## Finitely related objects in categories

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.)

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 a (finitely) related question: mathoverflow.net/questions/9922/…. this could be interesting for the readers here. but the approaches there don't really work in the absence of finitely generated ... – Martin Brandenburg Apr 22 2010 at 7:16