Quoting the wiki:- a set A is Dedekind-infinite if some proper subset B of A is equinumerous to A. Explicitly, this means that there is a bijective function from A onto some proper subset B of A. -.
There is a also a categorical definition of Dedekind infinite object, which runs as follows:
-An object $A$ in a category $C$ is Dedekind infinite if there a monomorphism $j:A\rightarrow A$ which is not an iso.-
Now, my question is:
is there some suitable category $C$ over some ground "category of sets" $E$ (for instance a topos with a NNO) and a Dedekind infinite object $A$ in $C$, whose image is finite (in Dedekind's sense, or even in the standard sense of being in a bijiection with a finite number) in $E$?
Basically, I am after some gadget that "looks" (Dedekind) infinite inside of $C$ but not "in reality", ie in the ground category $E$.
Why I am interested: assuming that such animals exists somewhere in the vast world of mathematics, they would possibly be good candidates for mirroring Cantor's paradise inside the finite world.