# Group theoretic Characterizations of Large Cardinals

Vopenka's principle is characterized (in the category theoretical definition) by no large full subcategory of a locally presentable category being discrete. In other words, proper classes of types of "algebra-like things" have one object with a homomorphism into another object.

On the other hand, Vopenka's principle can be characterized by just a single such locally presentable category having that property. Namely, Vopenka's principle is equivalent to no large full subcategory of $$\text{Graph}$$ being discrete. As a result, Vopenka's principle has analogies in other categories, for example $$\text{Grp}$$.

This is quite interesting, as it gives the following characterization of Vopenka cardinals:

$$\kappa$$ is Vopenka iff $$\kappa$$ is inaccessible and for every $$\kappa$$-sized family $$C$$ of nonisomorphic groups of order below $$\kappa$$, there are groups $$G$$ and $$H$$ in $$C$$ such that $$G$$ is a proper subgroup of $$H$$.

This is quite interesting. While this isn't exactly the most natural characterization of Vopenka cardinals, it is nonetheless a characterization of a large cardinal using group theory.

The question is when is this possible elsewhere in large cardinals? I really think it would be interesting if, say, weakly compact cardinals could be characterized in terms of infinite group theory. This is because infinite group theory is (in my opinion) studied without set theory. This is of course with reason; usually, whenever group theory has anything to do with infinite set theory, it ends up just being a byproduct of a relation to abstract algebra or category theory.

What other large cardinals can be characterized in terms of group theory?

• Large cardinal axiom. I'll specify in the page, sorry – Keith Millar Sep 23 '18 at 4:04
• Oh, right, I made a bit of an extra generalization. Although, the group characterization still holds. – Keith Millar Sep 23 '18 at 16:40