scholarly journals Homotopy theoretic models of identity types

2009 ◽  
Vol 146 (1) ◽  
pp. 45-55 ◽  
Author(s):  
STEVE AWODEY ◽  
MICHAEL A. WARREN
Keyword(s):  
Mathematical Logic ◽  
Type Theory ◽  
Theoretical Basis ◽  
Homotopy Theory ◽  
Model Category ◽  
Proof Assistants ◽  
Model Categories ◽  
Wide Range ◽  
Abstract Framework ◽  
Computer Proof

Quillen [17] introduced model categories as an abstract framework for homotopy theory which would apply to a wide range of mathematical settings. By all accounts this program has been a success and—as, e.g., the work of Voevodsky on the homotopy theory of schemes [15] or the work of Joyal [11,12] and Lurie [13] on quasicategories seem to indicate—it will likely continue to facilitate mathematical advances. In this paper we present a novel connection between model categories and mathematical logic, inspired by the groupoid model of (intensional) Martin–Löf type theory [14] due to Hofmann and Streicher [9]. In particular, we show that a form of Martin–Löf type theory can be soundly modelled in any model category. This result indicates moreover that any model category has an associated “internal language” which is itself a form of Martin-Löf type theory. This suggests applications both to type theory and to homotopy theory. Because Martin–Löf type theory is, in one form or another, the theoretical basis for many of the computer proof assistants currently in use, such asCoqandAgda(cf. [3] and [5]), this promise of applications is of a practical, as well as theoretical, nature.

scholarly journals Semantics of higher inductive types

2019 ◽  
Vol 169 (1) ◽  
pp. 159-208 ◽  
Author(s):  
PETER LEFANU LUMSDAINE ◽  
MICHAEL SHULMAN
Keyword(s):  
Type Theory ◽  
Homotopy Theory ◽  
Model Category ◽  
Suitable Model ◽  
Simplicial Sets ◽  
James Construction ◽  
Quillen Model Category ◽  
Wide Range ◽  
Inductive Types ◽  
Cartesian Closed

AbstractHigher inductive typesare a class of type-forming rules, introduced to provide basic (and not-so-basic) homotopy-theoretic constructions in a type-theoretic style. They have proven very fruitful for the “synthetic” development of homotopy theory within type theory, as well as in formalising ordinary set-level mathematics in type theory. In this paper, we construct models of a wide range of higher inductive types in a fairly wide range of settings.We introduce the notion ofcell monad with parameters: a semantically-defined scheme for specifying homotopically well-behaved notions of structure. We then show that any suitable model category hasweakly stable typal initial algebrasfor any cell monad with parameters. When combined with the local universes construction to obtain strict stability, this specialises to give models of specific higher inductive types, including spheres, the torus, pushout types, truncations, the James construction and general localisations.Our results apply in any sufficiently nice Quillen model category, including any right proper, simplicially locally cartesian closed, simplicial Cisinski model category (such as simplicial sets) and any locally presentable locally cartesian closed category (such as sets) with its trivial model structure. In particular, any locally presentable locally cartesian closed (∞, 1)-category is presented by some model category to which our results apply.

scholarly journals Locally constant functors

2009 ◽  
Vol 147 (3) ◽  
pp. 593-614 ◽  
Author(s):  
DENIS–CHARLES CISINSKI
Keyword(s):  
Homotopy Theory ◽  
Model Category ◽  
Model Categories ◽  
Combinatorial Model ◽  
Constant Coefficients ◽  
Kan Extensions ◽  
Bousfield Localization

AbstractWe study locally constant coefficients. We first study the theory of homotopy Kan extensions with locally constant coefficients in model categories, and explain how it characterizes the homotopy theory of small categories. We explain how to interpret this in terms of left Bousfield localization of categories of diagrams with values in a combinatorial model category. Finally, we explain how the theory of homotopy Kan extensions in derivators can be used to understand locally constant functors.

scholarly journals An interpretation of dependent type theory in a model category of locally cartesian closed categories

2021 ◽  
pp. 1-26
Author(s):  
Martin E. Bidlingmaier
Keyword(s):  
Type Theory ◽  
Model Category ◽  
Model Categories ◽  
Dependent Type Theory ◽  
Categorical Structure ◽  
Categorical Models ◽  
Cartesian Closed Categories ◽  
Cartesian Closed ◽  
Original Interpretation ◽  
Dependent Type

Abstract Locally cartesian closed (lcc) categories are natural categorical models of extensional dependent type theory. This paper introduces the “gros” semantics in the category of lcc categories: Instead of constructing an interpretation in a given individual lcc category, we show that also the category of all lcc categories can be endowed with the structure of a model of dependent type theory. The original interpretation in an individual lcc category can then be recovered by slicing. As in the original interpretation, we face the issue of coherence: Categorical structure is usually preserved by functors only up to isomorphism, whereas syntactic substitution commutes strictly with all type-theoretic structures. Our solution involves a suitable presentation of the higher category of lcc categories as model category. To that end, we construct a model category of lcc sketches, from which we obtain by the formalism of algebraically (co)fibrant objects model categories of strict lcc categories and then algebraically cofibrant strict lcc categories. The latter is our model of dependent type theory.

scholarly journals Arrow categories of monoidal model categories

2019 ◽  
Vol 125 (2) ◽  
pp. 185-198
Author(s):  
David White ◽  
Donald Yau
Keyword(s):  
Homotopy Theory ◽  
Model Category ◽  
Topological Spaces ◽  
Model Structure ◽  
Model Categories ◽  
Projective Model ◽  
Monoidal Structure ◽  
Chain Complexes

We prove that the arrow category of a monoidal model category, equipped with the pushout product monoidal structure and the projective model structure, is a monoidal model category. This answers a question posed by Mark Hovey, in the course of his work on Smith ideals. As a corollary, we prove that the projective model structure in cubical homotopy theory is a monoidal model structure. As illustrations we include numerous examples of non-cofibrantly generated monoidal model categories, including chain complexes, small categories, pro-categories, and topological spaces.

scholarly journals Model structure on the universe of all types in interval type theory

2020 ◽  
pp. 1-32
Author(s):  
Simon Boulier ◽  
Nicolas Tabareau
Keyword(s):  
Type Theory ◽  
Homotopy Theory ◽  
Model Structure ◽  
Model Categories ◽  
Model Interpretation ◽  
Slight Generalization ◽  
Cubical Sets ◽  
Definition Of ◽  
Interval Type ◽  
The Universe

Abstract Model categories constitute the major context for doing homotopy theory. More recently, homotopy type theory (HoTT) has been introduced as a context for doing syntactic homotopy theory. In this paper, we show that a slight generalization of HoTT, called interval type theory (⫿TT), allows to define a model structure on the universe of all types, which, through the model interpretation, corresponds to defining a model structure on the category of cubical sets. This work generalizes previous works of Gambino, Garner, and Lumsdaine from the universe of fibrant types to the universe of all types. Our definition of ⫿TT comes from the work of Orton and Pitts to define a syntactic approximation of the internal language of the category of cubical sets. In this paper, we extend the work of Orton and Pitts by introducing the notion of degenerate fibrancy, which allows to define a fibrant replacement, at the heart of the model structure on the universe of all types. All our definitions and propositions have been formalized using the Coq proof assistant.

Cubical Agda: A dependently typed programming language with univalence and higher inductive types

2021 ◽  
Vol 31 ◽  
Author(s):  
ANDREA VEZZOSI ◽  
ANDERS MÖRTBERG ◽  
ANDREAS ABEL
Keyword(s):  
Programming Language ◽  
Type Theory ◽  
Type Checking ◽  
Dependent Type Theory ◽  
Wide Range ◽  
General Schema ◽  
Direct Definition ◽  
Inductive Types ◽  
Definition Of ◽  
Cubical Type Theory

Abstract Proof assistants based on dependent type theory provide expressive languages for both programming and proving within the same system. However, all of the major implementations lack powerful extensionality principles for reasoning about equality, such as function and propositional extensionality. These principles are typically added axiomatically which disrupts the constructive properties of these systems. Cubical type theory provides a solution by giving computational meaning to Homotopy Type Theory and Univalent Foundations, in particular to the univalence axiom and higher inductive types (HITs). This paper describes an extension of the dependently typed functional programming language Agda with cubical primitives, making it into a full-blown proof assistant with native support for univalence and a general schema of HITs. These new primitives allow the direct definition of function and propositional extensionality as well as quotient types, all with computational content. Additionally, thanks also to copatterns, bisimilarity is equivalent to equality for coinductive types. The adoption of cubical type theory extends Agda with support for a wide range of extensionality principles, without sacrificing type checking and constructivity.

Mathematical Logic: Proof Theory, Type Theory and Constructive Mathematics

2005 ◽  
pp. 779-813
Author(s):  
Helmut Schwichtenberg ◽  
Vladimir Keilis-Borok ◽  
Samuel Buss
Keyword(s):  
Mathematical Logic ◽  
Type Theory ◽  
Proof Theory ◽  
Constructive Mathematics

FORMAL PROOFS OF FUNCTIONAL BSP PROGRAMS

2003 ◽  
Vol 13 (03) ◽  
pp. 365-376 ◽  
Author(s):  
FRÉDÉRIC GAVA
Keyword(s):  
Type Theory ◽  
Great Difficulty ◽  
Parallel Applications ◽  
Formal Proofs ◽  
Higher Order Logic ◽  
New Approach ◽  
Bulk Synchronous Parallel ◽  
Parallel Data ◽  
Wide Range ◽  
The Given

The Bulk Synchronous Parallel ML (BSML) is a functional language for BSP programming, a model of computing which allows parallel programs to be ported to a wide range of architectures. It is based on an extension of the ML language by parallel operations on a parallel data structure called parallel vector, which is given by intention. We present a new approach to certifying BSML programs in the context of type theory. Given a specification and a program, an incomplete proof of the specification (of which algorithmic contents corresponds to the given program) is built in the type theory, in which gaps would correspond to the proof obligation. This development demonstrates the usefulness of higher-order logic in the process of software certification of parallel applications. It also shows that the proof of rather complex parallel algorithms may be done with inductive types without great difficulty by using existing certified programs. This work has been implemented in the Coq Proof Assistant, applied on non-trivial examples and is the basis of a certified library of BSML programs.

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