Around definable types in $p$-adically closed fields

TL;DR

This paper advances the understanding of definable types in $p$-adically closed fields, showing how they can be represented and manipulated, with implications for definable groups, spaces, and compactness.

Contribution

It introduces new methods for representing definable types, proves their structural properties, and demonstrates how definable groups and spaces behave in this setting.

Findings

Definable types can be coded as real tuples in the field sort.

Global definable types are generated by countable unions of chains.

Definable compactness notions coincide and are definable in families.

Abstract

We prove some technical results on definable types in $p$-adically closed fields, with consequences for definable groups and definable topological spaces. First, the code of a definable $n$-type (in the field sort) can be taken to be a real tuple (in the field sort) rather than an imaginary tuple (in the geometric sorts). Second, any definable type in the real or imaginary sorts is generated by a countable union of chains parameterized by the value group. Third, if $X$ is an interpretable set, then the space of global definable types on $X$ is strictly pro-interpretable, building off work of Cubides Kovacsics, Hils, and Ye. Fourth, global definable types can be lifted (in a non-canonical way) along interpretable surjections. Fifth, if $G$ is a definable group with definable f-generics ($dfg$), and $G$ acts on a definable set $X$, then the quotient space $X/G$ is definable, not just interpretable. This explains some phenomena observed by Pillay and Yao. Lastly, we show that interpretable topological spaces satisfy analogues of first-countability and curve selection. Using this, we show that all reasonable notions of definable compactness agree on interpretable topological spaces, and that definable compactness is definable in families.