Vopenka's Principle and Woodin-like cardinals

Vopěnka’s Principle has evolved into one of the strongest connections among category theory, model theory and set theory. In this talk, I give a brief overview of the historical development of the principle, show some of its numerous characterisations and contrast it with large cardinal notions from set theory, especially with Woodin cardinals. I argue that this naturally leads to the definition of a new type of large cardinal, called “Woodin for strong compactness” and discuss some of its properties that arose during my PhD research.