Thickness, and a categoric view of type-space functors
Thickness, and a categoric view of type-space functors
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厚度和类型空间函子的分类视图
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发表时间：
2003
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通讯作者：
Itay Ben
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作者：
Itay Ben
我们定义了厚猫类（紧抽象理论，其中特别包含半Hausdor、Hausdor和一阶猫），并证明了在这类中简单性的行为与一阶理论相同.我们考虑众所周知的一阶概念，如可解释性或稳定的划分/约简，并提出类似的概念，可以自然地表示在类型空间函子之间的映射。我们证明了几个理想的新概念的性质，并显示它们之间的连接和他们的经典同行。最后，我们得出了几个关于猫和简单性的零散结果。导论.在（Ben 03）中，我们提出了猫（或紧抽象理论），它可以被看作是一个具有紧性但没有否定性的模型论框架。它比一阶框架更一般，并且可以容纳，例如，不允许一阶描述的各种分析结构。事实上，我们展示了几种否定猫的方法的等价性，每种方法都有自己的优点。猫的第一个和最具体的表示是通过一种特殊的普遍域，这种域在没有否定的语言中是同质的和紧凑的。类型、划分等，或多或少都和平常一样存在于一个宇宙的范围内。在（Benb）中，我们展示了在这种情况下，简单性是如何从非划分具有局部特征的假设中被分离出来的，尽管这并不意味着非划分扩张总是存在的。在本文的第一部分中，我们引入了厚猫类，即，猫的不可识别性是不可分类型的。我们表明，厚度的假设是非常温和的，并与此假设许多
We dene the class of thick cats (compact abstract theories, which con- tains in particular semi-Hausdor, Hausdor and rst order cats), and prove that in this class simplicity behaves as in rst order theories. We consider well-known rst order no- tions, such as interpretability or stable dividing/reduct, and propose analogous notions that can be naturally expressed in terms of maps between type-space functors. We prove several desirable properties of the new notions and show the connection between them and their classical counterparts. We conclude with several scattered results concerning cats and simplicity. Introduction. In (Ben03) we dened cats (or compact abstract theo- ries), which may be viewed as a model-theoretic framework with compact- ness but without negation. It is more general than the rst order framework, and can accommodate, for example, various kinds of analytic structures that do not admit a rst order description. In fact, we showed the equivalence of several quite dieren t approaches to the denition of cats, each having its own merits. The rst and most concrete presentation of a cat is via a particular kind of universal domains which are homogeneous and compact in a language without negation. Types, dividing, etc., are dened more or less as usual inside a universal domain. In (Benb) we showed how simplicity can be de- veloped in this context from the assumption that non-dividing has the local character, even though this does not imply that non-dividing extensions always exist. In the rst section of the present paper we introduce the class of thick cats, i.e., cats where indiscernibility is type-denable. We show that the thickness assumption is very mild, and that with this assumption many