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Q: What is the univalent principle in math?

The univalent principle, also known as the univalence axiom, is a foundational principle in mathematics that is associated with the field of type theory, particularly in the context of homotopy type theory.

Homotopy type theory is a branch of mathematics that seeks to establish a connection between type theory, which is a formal system for reasoning about types and their relationships, and homotopy theory, which deals with the study of continuous deformations of spaces.

The univalence axiom, introduced by Vladimir Voevodsky, is a principle that asserts a fundamental relationship between the notions of equivalence and identity in type theory. It states that two types are equivalent if and only if they are isomorphic in a suitable sense.

More specifically, the univalence axiom states that for any two types A and B, there exists an equivalence between the type A = B (the type of identifications or paths between A and B) and the type A ≃ B (the type of equivalences or isomorphisms between A and B). This equivalence is often denoted as “ua” or “iso”, and it allows for the interchange of equivalent types within the system.

The univalence axiom has far-reaching consequences and has been instrumental in the development of homotopy type theory. It provides a powerful tool for reasoning about types and enables connections between different areas of mathematics, including algebraic topology, category theory, and logic.

It is important to note that the univalence axiom is a subject of ongoing research and discussion within the field of homotopy type theory. Different variations and formulations of the axiom have been proposed, and its implications and relationship to other foundational principles continue to be explored.