## Martin-Löf Complexes (2009)

Citations: | 1 - 1 self |

### BibTeX

@MISC{Awodey09martin-löfcomplexes,

author = {S. Awodey and P. Hofstra and M. A. Warren},

title = {Martin-Löf Complexes},

year = {2009}

}

### OpenURL

### Abstract

In this paper we define Martin-Löf complexes to be algebras for monads on the category of (reflexive) globular sets which freely add cells in accordance with the rules of intensional Martin-Löf 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

### Citations

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Citation Context ...lopment of programming languages as well, a task for which it is especially well-suited in virtue of its combination of expressive strength and desirable proof-theoretic properties. (See the textbook =-=[15]-=- for a discussion.) The type theory has two variants: an intensional, and an extensional version. The difference between them lies mainly in the treatment of equality. In the intensional version (with... |

213 |
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Citation Context ...ced by the theory. The main technical difficulty then is to prove that there is still an equivalence of groupoids between the two. To this end, we employ a Tait-style computability predicate argument =-=[19]-=-, which allows us to prove that every term of the theory represents, up to propositional equality, an object or morphism of the free groupoid. This essentially shows that even though the theory forces... |

156 |
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Citation Context ...sent work in its context; especially for the reader coming from one field or the other, a brief summary is given of the essential concepts from the different subjects involved. Martin-Löf type theory =-=[14]-=- is a formal system originally intended to provide a rigorous framework in which to develop constructive mathematics. At heart,MARTIN-LÖF COMPLEXES 3 it is a calculus for reasoning about dependent ty... |

112 |
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Citation Context ...logy, in particular concerning identity types. We assume that the reader is somewhat familiar with at least simple type theory. For more background on (dependent) type theory we refer to the textbook =-=[10]-=-. The reader who is more familiar with higher-dimensional category theory or homotopy theory might also consult [1] for a “homotopical” view of type theory. In the last subsection we introduce the bas... |

63 |
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Citation Context ...theory. In the last subsection we introduce the basic categorical structures used in the paper, namely globular sets. A more detailed exposition of globular sets may be found in [17], or the textbook =-=[12]-=-. 2.1. Type dependency, contexts and judgements. Type dependency means that types may depend on variables of other types; for example one can has a type T(x) depending on a variable x of type S. Such ... |

55 |
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Citation Context ...tisfy the usual groupoid laws, but only up to propositional equality. We shall return to this point below. 1.1. Groupoid semantics. A good notion of a model for the extensional theory is due to Seely =-=[16]-=-, who showed that one can interpret type dependency in locally cartesian closed categories in a very natural way. (There are certain coherence issues related to this semantics, prompting a later refin... |

28 |
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Citation Context ...topical” view of type theory. In the last subsection we introduce the basic categorical structures used in the paper, namely globular sets. A more detailed exposition of globular sets may be found in =-=[17]-=-, or the textbook [12]. 2.1. Type dependency, contexts and judgements. Type dependency means that types may depend on variables of other types; for example one can has a type T(x) depending on a varia... |

27 |
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Citation Context ...y proofs and ordinary unit principles are clarified in the following lemma (the idea for the proof of which comes essentially from results, which are not “stratified” in the way considered here, from =-=[18]-=-). Lemma 3.10. Assuming the rules of Tω and the usual rules for identity types, the following implications hold: (1) TRn implies OUPn. (2) TRn implies UIPn+1. (3) UIPn implies TRn. for n ≥ 0. Proof. F... |

21 |
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Citation Context ...f terms. In the intensional version with which we shall be concerned here, it can be shown that the identity types IdA(a, b) satisfy certain conditions which were observed by Hofmann and Streicher in =-=[8]-=- to be analogous to the groupoid laws. Specifically, the posited reflexivity of propositional equality produces identity proofsr(a) : IdA(a, a) for any term a : A, playing the role of a unit arrow for... |

20 | Homotopy theoretic models of identity types
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Citation Context ...py theory, resulting in new examples of certain algebraic structures which are important in topology. This fascinating connection is currently under investigation from several different perspectives (=-=[1, 20, 6, 13, 3, 5]-=-), and these preliminary results confirm the significance of the link. Some of these results will be surveyed in this brief introduction in order to position the present work in its context; especiall... |

19 |
Strong stacks and classifying spaces, Category theory
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Citation Context ... carries essentially more information than a groupoid structure. We can, however, make use of the adjunction by transferring along it the standard Quillen model structure on the category of groupoids =-=[11]-=-, turning the category of 1-dimensional ML-complexes into a cofibrantly generated model category. Doing so requires a detailed analysis of colimits in the category of ML-complexes, and to this end we ... |

14 | Quillen closed model structures for sheaves
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(Show Context)
Citation Context ...roupoids are a model of homotopy 1-types it then follows that 1-truncated Martin-Löf complexes are also a model. The model structure on MLCx1 is obtained by transferring, using a theorem due to Crans =-=[4]-=-, the model structure on the category of groupoids along the adjunction (10). In order to transfer the model structure it is first necessary to understand the behavior of colimits in MLCx1. In order t... |

13 | Homotopy theoretic aspects of constructive type theory
- Warren
(Show Context)
Citation Context ...py theory, resulting in new examples of certain algebraic structures which are important in topology. This fascinating connection is currently under investigation from several different perspectives (=-=[1, 20, 6, 13, 3, 5]-=-), and these preliminary results confirm the significance of the link. Some of these results will be surveyed in this brief introduction in order to position the present work in its context; especiall... |

11 | The identity type weak factorisation system
- Gambino, Garner
(Show Context)
Citation Context ...py theory, resulting in new examples of certain algebraic structures which are important in topology. This fascinating connection is currently under investigation from several different perspectives (=-=[1, 20, 6, 13, 3, 5]-=-), and these preliminary results confirm the significance of the link. Some of these results will be surveyed in this brief introduction in order to position the present work in its context; especiall... |

11 | Two-dimensional models of type theory
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- 2009
(Show Context)
Citation Context |

11 |
Weak ω-categories from intensional type theory
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- 2009
(Show Context)
Citation Context |

10 |
Homotopy types, Handbook of Algebraic Topology
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(Show Context)
Citation Context ...of Gj which are obtainable from α by applying the source and target maps. A globular set G is said to be reflexive if it comes equipped with a family of maps in : Gn → Gn+1, such that tnin = 1 = snin =-=(2)-=- We think of in(x) as the identity edge from x to itself. In this paper we shall be working with reflexive globular sets only. For readability we often omit the dimension from the source, target and i... |

3 |
Syntax and semantics of dependent types, Semantics and Logics of Computation
- Hofmann
- 1997
(Show Context)
Citation Context ...es for sum types is used, taking the projection terms as primitive. In the presence of the η-rule both formulations are equivalent, but without the η-rule this latter approach is strictly weaker (see =-=[7]-=-). 2.4. Identity types. Let A be a type. For every pair of terms a, b of type A we may form a new type A(a, b). This type is thought of as the type of proofs of the fact that a and b are equal. A term... |

1 |
Types are weak ω-groupoids, Submitted
- Berg, Garner
- 2009
(Show Context)
Citation Context ...re satisfied: (1) Each element of Γ(A)n is a tuple of (2n + 1) elements of the set of terms of T. (2) If both (α; β) and (α; β ′ ) are in Γ(A)n, then ⊢ A n+1 (α; β, β ′ ) : type is derivable in T. =-=(3)-=- The source and target maps s, t : Γ(A)n+1 → Γ(A)n must send a tuple (α0, . . . , α2n) to (α0, . . . , α2n−2) and (α0, . . .,α2n−3, α2n−1), respectively. We begin by defining Γ(A)0 := {a | ⊢ a : A}, Γ... |