Question

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.

Answer 1

It seems reasonable to think that a topos $C$ defined over (i.e., equipped with a geometric morphism to) the category $E$ of sets counts as a "suitable category $C$ over some ground "category of sets" $E$." Then presumably the word "image" in the question would refer to the forward-part of the geometric morphism, i.e., the "global sections" functor Hom$(1,-)$ from $C$ to $E$. With this interpretation, you get the following example (and lots more like it). Take $C$ to be the topos of $G$-sets where $G$ is your favorite non-trivial group. Take $A$ to be the disjoint union of infinitely many copies, say indexed by the natural numbers, of the regular action of $G$ on itself. This is Dedekind-infinite because you can shift the $n$-th copy to the $(n+1)$-st. But its set of global sections (i.e., $G$-fixed points) is not only finite but empty.

This sort of example is unlikely to be what you wanted. So could you please clarify how I should interpret "suitable category," "over," and "image"?