Notion of infinity in categories Please excuse me if the question is too vague or uninteresting.
Let $\mathcal{C}$ be a category and $A$ an object in $\mathcal{C}$. Motivated by the equivalence of Dedekind-finiteness and finiteness for sets (assuming Choice), let us define that an object $A$ is $\mathcal{C}$-infinite if there is an object $A'$ in $\mathcal{C}$ corresponding to a proper subobject of $A$ and an isomorphism $f: A \to A'$ in $\mathcal{C}$.
In the category Set, we have the property that ($B$ is Set-infinite and $B \subseteq A) \implies A$ is Set-infinite. The analogous property does not hold (with this definition of infinity) over arbitrary categories. For instance, the additive group $\mathbb{Q}$ has $\mathbb{Z}$ as a proper subgroup, is finite in the category Grp (since any homomorphism $f: \mathbb{Q} \to \mathbb{Q}$ must actually be an isomorphism), but $\mathbb{Z}$ is Grp-infinite by $g:\mathbb{Z} \to 2\mathbb{Z}$ with $z \mapsto 2z$.
In which categories would a version of this transitivity property hold? What about other properties depending on a notion of infinity? Or determining whether a given object in a category is infinite or not?
A result somewhat related to this notion is that a Banach space isomorphic to all its infinite-dimensional closed subspaces is isomorphic to a separable Hilbert space. Any such Banach space is obviously Ban-infinite.
I don't know much Category Theory or Logic, but this has been troubling me for a while (I'm an undergraduate student). Thanks in advance for your contributions.
 A: This is an enlarged comment. See section $8.2$ of $[1]$.
Transitivity Property: Let $B$ be $\mathscr{C}$-infinite, and let $B\hookrightarrow A$. Then $A$ is $\mathscr{C}$-infinite. This is not satisfied for all $\mathscr{C}$; e.g., let $\mathscr{C}$ be the category of presheaves on $(\mathbb{N},\leq)$ (Counterexample $8.2.4$, $[1]$).
The following theorem gives a sufficient condition (there might be many more) for the transitivity property to hold.
Theorem 1 (Proposition $8.2.3$, $[1]$): Let $A$ be an object of a topos $\mathscr{C}$ such that there is a complemented subobject $B$ of $A$ that is $\mathscr{C}$-infinite. Then $A$ is itself $\mathscr{C}$-infinite.
Proof (taken from $[1]$): Let $C\hookrightarrow B$ be a subobject of $B$ not containing the global element $b:\mathbf{1}\to B$ and an isomorphism $C\to B$. As $\models b\in B$, it follows that $\models\neg(b\in \overline{B})$; hence $\models\neg(b\in S\cup\overline{B})$. Thus $S\coprod\overline{B}$ is a proper subobject of $A$. There is an isomorphism $A\to S\coprod \overline{B}$. Hence $A$ is $\mathscr{C}$-infinite. $\square$
Bibliography
$[1]$ Francis Borceux, Handbook of Categorical Algebra: Volume $3$, Sheaf Theory. $522$ pp. Cambridge University Press. Published $1994$.
