Modalities in homotopy type theory

Rijke, Egbert; Shulman, Michael; Spitters, Bas

doi:10.48550/arxiv.1706.07526

Cited by 16 publications

(65 citation statements)

References 23 publications

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“…In Section 4, we introduce and study L-connected maps (Definition 4.1). Our main result in this section is the following theorem, which is proven, in a different flavor and in HoTT, in [RSS17].…”

Section: Introductionmentioning

confidence: 89%

“…In Section 3.2, we show that the class of L-local maps form a local class of maps (Proposition 3.12), thus admitting a univalent classifying map (Theorem 3.15). We can use this observation to link reflective subfibrations on E to reflective subuniverses in homotopy type theory, as given in [CORS18] and in [RSS17].…”

Section: Reflective Subfibrations and Classifying Mapsmentioning

confidence: 99%

“…In Section 3, we take from [RSS17] the definition of reflective subfibrations on an ∞-topos E and explore some of their properties. A reflective subfibration L • on E is a pullback-compatible assignment of reflective subcategories D X ⊆ E /X , with induced localization functor L X , for every X ∈ E. The collection of all objects in D X , as X varies in E, form the L-local maps, and the objects of D := D 1 are called L-local objects.…”

Section: Introductionmentioning

confidence: 99%

“…Our take on localization theory is inspired by homotopy type theory, a dependent type theory with homotopy-theoretic features ([UF13]). More precisely, the definition of reflective subfibration on an ∞-topos E appears in [RSS17] as the external notion (in higher topos theory, HTT) that captures the internal description (in homotopy type theory, HoTT) of a reflective subuniverse. The work in [RSS17] is mainly concerned Σ-closed reflective subuniverses, also known as modalities.…”

Section: Introductionmentioning

confidence: 99%

“…More precisely, the definition of reflective subfibration on an ∞-topos E appears in [RSS17] as the external notion (in higher topos theory, HTT) that captures the internal description (in homotopy type theory, HoTT) of a reflective subuniverse. The work in [RSS17] is mainly concerned Σ-closed reflective subuniverses, also known as modalities. These correspond to reflective subfibrations L • for which the composite of two L-local maps is again an L-local map.…”

Section: Introductionmentioning

confidence: 99%

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Localization theory in an $\infty$-topos

Vergura¹

2019

Preprint

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Inspired by [CORS18], we develop the theory of reflective subfibrations on an ∞-topos E. A reflective subfibration L• on E is a pullbackcompatible assignment of a reflective subcategory D X ⊆ E /X , for every X ∈ E. Reflective subfibrations abound in homotopy theory, albeit often disguised, e.g., as stable factorization systems. We prove that L-local maps (i.e., those maps that belong to some D X ) admit a classifying map, and we introduce Lseparated maps, that is, those maps with L-local diagonal. L-separated maps are the local class of maps for a reflective subfibration L ′• on E. We prove this fact in the companion paper [Ver]. In this paper, we investigate some interactions between L• and L ′• and explain when the two reflective subfibrations coincide.• 7.1. Further interactions between L • and L ′ • 7.2. Self-separated reflective subfibrations References

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Section: Introductionmentioning

confidence: 89%

Section: Reflective Subfibrations and Classifying Mapsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Localization theory in an $\infty$-topos

Vergura¹

2019

Preprint

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Fitch-Style Modal Lambda Calculi

Clouston

2018

Lecture Notes in Computer Science

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Fitch-style modal deduction, in which modalities are eliminated by opening a subordinate proof, and introduced by shutting one, were investigated in the 1990s as a basis for lambda calculi. We show that such calculi have good computational properties for a variety of intuitionistic modal logics. Semantics are given in cartesian closed categories equipped with an adjunction of endofunctors, with the necessity modality interpreted by the right adjoint. Where this functor is an idempotent comonad, a coherence result on the semantics allows us to present a calculus for intuitionistic S4 that is simpler than others in the literature. We show the calculi can be extendedà la tense logic with the left adjoint of necessity, and are then complete for the categorical semantics.

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Cartan Geometry in Modal Homotopy Type Theory

Felix¹

2018

Preprint

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In this article, some Differential Geometry is developed synthetically in a Modal Homotopy Type Theory. While Homotopy Type Theory is used to reason about general ∞-toposes, the "Modal" extension we are using here, is concerned with special ∞-toposes with the extra structure of an idempotent monad with some additional properties. On the type theory side, the extension is realized by adding well known axioms of a monadic modality.In the applications we have in mind, this monadic modality corresponds to monads exhibiting infinitesimal information of the objects of special ∞toposes of spaces. There are two main lines of examples of these toposes, one containing smooth manifolds, the other algebraic varieties. Since we use Homotopy Type Theory, stacks from both of these worlds are naturally included in our discussion. We will make use of this in developing some new higher differential geometry. Much of the higher differential geometry in this article is aimed at making our construction of moduli spaces of Gstructures and torsionfree G-structures possible in a useful way. As a basic example, this abstract construction of moduli spaces may be instantiated for a topos containing manifolds and the orthogonal group to give a construction of the moduli stack of Riemannian Metrics on a smooth manifold. The G-structures are developed along the lines of Urs Schreiber's Higher Cartan Geometry.

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Modalities in homotopy type theory

Cited by 16 publications

References 23 publications

Localization theory in an $\infty$-topos

Localization theory in an $\infty$-topos

Fitch-Style Modal Lambda Calculi

Cartan Geometry in Modal Homotopy Type Theory

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