## Homotopy theoretic models of identity types (2008)

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Citations: | 20 - 5 self |

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@MISC{Awodey08homotopytheoretic,

author = {Steve Awodey and Michael A. Warren},

title = { Homotopy theoretic models of identity types},

year = {2008}

}

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### Abstract

### Citations

470 |
P.: Interactive Theorem Proving and Program Development. Coq’Art: The Calculus of Inductive Constructions
- Bertot, Castéran
- 2004
(Show Context)
Citation Context ...e 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 as Coq and Agda (cf. =-=[2]-=- and [4]), this promise of applications is of a practical, as well as theoretical, nature. The present paper provides a precise indication of this connection between homotopy theory and logic; a more ... |

391 | Explicit substitutions
- Abadi, Cardelli, et al.
- 1991
(Show Context)
Citation Context ...t interval” I in either Gpd or, when the object A is a Kan complex, in SSet. In Gpd, I is the connected groupoid with exactly two objects (i.e., the “arrow category”) and in SSet it is the 1-simplex ∆=-=[1]-=-. Path objects may also be fruitfully considered in the context of weak factorization systems, where the left class L is thought of as the acyclic cofibrations and thesIDENTITY TYPES 7 right class R a... |

327 |
Homotopical algebra
- Quillen
- 1967
(Show Context)
Citation Context ... rule additional higher-dimensional structure must be considered in the interpretation. One way to add such structure is via the device of weak-factorization systems and Quillen model categories (cf. =-=[16]-=- and [3]). 3.1. Weak factorization systems. In any category C, given maps f : A and g : C �� D, we write f ⋔ g to indicate that f has left-lifting property (LLP) with respect to g. I.e. for any commut... |

261 |
Programming in Martin-Löf’s Type Theory: An Introduction
- Nordstrom, Petersson, et al.
- 1990
(Show Context)
Citation Context ...t of as sets and terms as elements of sets or, respectively, as objects of a category and global sections thereof. Alternatively, under an interpretation known as the Curry-Howard correspondence (cf. =-=[15]-=-), a type A can be regarded as a proposition and a term a : A as a proof of A. The simply typed λ-calculus is the type theory obtained by admitting the construction of products (A × B) and exponential... |

202 | Model Categories
- Hovey
- 1999
(Show Context)
Citation Context ...tors injective on objects, fibrations the Grothendieck fibrations and weak equivalences the categorical equivalences. Here all objects are both fibrant and cofibrant. The reader should consult, e.g., =-=[10]-=- or [7] for further examples and details.�� � � � � 3·3. Path objects Identity types 7 Recall from [10] that in a model category C a (very good) path object AI for an object A consists of a factoriza... |

157 |
An intuitionistic theory of types: Predicative part
- Martin-Löf
- 1975
(Show Context)
Citation Context ...o 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 =-=[13]-=- due to Hofmann and Streicher [8]. 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 ha... |

143 |
Model categories, Mathematical surveys and monographs
- Hovey
- 1998
(Show Context)
Citation Context ...tors injective on objects, fibrations the Grothendieck fibrations and weak equivalences the categorical equivalences. Here all objects are both fibrant and cofibrant. The reader should consult, e.g., =-=[9]-=- or [6] for further examples and details. 3.3. Path objects. Recall from [9], that in a model category C a (very good) path object A I for an object A consists of a factorization A � A ��� ∆ ��� A × A... |

122 |
Homotopy theories and model categories, Handbook of algebraic topology
- Dwyer, Spaliński
- 1995
(Show Context)
Citation Context ...jective on objects, fibrations the Grothendieck fibrations and weak equivalences the categorical equivalences. Here all objects are both fibrant and cofibrant. The reader should consult, e.g., [9] or =-=[6]-=- for further examples and details. 3.3. Path objects. Recall from [9], that in a model category C a (very good) path object A I for an object A consists of a factorization A � A ��� ∆ ��� A × A I r ��... |

68 |
Model categories, volume 63 of Mathematical Surveys and Monographs
- Hovey
- 1999
(Show Context)
Citation Context ...tors injective on objects, fibrations the Grothendieck fibrations and weak equivalences the categorical equivalences. Here all objects are both fibrant and cofibrant. The reader should consult, e.g., =-=[9]-=- or [6] for further examples and details. 3.3. Path objects. Recall from [9], that in a model category C a (very good) path object A I for an object A consists of a factorization A � ��� ∆ � r � I A �... |

56 |
Locally cartesian closed categories and type theory
- Seely
- 1984
(Show Context)
Citation Context ...gories corresponding to dependent type theory are locally cartesian closed categories. We now present the syntax of Martin-Löf type theory in more detail together with an interpretation, due to Seely =-=[17]-=-, in locally cartesian closed categories. This interpretation is “non-split” in the sense that it does not model substitution on the nose, but only up to canonical natural isomorphism, due to the pseu... |

53 |
Homotopy theories and model categories
- Dwyer, Spaliński
- 1995
(Show Context)
Citation Context ...jective on objects, fibrations the Grothendieck fibrations and weak equivalences the categorical equivalences. Here all objects are both fibrant and cofibrant. The reader should consult, e.g., [9] or =-=[6]-=- for further examples and details. 3.3. Path objects. Recall from [9], that in a model category C a (very good) path object A I for an object A consists of a factorization A � ��� ∆ � r � I A �� p A ×... |

39 | On the interpretation of type theory in locally cartesian closed categories
- Hofmann
- 1995
(Show Context)
Citation Context ...non-split” in the sense that it does not model substitution on the nose, but only up to canonical natural isomorphism, due to the pseudo-functoriality introduced by a choice of pullbacks (cf. [5] and =-=[7]-=-). Because we are mostly interested in type theory as an internal language for categories this conflation of isomorphic objects will not concern us here. The homotopy theoretical interpretation will b... |

38 | Structured type theory
- Coquand, Coquand
- 1999
(Show Context)
Citation Context ... 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 as Coq and Agda (cf. [2] and =-=[4]-=-), this promise of applications is of a practical, as well as theoretical, nature. The present paper provides a precise indication of this connection between homotopy theory and logic; a more detailed... |

38 |
Fibred categories and the foundations of naive category theory
- Bénabou
- 1985
(Show Context)
Citation Context ...nd the behavior of substitution are in general only determined up to isomorphism. In the case of extensional type theory, these issues are resolved, as noted by Hofmann [8], using a result of Bénabou =-=[2]-=- which in essence yields choices of these operations satisfying the corresponding rules. The homotopical interpretation will be given in the Section 3. 2·1. Forms of judgement The syntax of type theor... |

33 |
Quasi-categories and Kan complexes
- Joyal
(Show Context)
Citation Context ...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 [14] or the work of Joyal =-=[10, 11]-=- and Lurie [12] on quasicategories seems to indicate — it will likely continue to facilitate mathematical advances. In this paper we present a novel connection between model categories and mathematica... |

33 |
A 1 -homotopy theory of schemes
- Morel, Voevodsky
- 1999
(Show Context)
Citation Context ...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 =-=[14]-=- or the work of Joyal [10, 11] and Lurie [12] on quasicategories seems to indicate — it will likely continue to facilitate mathematical advances. In this paper we present a novel connection between mo... |

28 |
Investigations into intensional type theory
- Streicher
- 1993
(Show Context)
Citation Context ...ls to be in the extensional theory obtained by adding (8) as a rule governing identity types. This fact is the principal motivation for studying intensional rather than extensional type theories (cf. =-=[18]-=- for a more thorough discussion of the phenomenon of intensionality and the difference between intensional and extensional forms of the theory). Under the general interpretation in locally cartesian c... |

23 |
Constructions of factorization systems in categories
- Bousfield
- 1977
(Show Context)
Citation Context ...itional higher-dimensional structure must be considered in the interpretation. One way to add such structure is via the device of weak-factorization systems and Quillen model categories (cf. [16] and =-=[3]-=-). 3.1. Weak factorization systems. In any category C, given maps f : A �� B and g : C � D, we write f ⋔ g to indicate that f has left-lifting property (LLP) with respect to g. I.e. for any commutativ... |

22 |
The groupoid interpretation of type theory. In Twenty-five years of constructive type theory 1995, volume 36 of Oxford Logic Guides
- Hofmann, Streicher
(Show Context)
Citation Context .... 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 [13] due to Hofmann and Streicher =-=[8]-=-. 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 languag... |

22 | The groupoid interpretation of type theory
- Hofmann, Streicher
- 1996
(Show Context)
Citation Context .... 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 [13] due to Hofmann and Streicher =-=[8]-=-. 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 languag... |

17 |
Substitution up to isomorphism
- Curien
- 1993
(Show Context)
Citation Context ...ion is “non-split” in the sense that it does not model substitution on the nose, but only up to canonical natural isomorphism, due to the pseudo-functoriality introduced by a choice of pullbacks (cf. =-=[5]-=- and [7]). Because we are mostly interested in type theory as an internal language for categories this conflation of isomorphic objects will not concern us here. The homotopy theoretical interpretatio... |

13 | Homotopy theoretic aspects of constructive type theory
- Warren
(Show Context)
Citation Context ... as theoretical, nature. The present paper provides a precise indication of this connection between homotopy theory and logic; a more detailed discussion of these and further results will be given in =-=[19]-=-. 2. Type Theory Type theory is concerned with (at least) two basic kinds of entities: types and terms. Types are written as A, B, . . . and terms as a, b, . . .. Every term has a unique type and we w... |

11 |
Bousfield, Constructions of factorization systems in categories
- K
- 1977
(Show Context)
Citation Context ...itional higher-dimensional structure must be considered in the interpretation. One way to add such structure is via the device of weak-factorization systems and Quillen model categories (cf. [16] and =-=[3]-=-). 3.1. Weak factorization systems. In any category C, given maps f : A and g : C �� D, we write f ⋔ g to indicate that f has left-lifting property (LLP) with respect to g. I.e. for any commutative sq... |

7 |
and J.-J.Lévy. Explicit substitution
- Abadi, Cardelli, et al.
- 1989
(Show Context)
Citation Context ...Section 3·2 above, we take I in Gpd to be the free connected groupoid with exactly two objects and one isomorphism between them (i.e., the “arrow category”) and in SSet we take I to be the 1-simplex ∆=-=[1]-=-. Path objects may also be fruitfully considered in the context of weak factorization systems, where the left class L is thought of as the acyclic cofibrations and the right class R as the fibrations.... |

2 |
A 1 -homotopy theory of schemes, Publications Mathématiques de l’IHES no. 90
- Morel, Voevodsky
- 1999
(Show Context)
Citation Context ...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 =-=[14]-=- or the work of Joyal [10, 11] and Lurie [12] on quasicategories seems to indicate — it will likely continue to facilitate mathematical advances. In this paper we present a novel connection between mo... |

2 |
Higher topos theory. Unpublished manuscript, available on the arXiv as arXiv:math/0608040
- Lurie
- 2007
(Show Context)
Citation Context ...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 ... |

1 |
Substitution up to isomorphism, Fundamenta Informaticae 19
- Curien
- 1993
(Show Context)
Citation Context ...ion is “non-split” in the sense that it does not model substitution on the nose, but only up to canonical natural isomorphism, due to the pseudo-functoriality introduced by a choice of pullbacks (cf. =-=[5]-=- and [7]). Because we are mostly interested in type theory as an internal language for categories this conflation of isomorphic objects will not concern us here. The homotopy theoretical interpretatio... |

1 |
on quasi-categories, Unpublished notes distributed during the Fields Institute program on Geometric Applications of Homotopy Theory
- Notes
- 2007
(Show Context)
Citation Context ...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 [14] or the work of Joyal =-=[10, 11]-=- and Lurie [12] on quasicategories seems to indicate — it will likely continue to facilitate mathematical advances. In this paper we present a novel connection between model categories and mathematica... |

1 |
Higher topos theory, available on the arXiv as arXiv:math/0608040
- Lurie
- 2007
(Show Context)
Citation Context ...de 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 [14] or the work of Joyal [10, 11] and Lurie =-=[12]-=- on quasicategories seems to indicate — it will likely continue to facilitate mathematical advances. In this paper we present a novel connection between model categories and mathematical logic, inspir... |

1 |
Notes on quasi-categories. Unpublished manuscript
- Joyal
- 2007
(Show Context)
Citation Context ...h 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 l... |

1 |
K.PETERSSON andJ.M.SMITH. Programming in Martin–Löf’s Type Theory. An Introduction (Oxford
- NORDSTRÖM
- 1990
(Show Context)
Citation Context ...t of as sets and terms as elements of sets or, respectively, as objects of a category and global sections thereof. Alternatively, under an interpretation known as the Curry–Howard correspondence (cf. =-=[16]-=-), a type A can be regarded as a proposition and a term a : A as a proof of A. The simply typed λ-calculus is the type theory obtained by admitting the construction of products (A × B) and exponential... |

1 |
cartesian closed categories and type
- Locally
- 1984
(Show Context)
Citation Context ...gories corresponding to dependent type theory are locally cartesian closed categories. We now present the syntax of Martin–Löf type theory in more detail together with an interpretation, due to Seely =-=[18]-=-, in locally cartesian closed categories. This interpretation is “non-split” in the sense that it does not model substitution on the nose, but only up to canonical natural isomorphism, due to the pseu... |

1 |
Notes on quasi-categories. Unpublished notes distributed during the Fields Institute program on Geometric Applications of Homotopy Theory
- Joyal
- 2007
(Show Context)
Citation Context ...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 [14] or the work of Joyal =-=[10, 11]-=- and Lurie [12] on quasicategories seems to indicate — it will likely continue to facilitate mathematical advances. In this paper we present a novel connection between model categories and mathematica... |

1 |
Higher topos theory. Unpublished e-print
- Lurie
- 2007
(Show Context)
Citation Context ...de 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 [14] or the work of Joyal [10, 11] and Lurie =-=[12]-=- on quasicategories seems to indicate — it will likely continue to facilitate mathematical advances. In this paper we present a novel connection between model categories and mathematical logic, inspir... |