The real projective spaces in homotopy type theory

Abstract

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 RP n , as the quotient space identifying antipodal points of the n-sphere, does not translate directly to homotopy type theory. Instead, we define RP n 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 RP n+1 to be the mapping cone of the projection map of the tautological bundle of RP n , and we use its universal property and the univalence axiom to define the tautological bundle on RP n+1 .By showing that the total space of the tautological bundle of RP n is the n-sphere S n , we retrieve the classical description of RP n+1 as RP n with an (n + 1)-disk attached to it. The infinite dimensional real projective space RP ∞ , defined as the sequential colimit of RP n 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.

Quotient Inductive-Inductive Types

“…Despite a lot of work making use of concrete HITs [4,[9][10][11]23,26,27], and despite the fact that it is usually -on some intuitive level -clear for the expert how the elimination principle for such a HIT can be derived, giving a general specification and a theoretical foundation for HITs has turned out to be a major difficulty. Several approaches have been proposed [6,18,28,33], and they do indeed give a satisfactory specification of HITs in the sense that they cover all HITs which have been used so far (see related work below).…”

Symmetries in reversible programming: from symmetric rig groupoids to reversible programming languages

“…Using BAut, we can talk about types that are equivalent to a finite set of specified cardinality, for example, the subuniverse of 2-element sets is given by BAut(2). This has been used to construct the real projective spaces in HoTT [Buchholtz and Rijke 2017], and also to give sound and complete denotational semantics to a 1-bit fragment of Π [Carette, Chen, Choudhury, and Sabry 2018].…”

Higher Structures in Homotopy Type Theory

“…These include joins and suspensions (and therefore, spheres), cofibers (and thus smash products), sequential colimits, the propositional truncation [22,31] and all the higher truncations [43], set quotients, and in fact, all non-recursive HITs specified using point-, 1-, and 2-constructors by a construction due to van Doorn [24]. We also get cell complexes [15], Eilenberg-MacLane spaces [33], and projective spaces [17], and so a lot of algebraic topology can be developed on this basis, and even a theory of ∞-groups [14] and spectra (and thus homology and cohomology theory), culminating in a proof that π 4 (S 3 ) = Z/2Z [13], and a formalized proof of the Serre spectral sequence for cohomology [23].…”