# Category Theory and Very Large Cardinals: Do large cardinal axioms join with any other branch of mathematics?

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:

1. Are there any other large cardinal axioms equivalent or equiconsistent to category theoretical principles?
2. Are there any other similar branches of math to category theory which also have easily statable principles equiconsistent to large cardinal axioms?
3. Is there any explanation for why category theory in specific exhibits this property but not, for example, group theory or the theory of magmas?
• Harvey Friedman has spent a lot of his career finding combinatorial facts equiconsistent with some very large cardinals. See this for an example with $n$-huge cardinals. – Wojowu Nov 13 '17 at 7:34
• I don't know if this will have anything new to add, but you may find Set theory for Category Theory an interesting resource. You've probably already read it. – Derek Elkins Nov 13 '17 at 10:03
• For your third question, though, I would say the difference is in the difference between how category theory and group theory are usually used. Categories aren't studied as an algebraic object so much. Instead, something like the category of rings is (e.g. from a Lawvere theory perspective) the category of all models of the (algebraic) theory of rings. Connections like these to logic/type theory and model theory make the connections to large cardinals far less surprising. "Category theory" is also a broad term. You could argue that category theory is magma theory if taken similarly broadly. – Derek Elkins Nov 13 '17 at 10:25
• "the most strong large cardinal that can be used to prove a useful claim about real numbers is strong inaccessibility ..." is inaccurate. The development of large cardinals is closely tied to studying the real numbers. Descriptive set theory which is all about the reals is abounding in use of strong large cardinals such as sharps, measurable, Woodin, etc. – William Nov 13 '17 at 22:44
• I meant in the theory of mathematics concerning more standard, non-set theory related ideas. For example, calculus, prime numbers, etc. I meant for the fields in question to be those not directly involving set theory. – Zetapology Nov 13 '17 at 23:39