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Homotopical Patch Theory
"... Homotopy type theory is an extension of MartinLöf type theory, based on a correspondence with homotopy theory and higher category theory. In homotopy type theory, the propositional equality type becomes proofrelevant, and corresponds to paths in a space. This allows for a new class of datatypes, ..."
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Homotopy type theory is an extension of MartinLöf type theory, based on a correspondence with homotopy theory and higher category theory. In homotopy type theory, the propositional equality type becomes proofrelevant, and corresponds to paths in a space. This allows for a new class of datatypes, called higher inductive types, which are specified by constructors not only for points but also for paths. In this paper, we consider a programming application of higher inductive types. Version control systems such as Darcs are based on the notion of patches—syntactic representations of edits to a repository. We show how patch theory can be developed in homotopy type theory. Our formulation separates formal theories of patches from their interpretation as edits to repositories. A patch theory is presented as a higher inductive type. Models of a patch theory are given by maps out of that type, which, being 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.
A Cubical Approach to Synthetic Homotopy Theory
"... 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 higherdimensional paths. While some aspects of homotopy theory have been developed synthetically ..."
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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 higherdimensional paths. While some aspects of homotopy theory have been developed synthetically and formalized in proof assistants, some seemingly straightforward 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 higherdimensional cube types, such as a type of squares, dependent on four points and four lines, and a type of threedimensional cubes, dependent on the boundary of a cube. Pathoverapath types and higher generalizations are used to describe cubes in a fibration over a cube in the base. These higherdimensional cube and pathover 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. I.
Homotopy Type Theory: Unified Foundations of Mathematics and Computation∗
, 2015
"... Homotopy type theory is a recentlydeveloped unification of previously disparate frameworks, which can serve to advance the project of formalizing and mechanizing mathematics. One framework is based on a computational conception of the type of a construction, the other is based on a homotopical con ..."
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Homotopy type theory is a recentlydeveloped unification of previously disparate frameworks, which can serve to advance the project of formalizing and mechanizing mathematics. One framework is based on a computational conception of the type of a construction, the other is based on a homotopical conception of the homotopy type of a space. The computational notion of type has its origins in Brouwer’s program of intuitionism, and Church’s λcalculus, both of which sought to ground mathematics in computation (one would say “algorithm ” these days). The homotopical notion comes from Grothendieck’s late conception of homotopy types of spaces as represented by ∞groupoids [12]. The computational perspective was developed most fully by Per MartinLöf, leading in particular to his Intuitionistic Theory of Types [23], on which the formal system of homotopy type theory is based. The connection to homotopy theory was first hinted at in the groupoid interpretation of Hofmann and Streicher [14, 13].1 It was then made explicit by several researchers, roughly simultaneously.2 The connection was clinched