The Game-Theoretic Kat\v{e}tov Order and Idealised Effective Subtoposes

Abstract

This paper addresses the longstanding problem of determining the structure of the $\leq_{\mathrm{LT}}$-order in the Effective Topos, known to effectively embed the Turing degrees. In a surprising discovery, we show that the $\leq_{\mathrm{LT}}$-order is in fact tightly controlled by the combinatorics of filters on $\omega$, raising deep questions about how combinatorial and computable complexity interact, both within this order and beyond it. To make the connection precise, we introduce a game-theoretic (''gamified'') variant of the Kat\v{e}tov order on filters over $\omega$, which turns out to exhibit a striking mix of coarseness and subtlety. For one, it is strictly coarser than the classical Rudin-Keisler order and, when viewed dually on ideals, collapses all MAD families to a single equivalence class. On the other hand, the order also supports a rich internal structure, including an infinite strictly ascending chain of ideal classes, which we identify by way of a new separation technique. From the computability-theoretic perspective, we show that a computable (and extended) variant of the gamified Kat\v{e}tov order is isomorphic to the original $\leq_{\mathrm{LT}}$-order. Moreover, our work brings into focus a new degree-spectrum invariant for filters $\mathcal{F}$, $$\mathcal{D}_{\mathrm{T}}(\mathcal{F}):=\{\,[f\colon\omega\to\omega] \mid f\leq_{\mathrm{LT}} \mathcal{F} \},$$ which is shown to always determine a proper initial segment of the Turing degrees. Extending this, given any $\Delta^1_1$ filter $\mathcal{F}$, we show that $\mathcal{D}_{\mathrm{T}}(\mathcal{F})$ is precisely the class of hyperarithmetic degrees. This significantly generalises previous results obtained by van Oosten \cite{vO14} and Kihara \cite{Kih23}. The proofs draw on ideas from general topology, descriptive set theory, and computability theory.