Date: Fri, 6 Oct 1995 11:00:12 -0300 (ADT) Subject: if a functor locally has a right adjoint does it have a global one ? Date: Fri, 06 Oct 1995 14:36:29 MEZ From: Thomas Streicher Subject: if a functor locally has a right adjoint does it have a global one ? Let F : A->B be a functor. We say that F has "locally right adjoints" iff for any object I of A the (obvious) functor F^(I) : A/I->B/FI has a right adjoint U^(I). My question now is : What are sufficient conditions guaranteeing that such an F has itself a right adjoint ? What I know is that if A has a terminal object and B has binary products, then if F has locally right adjoints then F has a right adjoint. The reason is that F = Sigma_FI o F^(1) and - as B has binary products the functor Sigma_F1 has right adjoint (F1)* - F has right adjoint U^(1) o (F1)*. The main case I am interested in is the cxase where both A and B are "partial lex", i.e. have binary(!) pullbacks and products. I vaguely remember that some time ago on this network there was a discussion on functors having locally right adjoints. Maybe one of the persons involved in this discussion coiuld provide me with a hint. Thanks, Thomas Streicher Date: Mon, 9 Oct 1995 10:38:46 -0300 (ADT) Subject: Re: if a functor locally has a right adjoint does it have a Date: Mon, 9 Oct 1995 10:16:16 +0100 From: BOERGER One sufficient condition can be found in my joint paper with Walter Tholen "Abschwaechungen des Adjunktionsbegriffs", manuscripta math. 19 (1976), 19-45 as Theorem 14, part 2: A lcally riht adjoint functor is right adjoint if it is weakly final and preserves binary products and if binary products exist in the domain category. Obviously, weak finality and product presevation are also necessary. Greetings Reinhard Boerger