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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
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
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.
2Dimensional Directed Dependent Type Theory
 SUBMITTED TO POPL 2011
, 2011
"... The groupoid interpretation of dependent type theory given by Hofmann and Streicher associates to each closed type a category whose objects represent the elements of that type and whose maps represent proofs of equality of elements. The categorial structure ensures that equality is reflexive (identi ..."
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The groupoid interpretation of dependent type theory given by Hofmann and Streicher associates to each closed type a category whose objects represent the elements of that type and whose maps represent proofs of equality of elements. The categorial structure ensures that equality is reflexive (identity maps) and transitive (closure under composition); the groupoid structure, which demands that every map be invertible, ensures symmetry. Families of types are interpreted as functors; the action on maps (equality proofs) ensures that families respect equality of elements of the index type. The functorial action of a type family is computationally nontrivial in the case that the groupoid associated to the index type is nontrivial. For example, one may identity elements of a universe of sets up to isomorphism, in which case the action of a family of types indexed by sets must respect set isomorphism. The groupoid interpretation is 2dimensional in that the coherence requirements on proofs of equality are required to hold “on the
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.
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.