What is a Higher Level Set?

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Tsementzis, Dimitris
preprint description
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.