Ming Ng (Université d’Artois) – What is a space, and why should a logician care?

The logician is familiar with the idea that filters are generalised points. Topos theorists push this idea further by asking: what is a generalised space?

This talk will focus on one such answer: LT-Topologies ("Lawvere-Tierney Topologies"). In the 1980s, Hyland introduced the Effective Topos, and showed that the Turing Degrees embed into its poset of LT-Topologies, what we call the Lawvere-Tierney order (LT-order). Progress was slow until 2013, where Lee-van Oosten showed that this topological framework reveals a meaningful interaction between combinatorics and computable complexity.

In joint work with T. Kihara, we advance this perspective considerably by introducing a game-theoretic version of Katetov's Order. Essentially, this is the usual Katetov order now closed under well-founded iterations of Fubini powers. The first major theorem of our recent paper (arXiv:2602.08138) was to show that a computable variant of the gamified Katetov order is isomorphic to the original 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 talk will discuss some of these interactions.