What is a Higher Level Set?

Philosophia Mathematica:nkw032 (2016)
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Abstract

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. I argue in favor of the former over the latter. First, I explain why to choose 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.

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A meaning explanation for HoTT.Dimitris Tsementzis - 2020 - Synthese 197 (2):651-680.

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What numbers could not be.Paul Benacerraf - 1965 - Philosophical Review 74 (1):47-73.
Abstract.[author unknown] - 2011 - Dialogue and Universalism 21 (4):447-449.
What Scientific Theories Could Not Be.Hans Halvorson - 2012 - Philosophy of Science 79 (2):183-206.

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