# Stronger (?) form of Vopenka's principle

A category $$\mathcal{C}$$ is called $$\textbf{discrete}$$ if the only morphisms are identity morphisms.

Consider the following weaker notion: a category $$\mathcal{C}$$ is called $$\textbf{totally disconnected}$$ if $$\text{Hom}_\mathcal{C}(C,D)=\varnothing$$ for all $$C\neq D$$.

Vopenka's principle ($$\textbf{VP}$$) states that a full large subcategory $$\mathcal{D}$$ of a locally presentable category $$\mathcal{C}$$ cannot be discrete. I want to consider an anaologus statement:

($$\textbf{VP2}$$) Full large subcategory $$\mathcal{D}$$ of a locally presentable category $$\mathcal{C}$$ cannot be totally disconnected.

Since discrete categories are totally disconnected, $$\textbf{VP2}$$ implies $$\textbf{VP}$$. Hence $$\textbf{VP2}$$ can't be proven in $$\textbf{ZFC}$$, because $$\textbf{VP}$$ can't. Now I would like to know:

(1) Is there a counter-example to $$\textbf{VP2}$$ in $$\textbf{ZFC}$$? As far as I know, no counter-example to $$\textbf{VP}$$ has been found but since $$\textbf{VP2}$$ is in principle a stronger statement, perhaps we can construct one here?

(2) If not, what is the relation between $$\textbf{VP}$$ and $$\textbf{VP2}$$ inside $$\textbf{ZFC}$$? Are they equivalent? Or does at least consistency of $$\textbf{VP}$$ implies that of $$\textbf{VP2}$$?

(I wrote $$\textbf{ZFC}$$ as my set theory of choice but if there are any problems in using such a weak theory, feel free to consider some other).

Thanks!

• The category of fields is a disconnected full large subcategory of the category or rings, which is locally finitely presentable. Or am I missing something? Mar 13 at 13:14
• @AlexKruckman The category of fields is "not connected" but is not "disconnected" in the sense I defined above. I guess better term for my "disconnected" would be "totally disconnected". I will edit my question appropriately. Mar 13 at 13:33
• Ah, I see! I've edited to make the quantifier on $C$ and $D$ more explicit. Mar 13 at 14:29