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

154Thus hQ is an isomorphism with inverse P C ρ Q and we can conclude by applying2). □8. Equivalence for (co)module categories8.<strong>1.</strong> Equivalence for module categories coming from copretorsor. In thissubsection we prove that, for given categories A and B, under the assumptions ofTheorem 6.29, there exist a monad A on A and a monad B on B such that theircategories of modules are equivalent. We outline that the assumptions quoted aboveare satisfied in the particular case of a regular coherd.First of all we need to define the functors A Q B and B ̂QA which will be used to setthe equivalence between these module categories.Using the functors Q and ̂Q, we construct the lifting functors A Q B : B B → A A andB ̂Q A : A A → B B.Proposition 8.<strong>1.</strong> In the setting of 6.29 there exists a functor A (Q B ) : B B → A Asuch that A U A (Q B ) = Q B where (Q B , p Q ) = Coequ Fun(µBQ BU, Q B Uλ B). Moreoverwe have(174) p Q ◦ (A µ Q BU ) = A µ QB ◦ (Ap Q )where A µ QB = A Uλ AA (Q B ) : AQ B → Q B .Proof. In view of Theorem 6.29, we can apply Proposition 3.30.8.<strong>2.</strong> In light of Proposition 8.1, a functor Q : B → A introduced in 6.29 inducesa functor A (Q B ) : B B → A A for the monads A and B. Our next task is to provethat ( the ) B-A-bimodule functor ̂Q, constructed in Proposition 7.6, induces a functor̂QA : A A → B B which yields the inverse of A (Q B ) .BProposition 8.3. Within the assumptions and notations of Theorem 6.29, thereexists a functor B ̂QA : AA → B B such that B U B ̂QA = ̂Q( )A where ̂QA , pb Q=(Coequ Fun µ A Q bAU, ̂Q)A Uλ A . Moreover we have) ( )(175)A µ QB ◦(Bpb Q= pb Q◦B µb Q AUwhere B µb QA= B Uλ BB ̂QA : B ̂Q A → ̂Q A , so thatfunctor.(̂QA , B µb QA)□is an B-left moduleProof. In view of Proposition 7.6, we can apply Proposition 3.30 where Q is ̂Q andwe exchange the role of A and B, A and B.□Now we want to prove the first isomorphism.Within the assumptions and notations of Theorem 6.29, we will construct a functorialisomorphism B ̂QAA Q B∼ = B B.Lemma 8.4. Within the assumptions and notations of Theorem 6.29 the followingequality(176) ν ′ 0 ◦ (u B P ) = l ◦ (P u A )