Good Fibrations through the Modal Prism

Myers, David Jaz

doi:10.48550/arxiv.1908.08034

Cited by 3 publications

(28 citation statements)

References 5 publications

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“…We may also define the truncated shape modalities S n to have as modal types the types which are both n-truncated and S-modal. It is not known whether S n X = S X n for general X, but it is true for crisp X (see Proposition 4.5 of [17]).…”

Section: Preliminariesmentioning

confidence: 99%

“…We may single out the covering maps as the S 1 -étale maps whose fibers are sets. For more on this point of view, see the last section of [17]. Here, however, we will be more concerned with S-étale maps, which we will call ∞-covers.…”

Section: The Universal ∞-Cover Of a Higher Groupmentioning

confidence: 99%

“…We call a 1-cover just a cover, or a covering map. Theorem 6.1 of [17] gives a useful way for proving that a map is an ∞-cover. Proposition 2.2.3 (Theorem 6.1 of [17]).…”

Section: The Universal ∞-Cover Of a Higher Groupmentioning

confidence: 99%

“…Theorem 6.1 of [17] gives a useful way for proving that a map is an ∞-cover. Proposition 2.2.3 (Theorem 6.1 of [17]). Let π : E → B and suppose that there is a crisp, discrete type F so that for all b : B, fib π (b) = F .…”

Section: The Universal ∞-Cover Of a Higher Groupmentioning

confidence: 99%

“…We will define the notions of universal ∞-cover and of infinitesimal remainder in Sections 2.2 and 2.3 respectively. Our proof of this theorem will make extensive use of the theory of modalities in homotopy type theory developed by Rijke, Shulman, and Spitters [19], as well as the theory of modal étale maps developed in [5] and the theory of modal fibrations developed in [17].…”

Section: Introductionmentioning

confidence: 99%

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Modal Fracture of Higher Groups

Myers¹

2021

Preprint

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In this paper, we examine the modal aspects of higher groups in Shulman's Cohesive Homotopy Type Theory. We show that every higher group sits within a modal fracture hexagon which renders it into its discrete, infinitesimal, and contractible components. This gives an unstable and synthetic construction of Schreiber's differential cohomology hexagon. As an example of this modal fracture hexagon, we recover the character diagram characterizing ordinary differential cohomology by its relation to its underlying integral cohomology and differential form data, although there is a subtle obstruction to generalizing the usual hexagon to higher types. Assuming the existence of a long exact sequence of differential form classifiers, we construct the classifiers for circle k-gerbes with connection and describe their modal fracture hexagon.

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Section: Preliminariesmentioning

confidence: 99%

Section: The Universal ∞-Cover Of a Higher Groupmentioning

confidence: 99%

“…We call a 1-cover just a cover, or a covering map. Theorem 6.1 of [17] gives a useful way for proving that a map is an ∞-cover. Proposition 2.2.3 (Theorem 6.1 of [17]).…”

Section: The Universal ∞-Cover Of a Higher Groupmentioning

confidence: 99%

Section: The Universal ∞-Cover Of a Higher Groupmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Modal Fracture of Higher Groups

Myers¹

2021

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Modal Descent

Cherubini,

Rijke

2020

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Any modality in homotopy type theory gives rise to an orthogonal factorization system of which the left class is stable under pullbacks. We show that there is a second orthogonal factorization system associated to any modality, of which the left class is the class of -equivalences and the right class is the class of -étale maps. This factorization system is called the reflective factorization system of a modality, and we give a precise characterization of the orthogonal factorization systems that arise as the reflective factorization system of a modality. In the special case of the n-truncation the reflective factorization system has a simple description: we show that the n-étale maps are the maps that are right orthogonal to the map 1 → S n+1 . We use the -étale maps to prove a modal descent theorem: a map with modal fibers into X is the same thing as a -étale map into a type X. We conclude with an application to real-cohesive homotopy type theory and remarks how -étale maps relate to the formally etale maps from algebraic geometry.

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Orbifolds as microlinear types in synthetic differential cohesive homotopy type theory

Myers¹

2022

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Informally, an orbifold is a smooth space whose points may have finitely many internal symmetries. Formally, however, the notion of orbifold has been presented in a number of different guises -from Satake's V -manifolds to Moerdijk and Pronk's proper étale groupoids -which do not on their face resemble the informal definition. The reason for this divergence between formalism and intuition is that the points of spaces cannot have internal symmetries in traditional, set-level foundations. The extra data of these symmetries must be carried around and accounted for throughout the theory. More drastically, maps between orbifolds presented in the usual ways cannot be defined pointwise, and instead must use intermediary spaces.In this paper, we will put forward a definition of orbifold in synthetic differential cohesive homotopy type theory: an orbifold is a microlinear type for which the type of identifications between any two points is properly finite. In homotopy type theory, a point of a type may have internal symmetries, and we will be able to construct examples of orbifolds by defining their type of points directly. All maps between these orbifolds may be defined pointwise; the mapping space between two orbifolds is merely the type of functions.We will justify this synthetic definition by proving, internally, that every proper étale groupoid is an orbifold. In this way, we will show that the synthetic theory faithfully extends the usual theory of orbifolds. Along the way, we will investigate the microlinearity of higher types such as étale groupoids, showing that the methods of synthetic differential geometry generalize gracefully to higher analogues of smooth spaces. We will also investigate the relationship between the Dubuc-Penon and open-cover definitions of compactness in synthetic differential geometry, and show that any discrete, Dubuc-Penon compact subset of a second-countable manifold is subfinitely enumerable.

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Good Fibrations through the Modal Prism

Cited by 3 publications

References 5 publications

Modal Fracture of Higher Groups

Modal Fracture of Higher Groups

Modal Descent

Orbifolds as microlinear types in synthetic differential cohesive homotopy type theory

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