Expressing ‘the structure of’ in homotopy type theory

Abstract

As a new foundational language for mathematics with its very different idea as to the status of logic, we should expect homotopy type theory to shed new light on some of the problems of philosophy which have been treated by logic. In this article, definite description, and in particular its employment within mathematics, is formulated within the type theory. Homotopy type theory has been proposed as an inherently structuralist foundational language for mathematics. Using the new formulation of definite descriptions, opportunities to express 'the structure of' within homotopy type theory are explored, and it is shown there is little or no need for this expression.

Homotopy type theory: the logic of space

“…Finally, in §5 we will mention some (∞, 1)-categories that combine topological axioms with univalence, thus satisfying neither LEM nor UIP. 34 To a homotopy theorist or higher category theorist, assuming univalence instead of UIP is obviously the right move; but it can be a difficult step for those used to thinking of types as sets. However, univalence can also be motivated from purely type-theoretic considerations, as giving a "correct" answer to the question "what are the identity types of a universe?…”

“…", just as function extensionality answers "what are the identity types of a function type?". And from a philosophical point of view, univalence says that all properties of types are invariant under equivalence, since we can make any equivalence into an equality and apply transport; thus it expresses a strong "structural" nature of type theory [7,34], in contrast to ZFC-style set theory.…”