Do results holding in all Grothendieck universes generalize to large categories?

Question

There are some tricky foundational issues related to category theory that seem to be best resolved with Grothendieck universes (at the cost of assuming the existence of arbitrarily large inaccessible cardinals). With this we assume that every set is contained in some Grothendieck universe $U$ modeling ZFC and then look at categories whose objects and arrows are taken from $U$, while only assuming that $U$ is a Grothendieck universe. Let's say that we can prove some theorem about the category of $U$-small abelian groups, whose objects and arrows live in $U$. According to my understanding, since $U$ models ZFC, applying that theorem to the category of all abelian groups gives us something consistent with ZFC. Can we go further and show that ZFC+Tarski Axiom actually proves the theorem to hold of the category of all abelian groups? I am wondering if a reflection principle might help with this.

Answer 1

In this case, the answer is often yes.

To see this, we have to do a little model theory. Suppose we have a small language $\mathcal{L}$, some set $T$ of sentences over $\mathcal{L}$, and some sentence $\phi$ over $\mathcal{L}$, such that for all small $\mathcal{L}$ structures $M$, if $M \models T$ then $M \models \phi$.

In this case, we see that by the completeness theorem relativised to the universe, we have $T \vdash \phi$. Then by the soundness theorem, we have $T \models \phi$.

So if we return to your concrete example of Abelian groups, we indeed see that any first-order property $\phi$ which holds for all small groups must also hold for all groups (and, in fact, there must be a proof of this property derivable in the first-order theory of groups). If we use a many-sorted language, we could also discuss first-order results about short exact sequences of Abelian groups, for example.

However, this is not the case for second-order properties. An easy way to see this is looking at second-order ZFC. If we look at the smallest Grothendieck universe $U$, we see there are no small models of second-order ZFC at all, so $\bot$ holds for all small second-order ZFC models. But there is a model of second-order ZFC, namely, the smallest Grothendieck universe.