Homotopy Type Theory The real projective spaces in homotopy type theory > history (changes)

arxiv

Abstract

Homotopy type theory is a version of Martin-Lof 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 $\mathbb{R}P^n$, as the quotient space identifying antipodal points of the $n$-sphere, does not translate directly to homotopy type theory. Instead, we define $\mathbb{R}P^n$ by induction? on $n$ simultaneously with its tautological bundle of $2$-element sets. As the base case, we take $\mathbb{R}P^{-1}$ to be the empty type. In the inductive step, we take $\mathbb{R}P^{n+1}$ to be the mapping cone? of the projection map of the tautological bundle? of $\mathbb{R}P^{n}$, and we use its universal property and the univalence axiom to define the tautological bundle? on $\mathbb{R}P^{n+1}$. By showing that the total space of the tautological bundle? of $\mathbb{R}P^n$ is the n-sphere?, we retrieve the classical description of $\mathbb{R}P^{n+1}$ as $\mathbb{R}P(n)$ with an $(n+1)$-cell attached to it. The infinite dimensional real projective space?, defined as the sequential colimit? of the $\mathbb{R}P^n$ with the canonical inclusion maps, is equivalent to the Eilenberg-MacLane space $K(\mathbb{Z}/2\mathbb{Z},1)$, 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.