Let $\kappa$ be a regular cardinal. A $\kappa$-Boolean algebra ($\kappa$-BA) is a Boolean algebra having joins and meets of any cardinality strictly less than $\kappa$. A well-studied question is representability by algebras of sets, in which the Boolean operations are the set-theoretic ones.

Stone's theorem says that every $\omega$-BA is isomorphic to an $\omega$-BA of sets, but the corresponding result fails for $\kappa>\omega$. The only general result for cardinals greater than $\omega$ is the Loomis-Sikorski theorem, which asserts that every $\omega_1$-BA is a homomorphic image of an $\omega_1$-algebra of sets. A corollary is that the Baire sets of the Cantor space $2^X$ forms the free $\omega_1$-BA on generators $X$. No satisfactory similar characterization is known for $\kappa > \omega_1$.

Several authors have studied various forms of distributivity and its relation to representability. Roughly speaking, an algebra is $(\kappa,\lambda)$-distributive if $\kappa$-ary meets distribute over $\lambda$-ary joins. There are results relating distributivity to representability, but nothing as definitive as the Loomis-Sikorski theorem for $\omega_1$.

In this talk I will introduce a stronger form of distributivity called \emph{superdistributivity}. I will give seven equivalent characterizations of superdistributivity, one of which is a $\kappa$-Rasiowa-Sikorski property involving ultrafilter extensions reminiscent of Martin's axiom. Another characterization provides an infinitary quasi-equational axiomatization, implying the existence of free models. The last characterization is a fun game-theoretic characterization involving a mathematical formulation of the game of Capture the Flag.

Our main result is that the $\kappa$-Baire sets of the Cantor space $2^X$, the smallest $\kappa$-algebra of sets generated by the basic clopen sets, is the free superdistributive $\kappa$-BA on generators $X$. A corollary is a generalized version of the Loomis-Sikorski theorem: Every superdistributive $\kappa$-BA is a homomorphic image of a $\kappa$-algebra of sets.