Does internalization loses informations everywhere?

Question

It is well known that a group object in Grp is necessarily abelian. This can be understood as "internalization loses information". Indeed, if one was to study group theory by looking at group objects in Grp, one would conclude that all groups are abelians.

It is also well known that mathematical logic is usually done in the sense of "let's internalize our theory inside itself and study it from there". Yet, it is highly possible than such internalized theory is much weaker than the original one we started with. That is, by looking at it, one would make wrong conclusions about the original theory.

So, my question is simple: is it known, in some specific sense, that internalization of set theory is different than the one we started from? This question could also be formulated as "Does a topos object in a Topos "loses" information ?".

Thanks.

Answer 1

You say that internalization in $\mathsf{Grp}$ "loses information". This presupposes that there is some natural way to compare group objects in $\mathsf{Grp}$ to group objects in $\mathsf{Set}$. Although there is a natural comparison in this case (induced by the forgetful functor $\mathsf{Grp} \to \mathsf{Set}$), there won't be such a natural comparison between groups internal to $\mathcal{C}$ and groups internal to $\mathcal{D}$ in general. So I would prefer to simply say that groups internal to $\mathsf{Grp}$ are "less rich" than groups internal to $\mathsf{Set}$.

That being said, I would say that it is definitely not generally true that group objects in $\mathcal{C}$ are less rich than group objects in $\mathsf{Set}$. For example, group objects in the category $\mathsf{Top}$ of topological spaces are topological groups, which are richer than group objects in $\mathsf{Set}$ (In fact, it contains the category of groups in $\mathsf{Set}$ as the subcategory of discrete groups). Group objects internal to the category $\mathsf{Sh}(X)$ of sheaves on a space $X$ are sheaves of groups, which are also richer than group objects in $\mathsf{Set}$ (and they contain the category of group objects in $\mathsf{Set}$ as the subcategory of constant sheaves).

Similar remarks apply to set theory. For example, the topos $\mathsf{Sh}(X)$ of sheaves on a space $X$ (a topos object in $\mathsf{Set}$) is a richer category than $\mathsf{Set}$ (and it contains $\mathsf{Set}$ as the subcategory of constant sheaves).

Sometimes things come out somewhere in the middle. For example, group monoid objects in the category $\mathsf{Ord}$ of ordered sets are ordered groups monoids; not every group monoid admits an order, so in that sense ordered groups monoids are less rich than groups monoids, but the order is an additional structure and the same group monoid can admit many different orders, so in that sense ordered groups monoids are more rich than groups monoids.