**May 1**

**Joan Bagaria**,
Universitat de Barcelona

**From Strong to Woodin cardinals: A level-by-level analysis of the Weak Vopenka Principle**

In May 2019 Trevor Wilson proved that the Weak Vopenka Principle (WVP), which asserts that the opposite of the category of Ordinals cannot be fully embedded into the category of Graphs, is equivalent to the class of ordinals being Woodin. In particular this implies that WVP is not equivalent to Vopenka’s Principle, thus solving an important long-standing open question in category theory. I will report on a joint ensuing work with Trevor Wilson in which we analyse the strength of WVP for definable classes of full subcategories of Graphs, obtaining exact level-by-level characterisations in terms of a natural hierarchy of strong cardinals.