What is a Higher-Level Set?

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. 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.

Abstraction, equality and univalence

“…We could say that constructive abstraction 'forgets' inessential information in a controlled manner, that is, in such a way that the abstraction data can be fully recovered from (or faithfully encoded in) the abstracta. 26 From a philosophical perspective, some authors-notably Awodey [20] and Tsementzis [8,73]-have analysed UF in the light provided by a family of interrelated trends in philosophy of mathematics enveloped by the term mathematical structuralism (and mainly developed, in its different eliminativist and non-eliminativist variants, by Bourbaki, Benacerraf, Putnam, Resnik, Shapiro, Hellman and Parsons among others; see [74] and references therein). The idiosyncratic presentation of UF that we have proposed here is intended to consider UF from the standpoint provided by an alternative (and maybe complementary 27 ) conceptual framework that stresses above all the constructivist elan of UF.…”

“…However, and differently from a mere proposition, a proposition might have different inequivalent proofs. 8 Given two proofs p, q : a = X b of the fact that a and b are equal (that is, two identifications between a and b), we can use the fact that a = X b is itself a type to iterate the formation of equality types. This means that we can form the higher equality type p = (a= X b) q, which might or not be inhabited.…”

“…I clearly recall the feeling of a breakthrough that I experienced when I understood that this idea is wrong. Categories are not ‘sets in the next dimension.’ They are ‘partially ordered sets in the next dimension’ and ‘sets in the next dimension’ are groupoids’ [6] (see also ([7], Ex.9.1.14, p.310) and [8]). Briefly, whereas a partial order set is a set X endowed with a reflexive, transitive and antisymmetric relation, a category can be understood as a groupoid of objects (considered up to equivalence of categories) endowed with a supplementary structure that assigns to any pair of objects x and y a set Homfalse(x,yfalse) such that this assignment satisfies reflexivity (existence of trivial identities), transitivity (existence of compositions) and a kind of antisymmetry condition that identifies the invertible elements of Homfalse(x,yfalse) with the isomorphisms in the underlying groupoid.…”

“…Heyting's 'intuitionist' logic), such as the rejection of both the axiom of choice (AC) and the law of the excluded middle (LEM) (see for instance the discussion in [7], pp. [8][9][10][11]. The stratification of types in h-levels opens new possibilities for both the AC and the LEM.…”

A meaning explanation for HoTT

“…For some of the earlier writing that led to the ideas for UF see [35][36][37]. For a discussion of the sense in which UF is a foundation of mathematics and connections to mathematical structuralism, see [4,32,33]. For some philosophical issues associated to HoTT see [11,12,19,28].…”