What can Topology tell us about Logical Complexity?

Abstract: In the 1980s, category theorists introduced the Lawvere-Tierney $(\leq_{\mathrm{LT}})$ order in the Effective Topos, known to effectively embed the Turing degrees. Understanding its structure is a longstanding open problem in the area. In particular, there was an informal sense that the $\leq_{\mathrm{LT}}$-order reflects certain shifts in combinatorial complexity, but a precise characterisation remained elusive for some time. Recent work by the authors has substantially clarified the picture. In arXiv:2602.08138, the authors introduced a game-theoretic (''gamified'') version of the Katětov order on filters over $ω$ -- essentially, this is the usual Katětov order now closed under well-founded iterations of Fubini powers. The first major theorem of the paper was to show that a computable variant of the gamified Katětov order is isomorphic to the original $\leq_{\mathrm{LT}}$-order. This was a surprising discovery, and opens up many challenging questions regarding the interplay between combinatorial and computable complexity, which informed the rest of the paper's investigations. This note gives an informal survey of some of these interactions explored in arXiv:2602.08138, and announces some forthcoming results. The guiding perspective is that different notions of complexity arising in different areas of logic can be seen to be controlled by the same mechanism -- once placed in the right topological framework.

• The paper demonstrates that a combinatorial, game-theoretic variant of the Katětov order is isomorphic to the LT order on basic topologies in Eff.
• The methodology leverages computable iterations of Fubini powers to reveal structural properties of logical complexity using effective categorization.
• The findings bridge topology and computability theory by establishing invariants such as Turing degree profiles for non-principal filters.

Topological Characterizations of Logical Complexity in the Effective Topos

Introduction

The paper "What can Topology tell us about Logical Complexity?" (2605.14086) explores the interplay between topology, category theory, and logical complexity, specifically through the structure of Lawvere-Tierney (LT) topologies in the Effective Topos ($Eff$). The central contribution is the demonstration that a combinatorial, game-theoretic variant of the Katětov order—when properly iterated and restricted to computable settings—yields an isomorphism with the Lawvere-Tierney order, which was originally developed to encode computability-theoretic complexity within categorical frameworks.

Lawvere-Tierney Topologies and Logical Complexity

LT topologies in $Eff$ abstract the notion of "localness" from classical topology into topos-theoretic and computability-theoretic contexts. An LT topology is an endomorphism $j: \mathcal{P}(\omega) \to \mathcal{P}(\omega)$ satisfying properties analogous to sheaf-theoretic gluing and preservation of logical implication under realizability. The LT order ($\leq_{LT}$) classifies these operators based on their logical strength: $j \leq_{LT} k$ iff $j(p) \to k(p)$ holds for all $p$, using the realizability interpretation.

It is known that the Turing degrees embed effectively in the LT order in $Eff$, revealing that this order encodes computability-theoretic data about logical constructs. Past research characterized only the minimal and maximal classes; the internal structure of the order remained unclear.

Gamified Katětov Order and Its Isomorphism

A pivotal theorem in the paper is the identification of a computable variant of the gamified Katětov order with the LT order. The classical Katětov order organizes filters and ideals over $\omega$ via combinatorial reducibility notions (namely, existence of functions mapping sets in one filter to another). The gamified Katětov order, introduced and developed in [KiNg26], augments the classical order by closing under well-founded iterations of Fubini powers, and by providing a game-theoretic operational description.

Major theorem: The computable gamified Katětov order on upper sets over $\omega$ is isomorphic to the LT order on basic topologies in the Effective Topos. This result establishes a direct, explicit combinatorial mechanism underlying logical complexity as measured in $Eff$0.

Structural Properties and Hierarchies

The paper rigorously analyzes the comparative structure of the gamified Katětov order relative to the Rudin-Keisler and Tukey orders. It is demonstrated that the gamified Katětov order is strictly coarser than both the Rudin-Keisler and Katětov orders and that it is fundamentally misaligned with the Tukey order—contradicting any presumption that coarseness corresponds to cofinality types. Specifically, the Tukey order and gamified Katětov order are shown to be incomparable on filters over $Eff$1 within ZFC.

A salient claim is that while classical Katětov classes distinguish a vast number of equivalence classes among maximal almost disjoint (MAD) families, the gamified Katětov order collapses all MAD families to a single equivalence class, yet still admits infinite strictly ascending chains and large antichains (including an embedding of $Eff$2 with chains of length $Eff$3 and antichains of size $Eff$4 [KiNg-nonlinear]).

Combinatorial and Computability-Theoretic Invariants

The unified framework enables a comparison of combinatorial and computability notions of complexity. Every filter $Eff$5 yields an initial segment of Turing degrees, introducing a new invariant—the Turing degree profile of $Eff$6. The analysis shows that:

• Every Turing degree $Eff$7 appears in the profile of some summable ideal’s dual filter.
• No single filter captures all Turing degrees.
• For non-principal $Eff$8-filters, the Turing degree profile coincides precisely with hyperarithmetic degrees.

These results generalize previous findings and establish a concrete methodology for interrogating the intersection of set-theoretic and computability-theoretic hierarchies via categorical/topological constructs.

Implications and Future Directions

This work constitutes a categorical unification of classical and computable hierarchies of logical complexity. The explicit isomorphism with gamified Katětov order provides a combinatorial toolkit for analyzing internal structure, separation of classes, and invariant relationships between filters, ideals, and Turing degrees.

From a theoretical standpoint, the research prompts further investigation into:

• The full classification of LT equivalence classes beyond basic and recursively joined topologies, leveraging game-theoretic separations.
• The categoricity and universality properties inherent in different orders, and their relation to categorical equivalence in logic.
• The translation and generalization of results to other realizability and arithmetic universes.

Practically, the insights foster new approaches to uniformity problems in computability, reverse mathematics, and model theory, and suggest potential applications in the categorification of learning-theoretic and complexity-theoretic constructions.

Conclusion

The paper achieves an explicit combinatorial characterization of Lawvere-Tierney logical complexity in the Effective Topos, connecting category-theoretic and set-theoretic hierarchies via the gamified Katětov order. The results clarify previously opaque structural features of logical complexity orders and establish a foundation for further studies into the interplay between topology, logic, and computability (2605.14086).