As the reader of this question may know, a surprising connection between large cardinals and categories can be drawn in some cases.
When it seems as though no easy connection can be made from group theory to any large cardinals, some of those axioms have a basis in category theory. However, one would expect that these axioms are weak, as they can be easily stated in a category theoretical form. Generally, large cardinals don't tend to connect with other branches of mathematics unless very weak; the most strong large cardinal that can be used to prove a useful claim about real numbers is strong inaccessibility which shows that there is a model of ZF in which no set of reals is Lebesgue measurable. Even then, Lebesgue measurability is almost purely set-theoretical.
However, Vopenka's principle, for example, is equivalent to every accessible functor $F:C\rightarrow Set$ for some category $C$ having an accessible subfunctor. It is also equivalent to every discrete full subcategory of a locally presentable category being small. Furthermore, it is equivalent to every locally presentable category $C$ and every full subcategory $D\hookrightarrow C$ (as long as it is closed under colimits) is a coreflective subcategory.
The existence of a measurable cardinal, for another example, is equivalent to an exact functor endofunctor which is not isomorphic to the identity. The existence of a measurable cardinal is also equivalent to $FinSet^{op}$ not being dense in $Set^{op}$.
Although not category theoretical, Jónsson cardinals are precisely those cardinals $\kappa$ such that every algebra of size $\kappa$ has a subalgebra of size $\kappa$.
Questions:
- Are there any other large cardinal axioms equivalent or equiconsistent to category theoretical principles?
- Are there any other similar branches of math to category theory which also have easily statable principles equiconsistent to large cardinal axioms?
- Is there any explanation for why category theory in specific exhibits this property but not, for example, group theory or the theory of magmas?