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!



Stronger (?) form of Vopenka's principle
VP2 is equivalent to VP because every set carries a rigid binary relation. This is similar as Lemma 6.3 in my book with Adámek. In fact, VP2 is an original formulation of VP (see Jech, Set Theory).


