Accessible categories admit a purely category-theoretic replacement for cardinality: the internal size. Generalizing results and methods from [LRV19b], we examine set-theoretic problems related to internal sizes and prove several Löwenheim–Skolem theorems for accessible categories. For example, assuming the singular cardinal hypothesis, we show that a large accessible category has an object in all internal sizes of high-enough co-finality. We also prove that accessible categories with directed colimits have filtrations: any object of sufficiently high internal size is (the retract of) a colimit of a chain of strictly smaller objects.

中文翻译：

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可访问类别中的尺寸和过滤

可访问类别承认基数的纯粹类别理论替代：内部大小。概括来自 [LRV19b] 的结果和方法，我们研究了与内部大小相关的集合论问题，并证明了可访问类别的几个 Löwenheim-Skolem 定理。例如，假设单一基数假设，我们表明一个大的可访问类别具有一个具有足够高共同终结性的所有内部大小的对象。我们还证明了具有定向 colimits 的可访问类别具有过滤器：任何具有足够高内部大小的对象都是（缩回）一系列严格较小对象的 colimit。