Structuralism, Invariance, and Univalence

S Awodey

doi:10.1093/philmat/nkt030

Abstract

The recent discovery of an interpretation of constructive type theory into abstract homotopy theory suggests a new approach to the foundations of mathematics with intrinsic geometric content and a computational implementation. Voevodsky has proposed such a program, including a new axiom with both geometric and logical significance: the Univalence Axiom. It captures the familiar aspect of informal mathematical practice according to which one can identify isomorphic objects. While it is incompatible with conventional foundations, it is a powerful addition to homotopy type theory. It also gives the new system of foundations a distinctly structural character.

Highlights

From a foundational perspective, the Univalence Axiom is certainly radical and unexpected, but it is not entirely without precedent.The first edition of Principia Mathematica used an intensional type theory, but the axiom of reducibility implied that every function has an extensionally equivalent predicative replacement
The unit interval [0, 1] is homeomorphic to the closed interval [0, 2], so topologically, these are the same space, say I. Theorem, or proof, it makes no practical difference which of two “isomorphic copies” are used, and so they can be treated as the same mathematical object for all practical purposes. This common practice is even sometimes referred to light-heartedly as “abuse of notation,” and mathematicians have developed a sort of systematic sloppiness to help them implement this principle, which is quite useful in practice, despite being literally false
That we have reduced identity of structures in general to the question of isomorphism of some “base category”, like Sets, we just need to know: what are the invariant properties of Sets? We already know that many sentences of conventional set theory express properties of sets that are not invariant, like ∅ ∈ X

Summary

The Principle of Structuralism

The following statement may be called the Principle of Structuralism: Isomorphic objects are identical. From one perspective, this captures a principle of reasoning embodied in everyday mathematical practice:. Theorem, or proof, it makes no practical difference which of two “isomorphic copies” are used, and so they can be treated as the same mathematical object for all practical purposes This common practice is even sometimes referred to light-heartedly as “abuse of notation,” and mathematicians have developed a sort of systematic sloppiness to help them implement this principle, which is quite useful in practice, despite being literally false.

Isomorphism

Objects

Identity

What does “are” mean?

Background