I was recently reminded about [this old question](https://math.stackexchange.com/q/629357/88262) on math.stackexchange.


Let $\operatorname{Mod}R$ be the category of (right) modules for a ring $R$. The questioner mistakenly thought that the Freyd-Mitchell embedding theorem implied that for every ring $R$ there was another ring $S$ and a full embedding $(\operatorname{Mod}R)^{\text{op}}\to\operatorname{Mod}S$, and asked for an explicit description of such an $S$.


I realized (see [this answer](https://math.stackexchange.com/a/955539/88262)) that general facts about locally presentable categories imply that, if Vopěnka's principle is true, then there is no such embedding (for nonzero $R$).


Is there a proof that doesn't assume Vopěnka's principle? Or could it be that this fact genuinely depends on very strong large cardinal assumptions?



Vopěnka's principle and contravariant full embeddings between module categories
Assuming (M) (= there is only a set of measurable cardinals), the category $\bf{Vec}$ of vector spaces
(over every field) has a small dense subcategory. This is an old result of Isbell (see also
<https://arxiv.org/pdf/1812.10649.pdf>). Hence $\bf{Vec}$$^{\text{op}}$ is boundable, i.e., it can be fully embedded
to a category of algebras (I do not know whether to a category of modules).
Hence the set-theoretical strength of "no opposite category of modules is boundable" lies between $\neg$(M) and VP.


Moreover, the existence of a full embedding of $\bf{Ab}^{op}$ to $\bf{Mod}$ $R$ implies WVP (weak Vopěnka's principle) is false.
Indeed, the accessible category of graphs with monomorphisms can be fully embedded to the category $\bf{Gra}$
of graphs. A. J. Przezdziecki <https://arxiv.org/pdf/1104.5689.pdf> constructed an embedding $G:\bf{Gra}\to\bf{Ab}$
such that $\bf{Ab}$$(GX,GY)$ is the free abelian group on $\bf{Gra}$$(X,Y)$. This yields $R$-modules $A\_i$
indexed by ordinals such that the only morphisms $A\_i\to A\_j$, $i<j$ are $0$ and there are non-zero morphisms
$A\_j\to A\_i$ for every $i<j$. Let $X\_i\subseteq A\_i$ consist of elements $x\_i\neq 0$ such that for every $i<j$
there is $y\_j\in A\_j$ and $f:A\_j\to A\_I$ with $x\_i=f(y\_j)$. We have $X\_i\neq\emptyset$ for every $i$. Otherwise,
every element $0\neq z\in A\_i$ would have a height $k$, i.e., the smallest ordinal such that $z$ is not
in the image of any $f:A\_k\to A\_i$. Since these heights are arbitrarily large and $A\_i$ is a set, we get
a contradiction. Add to $\bf{Ab}$ a unary relation and interpret it on $A\_i$ as $X\_i$. Homomorphisms $A\_j\to A\_i$,
$i<j$ preserve these relations but $0:A\_i\to A\_j$, $i<j$ do not. Hence sWVP (semiweak Vopěnka's principle) does
not hold. Following the recent result of T. M. Wilson <https://arxiv.org/pdf/1909.09333.pdf>, WVP does not hold.


