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Structuralism, Invariance, and Univalence

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journal contribution
posted on 04.03.2013, 00:00 authored by Steven AwodeySteven Awodey

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.


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Copyright The Author [2013]. Published by Oxford University Press. The version of record is available online at



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