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The identity type weak factorisation system
 U.U.D.M. REPORT 2008:20
, 2008
"... ... theory T with axioms for identity types admits a nontrivial weak factorisation system. After characterising this weak factorisation system explicitly, we relate it to the homotopy theory of groupoids. ..."
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... theory T with axioms for identity types admits a nontrivial weak factorisation system. After characterising this weak factorisation system explicitly, we relate it to the homotopy theory of groupoids.
TWODIMENSIONAL MODELS OF TYPE THEORY
, 2008
"... We describe a nonextensional variant of MartinLöf type theory which we call twodimensional type theory, and equip it with a sound and complete semantics valued in 2categories. ..."
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We describe a nonextensional variant of MartinLöf type theory which we call twodimensional type theory, and equip it with a sound and complete semantics valued in 2categories.
Types are weak ωgroupoids
, 2008
"... We define a notion of weak ωcategory internal to a model of MartinLöf type theory, and prove that each type bears a canonical weak ωcategory structure obtained from the tower of iterated identity types over that type. We show that the ωcategories arising in this way are in fact ωgroupoids. ..."
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We define a notion of weak ωcategory internal to a model of MartinLöf type theory, and prove that each type bears a canonical weak ωcategory structure obtained from the tower of iterated identity types over that type. We show that the ωcategories arising in this way are in fact ωgroupoids.
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
Extensional normalization in the logical framework with proof irrelevant equality
 In Workshop on Normalization by Evaluation, affiliated to LiCS 2009, Los Angeles
, 2009
"... We extend the Logical Framework by proof irrelevant equality types and present an algorithm that computes unique long normal forms. The algorithm is inspired by normalizationbyevaluation. Equality proofs which are not reflexivity are erased to a single object ∗. The algorithm decides judgmental eq ..."
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We extend the Logical Framework by proof irrelevant equality types and present an algorithm that computes unique long normal forms. The algorithm is inspired by normalizationbyevaluation. Equality proofs which are not reflexivity are erased to a single object ∗. The algorithm decides judgmental equality, its completeness is established by a PER model. 1.
2Dimensional Directed Type Theory
"... Recent work on higherdimensional type theory has explored connections between MartinLöf type theory, higherdimensional category theory, and homotopy theory. These connections suggest a generalization of dependent type theory to account for computationally relevant proofs of propositional equality ..."
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Recent work on higherdimensional type theory has explored connections between MartinLöf type theory, higherdimensional category theory, and homotopy theory. These connections suggest a generalization of dependent type theory to account for computationally relevant proofs of propositional equality—for example, taking IdSet A B to be the isomorphisms between A and B. The crucial observation is that all of the familiar type and term constructors can be equipped with a functorial action that describes how they preserve such proofs. The key benefit of higherdimensional type theory is that programmers and mathematicians may work up to isomorphism and higher equivalence, such as equivalence of categories. In this paper, we consider a further generalization of higherdimensional type theory, which associates each type with a directed notion of transformation between its elements. Directed type theory accounts for phenomena not expressible in symmetric higherdimensional type theory, such as a universe set of sets and functions, and a type Ctx used in functorial abstract syntax. Our formulation requires two main ingredients: First, the types themselves must be reinterpreted to take account of variance; for example, a Π type is contravariant in its domain, but covariant in its range. Second, whereas in symmetric type theory proofs of equivalence can be internalized using the MartinLöf identity type, in directed type theory the twodimensional structure must be made explicit at the judgemental level. We describe a 2dimensional directed type theory, or 2DTT, which is validated by an interpretation into the strict 2category Cat of categories, functors, and natural transformations. We also discuss applications of 2DTT for programming with abstract syntax, generalizing the functorial approach to syntax to the dependently typed and mixedvariance case. 1
MartinLöf Complexes
, 2009
"... In this paper we define MartinLöf complexes to be algebras for monads on the category of (reflexive) globular sets which freely add cells in accordance with the rules of intensional MartinLöf type theory. We then study the resulting categories of algebras for several theories. Our principal resu ..."
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In this paper we define MartinLöf complexes to be algebras for monads on the category of (reflexive) globular sets which freely add cells in accordance with the rules of intensional MartinLöf type theory. We then study the resulting categories of algebras for several theories. Our principal result is that there exists a cofibrantly generated Quillen model structure on
A Notion of Homotopy for the Effective Topos
, 2010
"... We define a notion of homotopy in the effective topos. AMS Subject Classification (2000): 18B25 (Topos Theory),55U35 (Abstract and axiomatic homotopy theory) ..."
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We define a notion of homotopy in the effective topos. AMS Subject Classification (2000): 18B25 (Topos Theory),55U35 (Abstract and axiomatic homotopy theory)
Foundations and Applications of HigherDimensional Directed Type Theory
"... Intuitionistic type theory [43] is an expressive formalism that unifies mathematics and computation. A central concept is the propositionsastypes principle, according to which propositions are interpreted as types, and proofs of a proposition are interpreted as programs of the associated type. Mat ..."
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Intuitionistic type theory [43] is an expressive formalism that unifies mathematics and computation. A central concept is the propositionsastypes principle, according to which propositions are interpreted as types, and proofs of a proposition are interpreted as programs of the associated type. Mathematical propositions are thereby to be understood as specifications, or problem descriptions, that are solved by providing a program that meets the specification. Conversely, a program can, by the same token, be understood as a proof of its type viewed as a proposition. Over the last quartercentury type theory has emerged as the central organizing principle of programming language research, through the identification of the informal concept of language features with type structure. Numerous benefits accrue from the identification of proofs and programs in type theory. First, it provides the foundation for integrating types and verification, the two most successful formal methods used to ensure the correctness of software. Second, it provides a language for the mechanization of mathematics in which proof checking is equivalent to type checking, and proof search is equivalent to writing a program to meet a specification.