## A closed model structure for n-categories, internal Hom, n-stacks and generalized Seifert-Van Kampen (1997)

Citations: | 23 - 6 self |

### BibTeX

@MISC{Simpson97aclosed,

author = {Carlos Simpson},

title = {A closed model structure for n-categories, internal Hom, n-stacks and generalized Seifert-Van Kampen },

year = {1997}

}

### Years of Citing Articles

### OpenURL

### Abstract

### Citations

361 | Homotopy limits completions and localizations - Bous, Kan - 1972 |

337 | associativity of H-Spaces I - Stasheff - 1963 |

298 | Simplicial objects in algebraic topology - May - 1967 |

238 |
homotopy theory
- Quillen
- 1969
(Show Context)
Citation Context ...-category. /// The proof of Theorem 3.1 We follow the proof of Jardine-Joyal that simplicial presheaves form a closed model category, as described in [13]. The proof is based on the axioms CM1–CM5 of =-=[21]-=-. Proof of CM1: The category of n-precats is a category of presheaves so it is closed under finite (and even arbitrary) direct and inverse limits. Proof of CM2: Given composable morphisms X f → Y g → ... |

146 | Higher-dimensional algebra & Topological quantum field theory
- Baez, Dolan
- 1995
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Citation Context ... several programs such as that outlined by Grothendieck [12], the generalization to n-stacks and n-gerbs of the work of Breen [7], or the program of Baez and Dolan in topological quantum field theory =-=[2]-=-. Once the theory of n-stacks is off the ground this will give an algebraic approach to the “geometric n-stacks” considered in [24]. We clarify the pretentions to rigor of the various sections of this... |

121 |
Simplicial presheaves
- Jardine
(Show Context)
Citation Context ... the elements of H 1 (G, V ) so we would expect to get H n (G, V ) in general, but for n > 1 there are no nontrivial strict morphisms from A to B. 1a closed model category developed by Jardine-Joyal =-=[13]-=- [15] in the case of simplicial presheaves. The cofibrations are essentially just monomorphisms (however we cannot— and don’t—require injectivity for top-degree morphisms, just as sets or categories w... |

117 | Champs Algébriques - Laumon, Moret-Bailly - 2000 |

99 |
The algebra of oriented simplexes
- Street
- 1987
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Citation Context ...n of nonstrict n-category from Breen, a definition which according to loc. cit “...has certainly the merit of existing...”. It is not clear whether this proposed construction was ever worked out. —In =-=[26]-=-, R. Street proposes a definition of weak n-category as a simplicial set satisfying a certain variant of the Kan condition where one takes into account the directions of arrows. —Kapranov and Voevodsk... |

75 | Higher-dimensional algebra III: ncategories and the algebra of opetopes
- Baez, Dolan
- 1998
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Citation Context ...lating n-categories and delooping machines). —J. Baez and J. Dolan have developed their theory originating in the letter refered to above, a definition of n-categories based on operads, in a preprint =-=[3]-=- of February 1997. In this preprint they discuss operads, give their definition of n-category and of certain morphisms of n-categories, and define the homotopy category of n-categories which they conj... |

50 |
des notions de n-catégorie et n-groupoïde non strictes via des ensembles multi-simpliciaux. K-Theory 16
- Sur
- 1999
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Citation Context ... 31062 Toulouse CEDEX, France. arXiv:alg-geom/9704006v1 10 Apr 1997 1. Introduction The purpose of this paper is to develop some additional techniques for the weak ncategories defined by Tamsamani in =-=[27]-=- (which he calls n-nerves). The goal is to be able to define the internal Hom(A, B) for two n-nerves A and B, which should itself be an n-nerve. This in turn is for defining the n + 1-nerve nCAT of al... |

47 | Abstract homotopy theory and generalized sheaf cohomology - Brown |

44 | Algebraic K-theory and generalized sheaf cohomology, Lecture - Brown, Gersten |

34 |
On the Classification of 2-gerbes and 2-stacks
- Breen
- 1994
(Show Context)
Citation Context ...theory of n-categories should open up the possibility to pursue any of the several programs such as that outlined by Grothendieck [12], the generalization to n-stacks and n-gerbs of the work of Breen =-=[7]-=-, or the program of Baez and Dolan in topological quantum field theory [2]. Once the theory of n-stacks is off the ground this will give an algebraic approach to the “geometric n-stacks” considered in... |

31 | The combinatorics of n-categorical pasting - Johnson - 1989 |

24 |
Homotopie Rationnelle: Modeles de
- Tanre
- 1983
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Citation Context ...d it is a groupoid with one isomorphism class, thus essentially a group in Ho(Top). This group is just the loop space based at any choice of point, viewed as a group in Ho(Top). It is well known ([1] =-=[28]-=-) that this object does not suffice to reconstitute the homotopy type of X, thus our functor from Top to the category of groupoids in Ho(Top) does not yield a factorization of the localization functor... |

16 |
Pursuing stacks, unpublished manuscript
- Grothendieck
- 1983
(Show Context)
Citation Context ...ing for (commonly called the “globular” case). —Gordon, Powers and Street have intensively investigated the cases n = 3 and n = 4 [11], following the path set out by Benabou for 2-categories [5]. —In =-=[12]-=- A. Grothendieck doesn’t seem to have hit upon any actual definition but gives a lot of nice intuition about n-categories. —On p. 41 of [12] starts a reproduction of a letter from Grothendieck to Bree... |

8 |
Coherence for tricategories Memoirs A.M.S
- Gordon, Power, et al.
- 1995
(Show Context)
Citation Context ...n − 1-category, so it isn’t quite the same as the approach we are looking for (commonly called the “globular” case). —Gordon, Powers and Street have intensively investigated the cases n = 3 and n = 4 =-=[11]-=-, following the path set out by Benabou for 2-categories [5]. —In [12] A. Grothendieck doesn’t seem to have hit upon any actual definition but gives a lot of nice intuition about n-categories. —On p. ... |

8 | une notion de 3-categorie adaptee a l’homotopie, U. Montpellier II preprint - Leroy, Sur |

8 | Algebraic (geometric) n-stacks - Simpson |

6 |
On the definition of weak ω-category. Macquarie mathematics report number 96/207
- Batanin
(Show Context)
Citation Context ...rect n + 1-category nCAT. As pointed out in the footnote above, one wonders in particular whether there is a closed model structure to go along with these strict n-categories. —In his recent preprint =-=[4]-=- M. Batanin develops some ideas towards a definition of weak ∞-category based on operads. In the introduction he mentions a letter from Baez and Dolan to Street dating to November 29, 1995 which conta... |

5 |
Introduction to Bicategories, Lect
- Bénabou
- 1967
(Show Context)
Citation Context ... are looking for (commonly called the “globular” case). —Gordon, Powers and Street have intensively investigated the cases n = 3 and n = 4 [11], following the path set out by Benabou for 2-categories =-=[5]-=-. —In [12] A. Grothendieck doesn’t seem to have hit upon any actual definition but gives a lot of nice intuition about n-categories. —On p. 41 of [12] starts a reproduction of a letter from Grothendie... |

5 |
groupoids and homotopy types
- KAPRANOV, VOEVODSKY
- 1991
(Show Context)
Citation Context ...Street proposes a definition of weak n-category as a simplicial set satisfying a certain variant of the Kan condition where one takes into account the directions of arrows. —Kapranov and Voevodsky in =-=[16]-=- construct, for a topological space X, a “Poincaré ∞- groupoid” which is a strictly associative ∞-groupoid but where the arrows are invertible only up to equivalence. This of course raises the questio... |

5 |
Homotopical algebra Springer LNM
- Quillen
- 1967
(Show Context)
Citation Context ...other hand one can see that these strict morphisms are not enough to reflect all of the “right” morphisms. 1 Our strategy to get around this problem will be based on the idea of closed model category =-=[20]-=-. We will construct a closed model category containing the n-nerves of Tamsamani. Then we can simply take as the “right” n-nerve of morphisms, the internal Hom(A, B) whenever A and B are fibrant objec... |

3 |
Letter to A. Grothendieck (refered
- Joyal
(Show Context)
Citation Context ...elements of H 1 (G, V ) so we would expect to get H n (G, V ) in general, but for n > 1 there are no nontrivial strict morphisms from A to B. 1a closed model category developed by Jardine-Joyal [13] =-=[15]-=- in the case of simplicial presheaves. The cofibrations are essentially just monomorphisms (however we cannot— and don’t—require injectivity for top-degree morphisms, just as sets or categories with m... |

3 | Homotopy everything H-spaces - Segal |

2 |
Flexible sheaves. Preprint available on q-alg
- Simpson
(Show Context)
Citation Context ...dn’t find an easy way to make this construction. The problem is somewhat analogous to the problem of finding a canonical inverse for a homotopy equivalence, solved in a certain topological context in =-=[23]-=- but which seems quite complicated to put into action here in view of the fact that our n-category A might not be fibrant (we don’t yet have the closed model structure!). Thus we will be happy with a ... |

2 | Cohomologie nonabélienne, Grundelehren der Wissenschaften in Einzeldarstellung 179 Springer-Verlag - Giraud - 1971 |

1 |
Letter to A. Grothendieck (refered to in Jardine’s paper). 68
- Joyal
- 1991
(Show Context)
Citation Context ...elements of H 1 (G, V ) so we would expect to get H n (G, V ) in general, but for n > 1 there are no nontrivial strict morphisms from A to B. 1a closed model category developed by Jardine-Joyal [13] =-=[15]-=- in the case of simplicial presheaves. The cofibrations are essentially just monomorphisms (however we cannot— and don’t—require injectivity for top-degree morphisms, just as sets or categories with m... |