关于预示模型论的一些贡献，II——往返
Some contributions to presheaf model theory, II -- back and forth
AI总结：
本文通过定义多种层次结构，展示了预示模型论中往返技术在不同条件下的统一性。
摘要：
我们在预示模型论的背景下讨论了往返技术。往返技术的本质在于展示各种校准两个模型之间相似性的层次结构，更一般地说，是两个由模型和其元组组成的对之间的关系。在本文中，我们为预示模型（及其从中选取的元组）定义了几种这样的层次结构：基于部分同构通过字面往返条件的可扩展程度，基于共享特定的抽象不变量（例如第§\ref{function_analysis}节中定义的$F^α_{M,\bar{a}}$），基于在给定数量的量词完备性范围内，对公式的实例化的真值达成一致，基于在某些埃伦费尔茨-弗拉杰斯型游戏中玩家II存在获胜策略，最后基于满足在构造斯科特句子时出现的某些无穷句子。我们最终证明了所有这些层次结构都是一致的。
英文摘要：
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 $F^α_{M,\bar{a}}$ of §\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-Fraïssé-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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
