MONADIC COMPLETION OF ADJUNCTIONS A famous theorem of Jon Beck says that an adjunction A ^ | | | F | -| | U | | | v C is monadic (equivalent to its category of Eilenberg--Moore algebras) iff U satisfies a certain condition involving "U-split coequalisers". Given any adjunction, of course one may "force" it to be monadic by replacing the category A by the category of algebras for the monad U.F, and there is a comparison functor. (1) My question is this: is there an explicit account in the literature somewhere of the construction obtained by "freely adding" the conditions of Beck's theorem (the U-split coequalisers) to A? (2) Further, has anyone considered what happens when one forces an adjunction to be BOTH monadic and comonadic? Steve Lack gave me a verbal answer to the second question, although he hasn't shown me his notes: (a) if we force U to be monadic, and then (the new) F to be comonadic, (the third) U need NOT be monadic, but (b) if we force U to be monadic again then (the fourth) F IS still comonadic. In my investigation of this question and similar ones that have arisen from my abstract Stone duality programme, I have found a construction due to Hayo Thielecke very useful, though it seems appropriate to generalise it considerably beyond either his or my interests: (3) Given an adjunction, or just the functor U, above, let H (for Hayo) be the category with objects: those of A morphisms: H(a, b) = C(Ua, Ub) Does anyone have an account of this category, maybe in the even more abstract framework of a 2-category? (4) The usefulness of the category H is that it provides a framework in which to lift an endofunctor S:A->A to the category of algebras, so long as T has a "de Morgan dual" P:C->C such that U.S=P.U. For example if S is (- x a) for some object a in A, or (which is what interests both Hayo and me) if U is Sigma^-: C^op -> C^op and S is (- x c), or the same situation with a symmetric monoidal (closed) structure. Making use of the construction of H, my sketch solution to question (2) is as follows: (i) As wide or lluf subcategories (all objects, some morphisms) of H we find the categories that freely make U faithful (f), or faithful and reflect invertibility (fri - this is the usage of "faithful" in Freyd & Scedrov); call these A_f and A_fri. (ii) F-|U extends to left adjoints F_f -| U_f : A_f -> C and F_fri -| U_fri : A_fri -> C and if F was faithful or fri then so is F_f or F_fri. (iii) Therefore the processes of making U and F faithful (or fri) commute, so we need only do each of them once. (iv) The auxillary category H is also convenient for representing freely adjoined U-split coequalisers, in a similar way to the "Karoubi completion" of a category to one with splittings of idempotents, but it's too messy to describe in this note. (v) Freely adding U-split coequalisers - commutes with freely forcing U (and F) to be faithful, and - commutes with freely adding F-split equalisers, and - preserves the properties that U or F are faithful or fri, but - DOES NOT COMMUTE with freely forcing U to reflect isomorphisms, and - DOES NOT COMMUTE with freely forcing F to reflect isomorphisms. (vi) This justifies Steve Lack's claims. (I have counterexamples in (v), though not actually to support (2a).) Resolving the construction of the monadic reflection of U (and the comonadic reflection of F) into forcing them to be fri and then to have split co/equalisers, we find that there is just one obstacle to the commutation of the four operations, but this is overcome by a single repetition as in (2b). Paul