博弈论的Katětov序与理想化的效果拓扑
The Game-Theoretic Katětov Order and Idealised Effective Subtoposes
AI总结：
本文通过博弈论方法研究滤子的Katětov序，揭示其与图灵度的深刻联系，并引入新的度谱不变量。
摘要：
本文针对有效拓扑中$\leq_{\mathrm{LT}}$序结构的长期问题展开研究，已知该序能够有效嵌入图灵度。令人惊讶的是，我们发现$\leq_{\mathrm{LT}}$序实际上由$\omega$上滤子的组合学严格控制，这引发了关于组合学与可计算复杂性如何相互作用的深刻问题，不仅限于该序本身，还延伸至更广泛的领域。为了使这种联系精确化，我们引入了一种博弈论（\"gamified\"）的滤子Katětov序变体，该序表现出粗略与细致的混合特性。一方面，它严格粗于经典的Rudin-Keisler序，且在对偶视角下，所有MAD族都坍缩为一个等价类。另一方面，该序也支持丰富的内部结构，包括无限严格上升的理想类链，我们通过一种新的分离技术加以识别。从可计算性理论的角度来看，我们证明了一个可计算（并扩展）的博弈化Katětov序变体与原始$\leq_{\mathrm{LT}}$序同构。此外，我们的研究聚焦于一个新的度谱不变量，即$$\mathcal{D}_{\mathrm{T}}(\mathcal{F}):=\{\,[f\colonω\toω] \mid f\leq_{\mathrm{LT}} \mathcal{F} \},$$该不变量始终确定图灵度的一个初始段。进一步地，对于任何$Δ^1_1$滤子$\mathcal{F}$，我们证明$\mathcal{D}_{\mathrm{T}}(\mathcal{F})$恰好是超算术度。这显著扩展了van Oosten \cite{vO14}和Kihara \cite{Kih23}之前获得的结果。证明过程借鉴了通用拓扑、描述集合论和可计算性理论中的思想。
英文摘要：
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 non-principal $Δ^1_1$ 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.
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