Covering Spaces in Homotopy Type Theory

Hou, Kuen-Bang

doi:10.1007/978-3-319-21284-5_15

Cited by 4 publications

(1 citation statement)

References 4 publications

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“…Another is higher inductive types (Lumsdaine, 2011;Shulman, 2011;Lumsdaine & Shulman, 2013), a new class of datatypes specified by constructors not only for points but also for paths. Higher inductive types were originally introduced to permit the type-theoretic definition of basic topological spaces such as circles and spheres, and have had significant applications in a line of work on using homotopy type theory to write computer-checked proofs of theorems from homotopy theory (Licata & Brunerie, 2013;Licata & Shulman, 2013;Univalent Foundations Program, 2013;Hou, 2014;Licata & Finster, 2014;Cavallo, 2015;Licata & Brunerie, 2015).…”

Section: Introductionmentioning

confidence: 99%

Homotopical patch theory

Angiuli¹,

Morehouse²,

Licata³

et al. 2016

J. Funct. Prog.

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Homotopy type theory is an extension of Martin-Löf type theory, based on a correspondence with homotopy theory and higher category theory. The propositional equality type becomes proofrelevant, and acts like paths in a space. Higher inductive types are a new class of datatypes which are specified by constructors not only for points but also for paths. In this paper, we show how patch theory in the style of the Darcs version control system can be developed in homotopy type theory. We reformulate patch theory using the tools of homotopy type theory, and clearly separate formal theories of patches from their interpretation in terms of basic revision control mechanisms. A patch theory is presented by a higher inductive type. Models of a patch theory are functions from that type, which, because function are functors, automatically preserve the structure of patches. Several standard tools of homotopy theory come into play, demonstrating the use of these methods in a practical programming context.

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Section: Introductionmentioning

confidence: 99%

Homotopical patch theory

Angiuli¹,

Morehouse²,

Licata³

et al. 2016

J. Funct. Prog.

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Adjoint Logic with a 2-Category of Modes

Licata

Shulman

2015

Lecture Notes in Computer Science

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A Cubical Approach to Synthetic Homotopy Theory

Licata¹,

Brunerie²

2015

2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science

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Abstract-Homotopy theory can be developed synthetically in homotopy type theory, using types to describe spaces, the identity type to describe paths in a space, and iterated identity types to describe higher-dimensional paths. While some aspects of homotopy theory have been developed synthetically and formalized in proof assistants, some seemingly easy examples have proved difficult because the required manipulations of paths becomes complicated. In this paper, we describe a cubical approach to developing homotopy theory within type theory. The identity type is complemented with higher-dimensional cube types, such as a type of squares, dependent on four points and four lines, and a type of three-dimensional cubes, dependent on the boundary of a cube. Path-over-a-path types and higher generalizations are used to describe cubes in a fibration over a cube in the base. These higher-dimensional cube and path-over types can be defined from the usual identity type, but isolating them as independent conceptual abstractions has allowed for the formalization of some previously difficult examples.

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Covering Spaces in Homotopy Type Theory

Cited by 4 publications

References 4 publications

Homotopical patch theory

Homotopical patch theory

Adjoint Logic with a 2-Category of Modes

A Cubical Approach to Synthetic Homotopy Theory

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