Finitely related objects, categorically

Finitely related objects, categorically

[Michel Hebert]

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


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