The Game-Theoretic Katětov Order and Idealised Effective Subtoposes
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866918327869243392 |
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| author | Kihara, Takayuki Ng, Ming |
| author_facet | Kihara, Takayuki Ng, Ming |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_08138 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Game-Theoretic Katětov Order and Idealised Effective Subtoposes Kihara, Takayuki Ng, Ming Logic Category Theory 03E05, 03E75, 03E15, 03D28, 18B25 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. |
| title | The Game-Theoretic Katětov Order and Idealised Effective Subtoposes |
| topic | Logic Category Theory 03E05, 03E75, 03E15, 03D28, 18B25 |
| url | https://arxiv.org/abs/2602.08138 |