Structuralist foundations of mathematics aim for an “invariant” conception of mathematics. But what should be their basic objects? Two leading answers emerge: higher groupoids or higher categories. In this paper I argue in favor of the former over the latter. First, I explain why to pick between them we need to ask the question of what is the correct “categorified” version of a set. Second, I argue in favor of groupoids over categories as “categorified” sets by introducing a pre-formal understanding of groupoids as abstract shapes. This conclusion lends further support to the perspective taken by the Univalent Foundations of mathematics.

中文翻译：

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什么是高级集？

数学的结构主义基础旨在建立一个“不变”的数学概念。但是它们的基本对象应该是什么？出现了两个主要答案：更高的 groupoids 或更高的类别。在本文中，我赞成前者而不是后者。首先，我解释为什么要在它们之间进行选择，我们需要问一个问题，什么是集合的正确“分类”版本。其次，我通过引入对 groupoids 作为抽象形状的预先形式的理解，支持 groupoids 而不是作为“分类”集合的类别。这一结论进一步支持了数学一元基础所采取的观点。