Constructing the propositional truncation using non-recursive HITs

Doorn, Floris van

doi:10.1145/2854065.2854076

Cited by 20 publications

(18 citation statements)

References 7 publications

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“…This means we have a function |−| : A → A , precomposition with which gives an equivalence ( A → B) (A → B) for any mere proposition B. There are several constructions of the propositional truncation in terms of pushouts and colimits due to Van Doorn [3], Kraus [7], and the second-named author [9].…”

Section: Introductionmentioning

confidence: 99%

The real projective spaces in homotopy type theory

Buchholtz

Rijke

2017

2017 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)

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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.

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Section: Introductionmentioning

confidence: 99%

The real projective spaces in homotopy type theory

Buchholtz

Rijke

2017

2017 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)

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“…As for (3), if A and P are sets we can consider it justified by UFP 2013, Theorem 10.1.5 (surjections of sets are regular epimorphisms). The general case requires a more refined notion of "quotient map", but is also true when suitably formulated (see Kraus 2016;Rijke 2017;van Doorn 2016).…”

Section: The ♯ Modality and Codiscretenessmentioning

confidence: 99%

Brouwer's fixed-point theorem in real-cohesive homotopy type theory

Shulman¹

2017

Math. Struct. Comp. Sci.

149

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We combine Homotopy Type Theory with axiomatic cohesion, expressing the latter internally with a version of "adjoint logic" in which the discretization and codiscretization modalities are characterized using a judgmental formalism of "crisp variables". This yields type theories that we call "spatial" and "cohesive", in which the types can be viewed as having independent topological and homotopical structure. These type theories can then be used to study formally the process by which topology gives rise to homotopy theory (the "fundamental ∞-groupoid" or "shape"), disentangling the "identifications" of Homotopy Type Theory from the "continuous paths" of topology. In a further refinement called "real-cohesion", the shape is determined by continuous maps from the real numbers, as in classical algebraic topology. This enables us to reproduce formally some of the classical applications of homotopy theory to topology. As an example, we prove Brouwer's fixed-point theorem.

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“…Most of the higher inductive types that are commonly used can be constructed just from pushouts and the other con-structions in MLTT with univalent universes. 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].…”

Section: Bmentioning

confidence: 99%

Higher Structures in Homotopy Type Theory

Buchholtz

2019

Synthese Library

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The intended model of the homotopy type theories used in Univalent Foundations is the ∞-category of homotopy types, also known as ∞-groupoids. The problem of higher structures is that of constructing the homotopy types needed for mathematics, especially those that aren't sets. The current repertoire of constructions, including the usual type formers and higher inductive types, suffice for many but not all of these. We discuss the problematic cases, typically those involving an infinite hierarchy of coherence data such as semi-simplicial types, as well as the problem of developing the meta-theory of homotopy type theories in Univalent Foundations. We also discuss some proposed solutions.

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Constructing the propositional truncation using non-recursive HITs

Cited by 20 publications

References 7 publications

The real projective spaces in homotopy type theory

The real projective spaces in homotopy type theory

Brouwer's fixed-point theorem in real-cohesive homotopy type theory

Higher Structures in Homotopy Type Theory

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