A stable functor has a left adjoint on each slice (multiajoint a` gauche, to quote Yves Diers). The word stable was first used in this sense by Gerard Berry (I believe), but it's most unfortunate as it already has at least one generic meaning in category theory, namely preservation by pullbacks. Can anyone suggest a better name? Here are some possibilities. 1) Stable endofunctors of the category of sets have power series expansions, so "analytic" (after Joyal) is a possibility. It goes nicely with continuous (in the sense of Dana Scott) although I don't want it to include preservation of filtered colimits. Also, as used by Andre' Joyal and Franc,ois Lamarche, analytic functors need only send pullbacks to weak pullbacks. 2) My definition of a functor U:M->C being stable is that every map X->UC in C factor as a *candidate* (diagonally universal map in Diers' terminology) followed by Uh for some h:A->B in M. cf factorisation systems. This suggests "quotate" or "democratic" (ie there are enough candidates). 3) Much of my work on stable functors as morphisms of domains is based on a technique of considering slices. This suggests "laminated functor" and similarly laminated category, laminated ccc, laminated topos for the corresponding well-behaved categories (the last for a category every slice of which is a topos, not necessarily with a terminal object). 4) Since they're functors with (non-chosen) adjoints on each slice, "slice-adjunctible" is another possibility. 5) I'm also interested in functors which just preserve (binary) pullbacks (between categories with them). I'd like to use the word "cartesian" to refer to anything concerned with pullback squares (and not just products), and call a pullback-preserving functor "cartesian" Any thoughts? Paul Taylor =========================================================================