The real projective spaces in homotopy type theory, Ulrik Buchholtz and Egbert Rijke, LICS 2017
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 , as the quotient space identifying antipodal points of the -sphere, does not translate directly to homotopy type theory. Instead, we define by induction? on simultaneously with its tautological bundle of -element sets. As the base case, we take to be the empty type. In the inductive step, we take to be the mapping cone? of the projection map of the tautological bundle? of , and we use its universal property and the univalence axiom to define the tautological bundle? on . By showing that the total space of the tautological bundle? of is the n-sphere?, we retrieve the classical description of as with an -cell attached to it. The infinite dimensional real projective space?, defined as the sequential colimit? of the with the canonical inclusion maps, is equivalent to the Eilenberg-MacLane space , which here arises as the subtype of the universe consisting of -element types. Indeed, the infinite dimensional projective space? classifies the -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.
Revision on October 14, 2018 at 17:12:31 by Ali Caglayan. See the history of this page for a list of all contributions to it.