Structuralism, Invariance, and Univalence

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. Recent advances in foundations of mathematics have led to some developments that are significant for the philosophy of mathematics, particularly structuralism. Specifically, the discovery of an interpretation of Martin-Löf’s constructive type theory into abstract homotopy theory [3] suggests a new approach to the foundations of mathematics, one with both intrinsic geometric content and a computational implementation [5]. Leading homotopy theorist Vladimir Voevodsky has proposed an ambitious new program of foundations on this basis, including a new axiom with both geometric and logical significance: the Univalence Axiom [4]. 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 the framework of homotopical type theory. ∗Thanks to Peter Aczel for many discussions on the subject of this paper, and to the Munich Center for Mathematical Philosophy, where this work was done and presented.