Andrew Brooke-Taylor (University of Leeds) – A free 2-generator shelf from large cardinals

One of the strongest known large cardinal axioms is I3, positing the existence of a non-trivial elementary embedding $j$ from $V_λ$ to $V_λ$ for some $λ$.  Given two such embeddings $j$ and $k$ for the same lambda, there is a natural "application" operation to yield a third, $j*k$, and elementarity shows that this operation is left self-distributive: $j*(k*l)=(j*k)*(j*l)$. Structures with such an operation are called LD-algebras or shelves. Laver showed that the algebra of embeddings generated by a single such $j$ under $*$ is in fact the free LD-algebra on 1 generator; and the set-theoretic context around this concrete (once you've assumed I3) instantiation of the free LD-algebra gives rise to various theorems about LD-algebras that are only known under this very strong large cardinal assumption.  Given I3, there will be many other embeddings from $V_λ$ to $V_λ$, and it is natural to ask if one can obtain from amongst them a free LD-algebra on more than one generator.  In joint work with Scott Cramer and Sheila Miller, we show that the answer is positive if one assumes a little more: from I2 we get a free 2-generator LD-algebra of embeddings.  This talk will focus on set-theoretic aspects of the proof; a week later I will be giving a talk in the ARTIN conference on the same topic, focusing more on the algebraic aspects.