## TYPE THEORY AND HOMOTOPY

Citations: | 4 - 0 self |

### BibTeX

@MISC{Awodey_typetheory,

author = {Steve Awodey},

title = {TYPE THEORY AND HOMOTOPY},

year = {}

}

### OpenURL

### Abstract

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 Martin-Löf into homotopy

### Citations

469 |
The formulae-as-types notion of construction
- Howard
- 1980
(Show Context)
Citation Context ... to provide a rigorous framework for constructive mathematics [ML75, ML98, ML84]. It is an extension of the typed λ-calculus admitting dependent types and terms. Under the Curry-Howard correspondence =-=[How80]-=-, one identifies types with propositions, and terms with proofs; viewed thus, the system is at least as strong as secondorder logic, and it is known to interpret constructive set theory [Acz74]. Indee... |

354 | Intuitionistic Type Theory - Martin-Löf - 1984 |

280 | Constructive mathematics and computer programming - Martin-Löf - 1982 |

274 | Programming in Martin-Löf’s Type Theory: an introduction - Nordström, Petersson, et al. - 1990 |

172 | An intuitionistic theory of types: predicative part, logic colloquium '73 , Rose and Shepherdson (eds - Martin-Lof - 1973 |

116 |
Monoidal globular categories as a natural environment for the theory of weak n-categories
- Batanin
- 1998
(Show Context)
Citation Context ...ory of 1-types; in this precise sense, groupoids are said to model homotopy 1-types. A famous conjecture of Grothendieck’s is that (arbitrary) homotopy types are modeled by weak ∞-groupoids (see e.g. =-=[Bat98]-=- for a precise statement). Recent work [AHW09] by the author, Pieter Hofstra, and Michael Warren has shown that the 1-truncation of the intensional theory, arrived at by adding the analogue of the Id-... |

68 | An intuitionistic theory of types - Martin-Löf - 1998 |

66 | Homotopy theories and model categories - Dwyer, Spalinski - 1995 |

61 |
Locally cartesian closed categories and type theory
- Seely
- 1984
(Show Context)
Citation Context ...nsional type theories (cf. [Str91] for a discussion of the difference between the intensional and extensional forms of the theory). A good notion of a model for the extensional theory is due to Seely =-=[See84]-=-, who showed that one can interpret type dependency in locally cartesian closed categories in a very natural way. (There are certain coherence issues, prompting a later refinement by Hofmann [Hof97], ... |

59 | Generalised algebraic theories and contextual categories - Cartmell - 1986 |

52 | Higher Topos Theory
- Lurie
- 2009
(Show Context)
Citation Context ..., it is also applicable not only in spaces and simplicial sets, but also in new settings, as in the work of Voevodsky on the homotopy theory of schemes [MV99], or that of Joyal [Joy02, Joy] and Lurie =-=[Lur09]-=- on quasicategories. In the work under consideration here (subsection 2.3), it is shown that Martin-Löf type theory can be interpreted in anyTYPE THEORY AND HOMOTOPY 5 model category. This allows the... |

51 | Extensional concepts in intensional type theory - Hofmann - 1995 |

45 |
Theory of quasi-categories
- Joyal
(Show Context)
Citation Context ...sheaves of homotopy types, higher groupoids, quasi-categories, and the like. Two important works in this area have just appeared (Lurie, Higher Topos Theory [Lur09]; Joyal, Theory of Quasi-Categories =-=[Joy]-=-). It may be said, somewhat roughly, that the notion of a “higher-dimensional topos” is to homotopy what that of a topos is to topology (as in [JT91]). This concept also has a clear categorical-algebr... |

42 | On the interpretation of type theory in locally cartesian closed categories - Hofmann - 1994 |

42 | Syntax and semantics of dependent types
- Hofmann
- 1997
(Show Context)
Citation Context ...y [See84], who showed that one can interpret type dependency in locally cartesian closed categories in a very natural way. (There are certain coherence issues, prompting a later refinement by Hofmann =-=[Hof97]-=-, but this need not concern us here.) Of course, intensional type theory can also be interpreted this way, but then the interpretation of the identity types necessarily becomes trivial in the above se... |

40 | Internal type theory - Dybjer - 1996 |

38 | Wellfounded trees in categories - Moerdijk, Palmgren - 2000 |

37 | Type theories, toposes and constructive set theory: predicative aspects of AST - Moerdijk, Palmgren - 2002 |

35 | Quasi-categories and Kan complexes - Joyal - 2002 |

32 |
A1-homotopy theory of schemes
- Morel, Voevodsky
- 1999
(Show Context)
Citation Context ... of any one specific setting. Thus, for instance, it is also applicable not only in spaces and simplicial sets, but also in new settings, as in the work of Voevodsky on the homotopy theory of schemes =-=[MV99]-=-, or that of Joyal [Joy02, Joy] and Lurie [Lur09] on quasicategories. In the work under consideration here (subsection 2.3), it is shown that Martin-Löf type theory can be interpreted in anyTYPE THEO... |

30 |
Semantics of type theory
- Streicher
- 1991
(Show Context)
Citation Context ...n the extensional theory obtained by adding Id-reflection as a rule governing identity types. This fact is the principal motivation for studying intensional rather than extensional type theories (cf. =-=[Str91]-=- for a discussion of the difference between the intensional and extensional forms of the theory). A good notion of a model for the extensional theory is due to Seely [See84], who showed that one can i... |

29 | The petit topos of globular sets - Street |

28 | Construction of factorization systems in categories - Bousfield - 1977 |

23 | Homotopy theoretic models of identity types
- Awodey, Warren
- 2009
(Show Context)
Citation Context ...poids, ∞-groupoids, simplicial sets, etc., or even spaces themselves. The basic result in this connection states that it is possible to model the intensional type theory in any Quillen model category =-=[AW09]-=- (see also [War08]). The idea is that a type is interpreted as an abstract “space” X and a term x : X ⊢ a(x) : A as a continuous function a : X � A. Thus e.g. a closed term a : A is a point a of A, an... |

23 | The groupoid interpretation of type theory
- Hofmann, Streicher
- 1996
(Show Context)
Citation Context ...ogic, e.g.TYPE THEORY AND HOMOTOPY 15 sheaf-theoretic independence proofs, topological semantics for many non-classical systems, and an abstract treatment of realizability (see the encyclopedic work =-=[Joh03]-=-). An important and lively research program in current homotopy theory is the pursuit (again following Grothendieck [Gro83]) of a general concept of “stack,” subsuming sheaves of homotopy types, highe... |

19 |
Strong stacks and classifying spaces
- Joyal, Tierney
- 1991
(Show Context)
Citation Context ...s Theory [Lur09]; Joyal, Theory of Quasi-Categories [Joy]). It may be said, somewhat roughly, that the notion of a “higher-dimensional topos” is to homotopy what that of a topos is to topology (as in =-=[JT91]-=-). This concept also has a clear categorical-algebraic component via Grothendieck’s “homotopy hypothesis”, which states that n-groupoids are combinatorial models for homotopy n-types, and ∞-groupoids ... |

19 |
Homotopical Algebra, volume 43
- Quillen
- 1967
(Show Context)
Citation Context ...features of homotopy of topological spaces, enabling one to “do homotopy” in different mathematical settings, and to express the fact that two settings carry the same homotopical information. Quillen =-=[Qui67]-=- introduced model categories as an abstract framework for homotopy theory which would apply to a wide range of mathematical settings. Such a structure consists of the specification of three classes of... |

14 |
An Extension of the Galois Theory of Grothendieck, volume 309
- Joyal, Tierney
- 1984
(Show Context)
Citation Context ...by Grothendieck as an abstract framework for sheaf cohomology, the notion of a topos was soon discovered to have a logical interpretation, admitting the use of logical methods into topology (see e.g. =-=[JT84]-=- for just one of many examples). Equally important was the resulting flow of geometric and topological ideas and methods into logic, e.g.TYPE THEORY AND HOMOTOPY 15 sheaf-theoretic independence proof... |

14 | Homotopy Theoretic Aspects of Constructive Type Theory
- Warren
- 2008
(Show Context)
Citation Context ...o model intensional type theory in 2-groupoids, and shows that when various truncation axioms are added, the resulting theory is sound and complete with respect to this semantics. In his dissertation =-=[War08]-=-, Warren showed that infinite-dimensional groupoids also give rise to models, which validate no such additional truncation axioms (see also [War10]). Such models do, however, satisfy type-theoreticall... |

13 | Two-dimensional models of type theory
- Garner
(Show Context)
Citation Context ...ls admitting non-trivial higher identity types. Such higher groupoids occur naturally as the (higher) fundamental groupoids of spaces (as discussed above). A step in this direction was made by Garner =-=[Garar]-=-, who uses a 2-dimensional notion of fibration to model intensional type theory in 2-groupoids, and shows that when various truncation axioms are added, the resulting theory is sound and complete with... |

13 | The identity type weak factorisation system
- Gambino, Garner
- 2008
(Show Context)
Citation Context ...s interpretation, one uses abstract “fibrations” to interpret dependent types, and abstract “path spaces” to model identity types, recovering the groupoid model and its relatives as special cases. In =-=[GG08]-=- it was then shown that the type theory itself carries a natural homotopy structure (i.e. a weak factorization system), so that the theory is not only sound, but also logically complete with respect t... |

13 |
Weak ω-categories from intensional type theory, Submitted
- Lumsdaine
- 2009
(Show Context)
Citation Context ...tion entirely within the logical system — it belongs, as it were, to the logic of homotopy theory, as we now proceed to explain. 2.4.1. Weak ω-groupoids. It has recently been shown by Peter Lumsdaine =-=[Lum09]-=- and, independently, Benno van den Berg and Richard Garner [BG09, vdB], that the tower of identity types over any fixed base type A in the type theory bears an infinite dimensional algebraic structure... |

11 | A survey of definitions of n-category - Leinster |

11 | Higher operads, higher categories. Number 298 - Leinster - 2004 |

11 | Investigations into intensional type theory. Habilitiation Thesis - Streicher - 1993 |

10 |
The strength of Martin–Löf type theory with one universe
- Aczel
- 1977
(Show Context)
Citation Context ...ndence [How80], one identifies types with propositions, and terms with proofs; viewed thus, the system is at least as strong as secondorder logic, and it is known to interpret constructive set theory =-=[Acz74]-=-. Indeed, Martin-Löf type theory has been used successfully to formalize large parts of constructive mathematics, such as the theory of generalized recursive definitions [NPS90, ML79]. Moreover, it is... |

9 | Cofibrantly generated natural weak factorisation systems - Garner - 2007 |

8 | Types are weak ω-groupoids - Berg, Garner |

7 |
Topological and simplicial models of identity types
- Berg, Garner
(Show Context)
Citation Context ...ally complete with respect to such abstract homotopical semantics. While some “coherence” issues regarding the strictness of the interpretation remain to be worked out (again, see [War08], as well as =-=[BG10]-=-), together these results clearly establish not only the viability of the homotopical interpretation as a semantics for type theory, but also the possibility of using type theory to reason in Quillen ... |

6 | Twenty-Five Years of Constructive Type Theory, volume 36 of Oxford Logic Guides - Smith - 1998 |

3 | From groups to groupoids - Brown - 1987 |

2 |
Homotopy types
- Baues
- 1995
(Show Context)
Citation Context ...pological “truncation” of the higher fundamental groups. Spaces for which the homotopy type is already completely determined by the fundamental groupoid are called homotopy 1-types, or simply 1-types =-=[Bau95]-=-. More generally, one has n-types, which are thought of as spaces which have no homotopical information above dimension n. One of the goals of homotopy theory is to obtain good models of homotopy n-ty... |

2 | An ω-category with all duals is an ω-groupoid - Cheng |

2 |
Voevodsky,∞-groupoids and homotopy types, Cahiers de Topologie et Géométrie Différentielle Catégorique 32
- Kapranov, A
- 1991
(Show Context)
Citation Context ... even higher homotopies. The resulting structure π∞(X) is called the fundamental weak ∞-groupoid of X. Such higherdimensional algebraic structures now play a central role in homotopy theory (see e.g. =-=[KV91]-=-); they capture much more of the homotopical information of a space than does the fundamental group π(X, p), or the groupoid π(X) = π1(X), which is a quotient of π∞(X) by collapsing the higher homotop... |

2 |
A very short note on the homotopy λ-calculus. Unpublished note
- Voevodsky
- 2006
(Show Context)
Citation Context ..., Vladimir Voevodsky gave a series of lectures at Stanford University entitled “Homotopy lambda-calculus”, in which an interpretation of intensional type theory into simplicial sets was proposed (see =-=[Voe06]-=-). At the same time, and independently, the author and his doctoral student Michael Warren established the interpretation of intensional type theory in Quillen model structures, following a suggestion... |

1 | Martin-Löf complexes
- Awodey, Hofstra, et al.
- 2009
(Show Context)
Citation Context ...ds are said to model homotopy 1-types. A famous conjecture of Grothendieck’s is that (arbitrary) homotopy types are modeled by weak ∞-groupoids (see e.g. [Bat98] for a precise statement). Recent work =-=[AHW09]-=- by the author, Pieter Hofstra, and Michael Warren has shown that the 1-truncation of the intensional theory, arrived at by adding the analogue of the Id-reflection rule for all terms of identity type... |

1 |
Pursuing stacks. Unpublished letter to Quillen
- Grothendieck
- 1983
(Show Context)
Citation Context ... p ∈ X, and the paths in X beginning and ending at p, and identify such paths up to homotopy, the result is the fundamental group π(X, p) of the space at the point. Pursuing an idea of Grothendieck’s =-=[Gro83]-=-, modern homotopy theory generalizes this classical construction in several directions: first, we remove the dependence on the base-point p by considering the fundamental groupoid π(X), consisting of ... |

1 |
Model Categories, volume 63
- Press
- 1999
(Show Context)
Citation Context ...A consists of a factorization r A �� �� A ���� � ∆ �� I p �� A × A, (1)� � � � � TYPE THEORY AND HOMOTOPY 9 of the diagonal map ∆ : A �� A×A as a trivial cofibration r followed by a fibration p (see =-=[Hov99]-=-). Paradigm examples of path objects are given by exponentiation by a suitable “unit interval” I in either Gpd or, when the object A is a Kan complex, in SSet. In e.g. the former case, GI is just the ... |

1 |
Groupoids and local cartesian closure. Department of Mathematics
- Palmgren
- 2003
(Show Context)
Citation Context ...ed to the intensional theory, which never appeared. Their intention was presumably to make use of higher categories and, perhaps, Quillen model structures. No preliminary results were stated, but see =-=[Pal03]-=-. In 2006, Vladimir Voevodsky gave a series of lectures at Stanford University entitled “Homotopy lambda-calculus”, in which an interpretation of intensional type theory into simplicial sets was propo... |

1 |
Algebraic model structures. 2010. on the archive under arXiv:0910.2733v2
- Riehl
(Show Context)
Citation Context ...a “coherent way”, i.e. respecting substitutions of terms for variables. Some solutions to this problem are discussed in [AW09, War08, Gar07]. One neat solution is implicit in the recent work of Riehl =-=[Rie10]-=- on “algebraic” Quillen model structures. A systematic investigation of the issue of coherence, along with several examples of coherent models derived from homotopy theory, can be found in the recent ... |