The Game-Theoretic Katětov Order and Idealised Effective Subtoposes
Takayuki Kihara, Ming Ng
TL;DR
This work identifies and develops the Gamified Katětov order, a game-theoretic variant of the Katětov order on upper sets over $\omega$, as a central tool to calibrate the LT-order on LT-topologies in the Effective Topos. It demonstrates that the GTK is strictly coarser than Rudin–Keisler yet incomparable with Tukey, while also encoding rich internal structure such as an infinite ascending chain above $\text{Fin}$ and a robust interaction with Fubini powers via $\delta$-Fubini constructions. A key bridge is that the computable GTK coincides with the LT-order on upper sets, and that a new invariant $\mathcal{D}_T(\mathcal{F})$ captures the Turing-degree spectrum of a filter $\mathcal{F}$, with nonprincipal $\Delta^1_1$ filters yielding hyperarithmetic degrees. The paper further connects these combinatorial insights to topos-theoretic subtoposes (idealised effective subtoposes) and to generalised Phoa-type phenomena, establishing a two-dimensional complexity landscape that couples combinatorial set theory with computability theory and descriptive set theory. This framework enables a uniform approach to questions about the complexity of LT-topologies, filters on $\omega$, and their interactions with Turing and hyperarithmetic degrees, suggesting a broad program of computability by majority and related notions.
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 $ω$, 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 Katětov order on filters over $ω$, 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 Katětov order is isomorphic to the original $\leq_{\mathrm{LT}}$-order. Moreover, our work brings into focus a new degree-spectrum invariant for filters $\mathcal{F}$, $$\mathcal{D}_{\mathrm{T}}(\mathcal{F}):=\{\,[f\colonω\toω] \mid f\leq_{\mathrm{LT}} \mathcal{F} \},$$ which is shown to always determine a proper initial segment of the Turing degrees. Extending this, given any $Δ^1_1$ 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.