The real projective spaces in homotopy type theory

Homotopy type theory is a version of Martin-Löf type theory taking advantage of its homotopical models. In particular, we can use and construct objects of homotopy theory and reason about them using higher inductive types. In this article, we construct the real projective spaces, key players in homotopy theory, as certain higher inductive types in homotopy type theory. The classical definition of ℝPⁿ, as the quotient space identifying antipodal points of the n-sphere, does not translate directly to homotopy type theory. Instead, we define ℝPⁿ by induction on n simultaneously with its tautological bundle of 2-element sets. As the base case, we take ℝP^-1 to be the empty type. In the inductive step, we take ℝPⁿ⁺¹ to be the mapping cone of the projection map of the tautological bundle of ℝPⁿ, and we use its universal property and the univalence axiom to define the tautological bundle on ℝPⁿ⁺¹. By showing that the total space of the tautological bundle of ℝPⁿ is the n-sphere Sⁿ, we retrieve the classical description of ℝPⁿ⁺¹ as ℝPⁿ with an (n + 1)-disk attached to it. The infinite dimensional real projective space ℝP^∞, defined as the sequential colimit of ℝPⁿ with the canonical inclusion maps, is equivalent to the Eilenberg-MacLane space K(ℤ/2ℤ, 1), which here arises as the subtype of the universe consisting of 2-element types. Indeed, the infinite dimensional projective space classifies the 0-sphere bundles, which one can think of as synthetic line bundles. These constructions in homotopy type theory further illustrate the utility of homotopy type theory, including the interplay of type theoretic and homotopy theoretic ideas.