Some contributions to presheaf model theory, II -- back and forth

TL;DR

This paper explores the back and forth technique in presheaf model theory, establishing the equivalence of various hierarchies that measure model similarity through different criteria.

Contribution

It introduces and compares multiple hierarchies for presheaf models based on extendibility, invariants, formula agreement, game strategies, and Scott sentences, showing their alignment.

Findings

All hierarchies are shown to be equivalent.

The paper formalizes new invariants for presheaf models.

It connects model-theoretic techniques with presheaf structures.

Abstract

We discuss the back and forth technique in the context of presheaf model theory. The essence of the back and forth technique lies in showing the relationship between various hierarchies which calibrate similarity between two models and, more generally, between two pairs consisting of a model and a tuple from it. In this paper we define several such hierarchies for presheaf models (and tuples of sections from them): those based on the degree of extendibility of partial isomorphisms through literal back and forth conditions, on sharing specific, abstract invariants which we define (the $F^\alpha_{M,\bar{a}}$ of \S{}\ref{function_analysis} for example), on agreeing on the (truth) values of instantiations of formulae up to a given amount of quantifier completity, on the existence of winning strategies for player II in certain Ehrenfeucht-Fra\"iss\'e-type games and, finally, on satisfying certain infinitary sentences that arise in the construction of Scott sentences. We ultimately show that all of these hierarchies align.