ALGEBRAIC CATEGORIES WHOSE PROJECTIVES ARE EXPLICITLY FREE
Let M = (M;m;u) be a monad and let (MX;m) be the free M-algebra on the object X. Consider an M-algebra (A;a), a retraction r : (MX;m)! (A;a) and a section t : (A;a)! (MX;m) of r. The retract (A;a) is not free in general. We observe that for many monads with a 'combinatorial avor' such a retract is not only a free algebra (MA0;m), but it is also the case that the object A0 of generators is determined in a canonical way by the section t. We give a precise form of this property, prove a characterization, and discuss examples from combinatorics, universal algebra, convexity and topos theory.