beck’s Theorem for monads). Let (L, R) be an adjunction whereL : B → A and R : A → B. Let r = Θ (Id RL ) = ɛL and assume that, for every(Y, RL µ Y)∈ RL B, there exists Coequ A(ɛLY, L RL µ Y). Then we can consider thefunctor K = Υ (Id RL ) : A → RL B and its left adjoint D : RL B → A. The functor Kis an equivalence of categories if and only if1) R preserves the coequalizer2) R reflects isomorphisms.(D, d) = Coequ Fun (ɛL RL U, L RL Uλ RL ) .Definition 3.63. Let A = (A, m A , u A ) be a monad on the category B and letR : A → B be a functor. The functor R is called ψ-monadic if it has a left adjointL : B → A for which there exists ψ : A → RL a monad morphism such that thefunctor K ψ = Υ (ψ) : A → A B is an equivalence of categories.43