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arxiv: 2605.14086 · v1 · pith:43G77UWRnew · submitted 2026-05-13 · 🧮 math.LO · math.CT

What can Topology tell us about Logical Complexity?

Takayuki Kihara , Ming Ng This is my paper

Pith reviewed 2026-05-15 02:17 UTC · model grok-4.3

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The pith

A computable variant of the gamified Katětov order on filters is isomorphic to the Lawvere-Tierney order, linking combinatorial complexity measures to computability in topos theory.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The Effective Topos is a mathematical setting where objects behave like computable sets and functions. Inside it, the Lawvere-Tierney order compares these objects by a kind of logical complexity. The authors previously showed that this order matches a version of the Katětov order made into a game that allows repeated Fubini-style iterations. Placing both notions inside the same topological framework makes different kinds of complexity from logic and combinatorics appear to follow the same rules. The note explores these links informally and flags some new results that will appear later.

Core claim

a computable variant of the gamified Katětov order is isomorphic to the original ≤_LT-order

Load-bearing premise

That the gamified Katětov order, closed under well-founded iterations of Fubini powers, precisely captures the combinatorial complexity shifts already present in the Lawvere-Tierney order.

read the original 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 Kat\v{e}tov order on filters over $\omega$ -- essentially, this is the usual Kat\v{e}tov 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 Kat\v{e}tov 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. This informal survey note discusses interactions between combinatorial and computable complexity, centered on the authors' prior result (arXiv:2602.08138) that a computable variant of the gamified Katětov order—filters over ω closed under well-founded iterations of Fubini powers—is isomorphic to the Lawvere-Tierney order ≤_LT in the effective topos (which effectively embeds the Turing degrees). It frames different notions of logical complexity as controlled by the same topological mechanism and announces forthcoming results.

Significance. If the asserted isomorphism holds, the note would supply a concrete combinatorial characterization of the structure of ≤_LT, linking shifts in complexity to game positions and winning strategies in a way that could resolve aspects of the longstanding open problem on the order's structure and bridge computability theory with category-theoretic topology.

major comments (1)
  1. The central claim—that the computable gamified Katětov order is isomorphic to ≤_LT—is stated in the abstract and introduction but supplies neither an explicit isomorphism (mapping game positions/strategies to subobject classifiers or topology shifts) nor a verification that the computable restriction and well-founded Fubini iteration closure preserve the effective embedding of Turing degrees; this correspondence is load-bearing for the survey's guiding perspective yet remains unstated in the manuscript.
minor comments (2)
  1. The manuscript employs informal phrasing such as 'surprising discovery' and 'opens up many challenging questions'; these could be replaced with more precise statements for a formal note.
  2. The title is broad; consider adding a subtitle referencing the effective topos and the gamified Katětov order to better reflect the specific content.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are introduced or quantified in the abstract; the note relies on background notions from effective topos theory and filter orders already present in the cited literature.

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Showing first 80 references.