*Weak Vopěnka's principle* says that



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> the opposite of the category of ordinals cannot be fully embedded in any locally presentable category.
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Recall that one form of Vopěnka's principle says that the category of ordinals cannot be fully embedded in any locally presentable category.


Adámek and Rosický show that weak Vopěnka's principle follows from Vopěnka's principle and implies the existence of a proper class of measurables, giving a rough indication of its consistency strength. Its interest lies in its equivalence to the principle that any full subcategory of a locally presentable category closed under limits is reflective, or its equivalence to the principle that every orthogonal subcategory of a locally presentable category is reflective.


But these results date from Adámek and Rosický's 1994 book *Locally Presentable and Accessible Categories*.


**Question:** Is a more precise understanding of weak Vopěnka's principle available today? In particular, is its consistency strength well-calibrated?



What is the consistency strength of weak Vopenka's principle?
The weak Vopěnka principle (WVP) and semi-weak Vopěnka principle (SWVP) are both equivalent to the large cardinal principle "Ord is Woodin". These results now appear in a [paper on the arXiv](https://arxiv.org/pdf/1907.00284.pdf). I kept the proof that Ord is Woodin implies SWVP as part of this MathOverflow answer (see below), but removed the outline of the proof that WVP implies Ord is Woodin because the paper contains a better proof of that now.


I will use the form of WVP which says that the opposite of the category of ordinals cannot be fully embedded in the category of graphs. In other words, there is no sequence of graphs $(G\_\alpha : \alpha < \mathrm{Ord})$ such that for all $\alpha < \beta$ there is no homomorphism from $G\_\alpha$ to $G\_\beta$, and for all $\alpha \le \beta$ there is a unique homomorphism from $G\_\beta$ to $G\_\alpha$.


SWVP (see Adámek and Rosický, *On Injectivity in Locally Presentable Categories*) says that there is no sequence of graphs $(G\_\alpha : \alpha < \mathrm{Ord})$ such that for all $\alpha < \beta$ there is no homomorphism from $G\_\alpha$ to $G\_\beta$, and for all $\alpha \le \beta$ there is at least one homomorphism from $G\_\beta$ to $G\_\alpha$.


For every similarity type consisting of finitary relation symbols, the category of all structures of this type can be fully embedded into the category of graphs (see Hedrlín and Pultr, *On full embeddings of categories of algebras*,) so replacing graphs by more general relational structures yields equivalent statements of WVP and SWVP.


"Ord is Woodin" means that for every class $A$ there is an $A$-strong cardinal. See Kanamori, *The Higher Infinite* for the definition of $A$-strong cardinal. For maximum generality, we work in GB + AC. (As a special case, the results hold in ZFC for definable classes $A$.)


**Proof that Ord is Woodin implies SWVP.**


Assume Ord is Woodin and let $(G\_\alpha : \alpha \in \mathrm{Ord})$ be a sequence of graphs such that for all $\alpha \le \beta$ there is at least one homomorphism from $G\_\beta$ to $G\_\alpha$. We will show that for some ordinal $\kappa$ there is a homomorphism from $G\_\kappa$ to $G\_{\kappa+1}$.


We can write $G\_\alpha = G(\alpha)$ where $G$ is a class function with domain Ord. Because Ord is Woodin, some cardinal $\kappa$ is $G$-strong. 
Take a cardinal $\beta$ greater than $\kappa$ and greater than the von Neumann rank of the graph $G(\kappa+1)$, meaning $G(\kappa+1) \in V\_\beta$. Because $\kappa$ is $\beta$-$G$-strong, there is a transitive class $M$ containing $V\_\beta$ and an elementary embedding $j : (V;\in) \to (M;\in)$ such that the critical point of $j$ is $\kappa$ and $V\_\beta \subset M$ and $j(\kappa) > \beta$ and $j(G) \restriction \beta = G \restriction \beta$. We describe the latter condition by saying that $j$ *coheres with* $G$.


Because $j$ coheres with $G$, the graph $G(\kappa+1)$ is equal to $j(G)(\kappa+1)$, so it is on the sequence $j(G)$, which is an Ord-sequence of graphs in $M$. Note that $j(G(\kappa)) = j(G)(j(\kappa))$ by the elementarity of $j$, so the graph $j(G(\kappa))$ is also on the sequence $j(G)$. We have $j(\kappa) > \kappa+1$, so by our assumption on the sequence $G$ and the elementarity of $j$, we have a homomorphism $j(G(\kappa)) \to G(\kappa+1)$ in $M$ and therefore in $V$. We also have a homomorphism $G(\kappa) \to j(G(\kappa))$ in $V$ given by $j \restriction G(\kappa)$. Composing these, we obtain a homomorphism $G(\kappa) \to G(\kappa+1)$, as desired.


