We show that Weak Vopěnka’s Principle, which is the statement that the opposite category of ordinals cannot be fully embedded into the category of graphs, is equivalent to the large cardinal principle Ord is Woodin, which says that for every class C there is a C-strong cardinal. Weak Vopěnka’s Principle was already known to imply the existence of a proper class of measurable cardinals. We improve this lower bound to the optimal one by defining structures whose nontrivial homomorphisms can be used as extenders, thereby producing elementary embeddings witnessing C-strongness of some cardinal.

中文翻译：

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弱 Vopenka 原理的大基数

我们证明了弱 Vopěnka 原则，即相反的序数类别不能完全嵌入到图的类别中的陈述，等价于大基数原则 Ord 是 Woodin，它说对于每个类C有一个C- 强大的红衣主教。众所周知，弱 Vopěnka 原理暗示存在适当类别的可测量基数。我们通过定义其非平凡同态可用作扩展器的结构，将这个下限改进为最优，从而产生基本嵌入见证C- 一些红衣主教的坚强。