Locally presentable categories, universes, and Vopenka's principle

Some aspects of the theory of locally presentable categories depends on set-theoretic assumptions. Namely, a large cardinal axiom known as Vopenka's principle implies several nice properties of such categories (and equivalent to some of them). For example, it implies that every cocomplete category with a dense small subcategory is locally presentable.

Now, let $$\mathcal{U}$$ be some set-theoretic universe such as a Grothendieck universe. Then we can consider the theory of locally presentable $$\mathcal{U}$$-categories, where, roughly speaking, we replace all small sets with $$\mathcal{U}$$-small sets. There is a preprint by Zhen Lin Low, Universes for category theory (arXiv:1304.5227), that studies such categories, but it does not consider properties that depend on Vopenka's principle. So, are these properties always true for locally presentable $$\mathcal{U}$$-categories or do they still depend on set-theoretic assumptions?

• It depends on $\cal U$: is the corresponding inaccessible cardinal a "Vopenka cardinal" or not? ncatlab.org/nlab/show/… – Mike Shulman Feb 24 at 18:31

In the Grothendieck universe approach to category theory, as you say, we replace all small sets with $$\mathcal{U}$$-small sets. Let's look at the definition of a locally presentable $$\mathcal{U}$$-category, for example, since that's in the conclusion of the theorem you mentioned: a locally presentable $$\mathcal{U}$$-category is a category with $$\mathcal{U}$$-small homsets which has all $$\mathcal{U}$$-small colimits and for which there exists a $$\mathcal{U}$$-small regular cardinal $$\lambda$$ and a $$\mathcal{U}$$-small set of $$\lambda$$-compact objects which generate the category under $$\lambda$$-filtered $$\mathcal{U}$$-small colimits. The size relationships here are key. For example, the category of all sets is not a locally presentable $$\mathcal{U}$$-category because its hom-sets are not $$\mathcal{U}$$-small, while the category of sets of cardinality less than some $$0 < \kappa < \mathcal{U}$$ is not a locally presentable $$\mathcal{U}$$-category because it lacks $$\mathcal{U}$$-small colimits.

Now one formulation of Vopěnka's principle is

There is no proper class-sized discrete full subcategory of $$\mathrm{Gra}$$ (some category of graphs)

or in a more positive form

Any discrete full subcategory of $$\mathrm{Gra}$$ has only a set of objects

The proof of a theorem like the one you mentioned (in the non-universe setting) is going to go something like this. We use this axiom to conclude that some particular transfinite construction of new objects in a category cannot go on "forever". It has to stop at stage $$\gamma$$ for some ordinal $$\gamma$$, and from $$\gamma$$ we'll somehow produce the cardinal $$\lambda$$ in the definition of a locally presentable category and the set of generators.

If we now adopt universes, you may be thinking that

Any discrete full subcategory of $$\mathrm{Gra}_{\mathcal{U}}$$ has only a set of objects

is just (trivially) true, even without Vopěnka's principle. But if we tried to repeat the above argument using $$\mathcal{U}$$-categories, we'd just learn that there is some $$\gamma$$ and $$\lambda$$ as before, but we have no control over their sizes. For example, maybe the $$\lambda$$ we end up with would be larger than the cardinality of our category and the set of generators just all the objects of our category. In order to show that $$\lambda \in \mathcal{U}$$ we would need

Any discrete full subcategory of $$\mathrm{Gra}_{\mathcal{U}}$$ has only a $$\mathcal{U}$$-small set of objects

and I guess this is what it means for $$\mathcal{U}$$ to be a Vopěnka cardinal, as Mike mentioned in a comment. If you believe in such cardinals then you believe in Con(ZFC + Vopěnka’s principle), so you are still relying on set-theoretic assumptions (and much more so than simply believing in universes in the first place).