Dear categorists, 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. A suggestion is to ask the canonical maps: colim_I(*C*(X,C_i))--> *C*(X,colim_I(C_i)) for filtered diagrams of strong epis, only to be surjective; in a finitary variety, this gives precisely the retracts of the (classical) finitely related algebras. Note that this surjectivity is all what is in fact needed in the analogous definitions of the finitely presented and the finitely generated objects, even in a finitely accessible category. So there is some uniformity. Note also such an X is finitely presentable iff it is finitely generated (in locally finitely presentable cats). Another suggestion (closer to the classical definition) is to ask them to be the coproducts of finitely presentables, all but at most one of which are projective. This corresponds to the (classical) finitely related algebras with respect to the canonical theory of an algebraic category.) Any opinion on the (ir)relevance of such a pursuit is most welcome! Thank you, Michel Hebert [For admin and other information see: http://www.mta.ca/~cat-dist/ ]