Two ideas underpin categorical logic: first, that a theory can be identified with its associated syntactic category; secondly, that the set-theoretic models of a theory can be identified with functors from its syntactic category to the category of sets and functions preserving suitable structure. This point of view is useful because it often helps us to establish existence of free models. After reviewing these ideas, I will present joint work with Giacomo Tendas on variant in the context of 2-dimensional category theory, in which we are interested in categories, rather than sets, equipped with additional structure (often subject to universal properties). Extra subtleties emerge, as one needs to deal with a form of quantification in between `there exists’ and `there exists a unique’. As an application, we obtain a variety of free 2-categorical constructions, including Joyal’s free bicompletions.

7 years ago, also in Oxford, Sylvy Anscombe and I asked this question, which is part of the general effort to try and understand the model theory of henselian valued fields through dividing lines. In 2024, Sylvy Anscombe and Franziska Jahnke completely classified NIP henselian valued fields. Their methods can be extended, with the help of works of Chernikov, Kaplan and Simon and of Kuhlmann and Rzepka, to NTP2 henselian valued fields, obtaining the following:

• if a henselian valued field is NTP2, then it is semitame and its residue field is NTP2;
• if a henselian valued field is separably algebraically maximal Kaplansky and its residue field is NTP2, then it is NTP2.

This covers a large class of fields, but there is still a gap. Notably, Fp((Q)) is in the middle: it is semitame but not Kaplansky.

To answer this question, we studied so called tame henselian fields with finite residue field, and derived quantifier elimination results, namely, we prove that any formula in the language of valued fields reduces to a formula of the form (∃y f(x,y)=0) ∧ φ(v(x)) ∧ ψ(res(x)), where φ and ψ are formulas in the language of ordered groups and of rings, respectively.

In Fp((Q)) specifically, the valuation ring itself is definable with a diophantine formula (ie of the form ∃y f(x,y)=0), reducing further our quantifier elimination result.

Finally, a large chunk of these formulas are known to be NTP2: when f(x,y) is additive in y, the formula ∃y f(x,y)=z is NTP2 (with respect to x and z). Unfortunately, that does not cover all formulas, so the answer to the titular question is still unknown.

Non-trivial henselian valuations are often so closely related to the arithmetic of the underlying field that they are encoded in it, i.e., that their valuation ring is first-order definable in the language of rings. In this talk, I will survey and present old and new results around the definability of henselian valuations, also with a view towards parameters and uniformity of definitions.

A field K is "large" if every smooth curve over K with at least one K-rational point has infinitely many K-rational points. In this talk, I'll discuss what we know about the relations between the arithmetic condition of largeness and the model-theoretic conditions of stability and NIP. Stable large fields are separably closed. For NIP large fields, we know something much weaker: there is a canonical field topology satisfying a weak form of the implicit function theorem for polynomials. Conjecturally, any stable or NIP infinite field should be large. I will discuss these results, as well as the following conjecture: if K is a field and p is a prime and every separable extension of K has degree prime to p, then K is large. This conjecture would imply that NIP fields of positive characteristic are large, and would classify stable fields of positive characteristic. I will present some (very weak) evidence for this conjecture.