Stability in Model Theory and Category Theory

模型理论研究可以用一阶逻辑公理(理论)来描述的数学结构(模型)。然而，许多重要的数学概念不能在一阶逻辑的范围内讨论，因此研究人员越来越有兴趣将模型理论的成功技术扩展到更一般的框架。一种这样的框架是连续逻辑，在连续逻辑中，我们通过0到1之间的某个值来衡量一项陈述的真实性，而不仅仅是将其归类为“真”或“假”。另一个例子是肯定逻辑，它没有内置的逻辑否定，它包含了连续逻辑。最后，还有可及范畴，在这里，人们用范式论的方法研究让人想起逻辑理论模型范畴的范畴。稳定性理论由谢拉在20世纪70年代创立，几十年来一直是模型理论中最深刻的部分之一，在20世纪90年代中期赫鲁晓夫斯基著名的数论马宁-芒福德和莫德尔-朗猜想的模型理论证明中达到顶峰。因此，将稳定性理论概括为正逻辑和包含更广泛类别的基本数学例子的可及范畴的背景是完全自然的。Kim-Pillay自20世纪90年代以来的工作表明，稳定性理论可以通过各种独立关系来研究，这些关系告诉我们，给定结构的哪些部分是相关的，哪些部分是无关的。最近，研究人员意识到，即使在正逻辑和可达范畴中也可以使用独立关系，稳定性理论的某些部分已被推广到这些背景下。我们的主要目标是将独立性关系的研究推广到尽可能广泛的可达范畴，包括简单范畴，以及最近新稳定性理论中感兴趣的某一类范畴。与此同时，我们将通过证明它的一个绝对版本来阐明20世纪90年代以来的稳定分叉猜想。采用一种略有不同的方法，范畴逻辑研究称为拓扑论的任意宇宙中的几何理论(包括实证理论)的模型。正因为如此，给定一个几何理论，人们就可以构造它的分类拓扑，它的“点”对应于模型，并提供该理论的某种普遍模型。我们将通过范畴逻辑的方法来研究某些实证理论，甚至一些可达范畴作为几何理论，而在这种背景下对稳定性的探索将为未来的研究开辟新的方向。我们将努力加强模型理论和绝对逻辑之间的思想交流，并在英国和国际上这两个非常强大的社区之间架起桥梁。

Model theory studies mathematical structures (models) which can be characterised by first-order logical axioms (theories). A number of important mathematical concepts, however, cannot be discussed within the confines of first-order logic, so researchers have been increasingly interested to extend successful techniques of model theory to more general frameworks. One such framework is continuous logic, where we measure the truth of a statement by some value between 0 and 1 rather than just classifying it as "true" or "false". Another example is positive logic, which does not have logical negation built in and it subsumes continuous logic. Finally, there are accessible categories, where one studies categories reminiscent of the category of models of a logical theory by methods of category theory. Stability theory, founded by Shelah in 1970s, has been one of the deepest parts of model theory for decades, culminating in Hrushovski's celebrated model-theoretic proofs of the number-theoretic Manin-Mumford and Mordell-Lang conjectures in the mid-1990s. Hence the desire to generalise stability theory to positive logic and the context of accessible categories that encompass a much wider class of fundamental mathematical examples is perfectly natural. The work of Kim-Pillay from 1990s has shown that stability theory can be studied through various independence relations that tell us which parts of a given structure are related, and which are not. Recently researchers realised that it is possible to use independence relations even in positive logic and accessible categories, and certain parts of stability theory have been generalised to those contexts. Our main goal is to generalise the study of independence relations to the widest possible class of accessible categories including the simple ones, and a certain class of interest in the recent neo-stability theory. At the same time, we will shed light on the stable forking conjecture from the 1990s, by proving a categorical version of it. Adopting a slightly different approach, categorical logic studies models of geometric theories (which includes positive theories) in arbitrary universes called toposes. Because of that, given a geometric theory, one can construct its classifying topos, whose `points' correspond to models, and which affords a certain universal model of the theory. We will study certain positive theories and even some accessible categories as geometric theories through methods of categorical logic, and the exploration of stability in that context will open novel directions for future research. We will strive to enhance the exchange of ideas between model theory and categorical logic, and to build bridges between the two very strong communities in the UK and internationally.