Is anyone aware of some work mentioning a plausible categorical definition of "finitely related" object ? I mean an intrinsic one, which does not depend on some forgetful functor.
(I am aware that even in algebraic categories, there cannot be one which fits with the classical definition - independent of a given forgetful functor- because even free objects are not preserved by categorical equivalences.I am also aware of Paul Taylor's suggestion in his CUP 1999's Practical Foundations (Exercises VII 23-24), however we discussed this and he agrees that his Exercise 24 seems incorrect: there he defines a finitely related object X in C as one for which C(X,-) preserves filtered colimits of strong epis, but this is really too strong (infinite sets do not satisfy in Set), and is not equivalent to his definition in Exercise 23.)
