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TYPE THEORY AND HOMOTOPY
"... The purpose of this informal survey article is to introduce the reader to a new and surprising connection between Geometry, Algebra, and Logic, which has recently come to light in the form of an interpretation of the constructive type theory of Per MartinLöf into homotopy ..."
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The purpose of this informal survey article is to introduce the reader to a new and surprising connection between Geometry, Algebra, and Logic, which has recently come to light in the form of an interpretation of the constructive type theory of Per MartinLöf into homotopy
A Syntactical Approach to Weak ωGroupoids
"... Abstract—When moving to a Type Theory without proof irrelevance the notion of a setoid has to be generalized to the notion of a weak ωgroupoid. As a first step in this direction we study the formalisation of weak ωgroupoids in Type Theory. This is motivated by Voevodsky’s proposal of univalent typ ..."
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Abstract—When moving to a Type Theory without proof irrelevance the notion of a setoid has to be generalized to the notion of a weak ωgroupoid. As a first step in this direction we study the formalisation of weak ωgroupoids in Type Theory. This is motivated by Voevodsky’s proposal of univalent type theory which is incompatible with proofirrelevance and the results by Lumsdaine and Garner/van de Berg showing that the standard eliminator for equality gives rise to a weak ωgroupoid.
Calculating the Fundamental Group of the Circle in Homotopy Type Theory
"... Abstract—Recent work on homotopy type theory exploits an exciting new correspondence between MartinLof’s dependent type theory and the mathematical disciplines of category theory and homotopy theory. The category theory and homotopy theory suggest new principles to add to type theory, and type theo ..."
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Abstract—Recent work on homotopy type theory exploits an exciting new correspondence between MartinLof’s dependent type theory and the mathematical disciplines of category theory and homotopy theory. The category theory and homotopy theory suggest new principles to add to type theory, and type theory can be used in novel ways to formalize these areas of mathematics. In this paper, we formalize a basic result in algebraic topology, that the fundamental group of the circle is the integers. Though simple, this example is interesting for several reasons: it illustrates the new principles in homotopy type theory; it mixes ideas from traditional homotopytheoretic proofs of the result with typetheoretic inductive reasoning; and it provides a context for understanding an existing puzzle in type theory—that a universe (type of types) is necessary to prove that the constructors of inductive types are disjoint and injective. I.
WRITTEN IN LJUBLJANA, SLOVENIA.
, 2012
"... 1 A short guide to constructive type theory 7 1.1 A dependent type over a type........................ 8 1.1.1 Dependent products......................... 9 ..."
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1 A short guide to constructive type theory 7 1.1 A dependent type over a type........................ 8 1.1.1 Dependent products......................... 9
The miniworkshop The Homotopy Interpretation of Constructive Type Theory,
, 2011
"... Abstract. Over the past few years it has become apparent that there is a surprising and deep connection between constructive logic and higherdimensional structures in algebraic topology and category theory, in the form of an interpretation of the dependent type theory of Per MartinLöf into classica ..."
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Abstract. Over the past few years it has become apparent that there is a surprising and deep connection between constructive logic and higherdimensional structures in algebraic topology and category theory, in the form of an interpretation of the dependent type theory of Per MartinLöf into classical homotopy theory. The interpretation results in a bridge between the worlds of constructive and classical mathematics which promises to shed new light on both. This miniworkshop brought together researchers in logic, topology, and cognate fields in order to explore both theoretical and practical ramifications of this discovery.