Naturality and Definability III

TL;DR

This paper explores the relationship between naturality in category theory and definability in model theory, presenting new results on their interaction, model construction, and definability without parameters.

Contribution

It proves that naturality implies definability under certain conditions, constructs models where all uniformisable constructions are weakly natural, and shows natural constructions are parameter-free definable.

Findings

Naturality implies definability under mild conditions.

Constructed a model where all uniformisable constructions are weakly natural.

Natural constructions represented by formulas are parameter-free definable.

Abstract

In this paper, we deal with the notions of naturality from category theory and definablity from model theory and their interactions. In this regard, we present three results. First, we show, under some mild conditions, that naturality implies definablity. Second, by using the reverse Easton iteration of Cohen forcing notions, we construct a transitive model of ZFC in which every uniformisable construction is weakly natural. Finally, we show that if F is a natural construction on a class K of structures which is represented by some formula, then it is uniformly definable without any extra parameters. Our results answer some questions by Hodges and Shelah.