## Limits in n-categories (1997)

Citations: | 4 - 0 self |

### BibTeX

@TECHREPORT{Simpson97limitsin,

author = {Carlos Simpson},

title = {Limits in n-categories},

institution = {},

year = {1997}

}

### Years of Citing Articles

### OpenURL

### Abstract

One of the main notions in category theory is the notion of limit. Similarly, one of the most commonly used techniques in homotopy theory is the notion of “homotopy limit” commonly called “holim ” for short. The purpose of the this paper is to begin to develop

### Citations

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375 |
The irreducibility of the space of curves of given genus
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Citation Context ...r homotopies” and the like. This brings the ideas much closer to the relatively simple notion of limits in a category. I first learned of the notion of “2-limit” from the paper of Deligne and Mumford =-=[13]-=-, where it appears at the beginning with very little explanation. Unfortunately at the writing of the present paper I have not been able to investigate the history of the notion of n-limits, and I apo... |

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

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

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238 | homotopy theory - Quillen - 1969 |

117 |
Champs Algébriques
- Laumon, Moret-Bailly
- 2000
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Citation Context ...hich F maps. Of course this needs to be investigated some more in any specific case, in order to get useful information. When C is the 2-category of categories we obtain the classical notion of stack =-=[22]-=- [2]. When C is the ∞-category of simplicial sets we obtain the notion of “homotopy sheaf” which is equivalent in Jardine’s terminology to a simplicial presheaf which is flasque with respect to each o... |

116 |
Monoidal globular categories as a natural environment for the theory of weak n-categories
- Batanin
- 1998
(Show Context)
Citation Context ...t and the homotopical notion of holim. We treat Tamsamani’s notion of n-category [36], but similar arguments and results should hold for the Baez-Dolan approach [3], [5], or the Batanin approach [6], =-=[7]-=-. We define the notions of direct and inverse limits in an arbitrary (fibrant cf [32]) ncategory C. Suppose A is an n-category, and suppose ϕ : A → C is a morphism, which we think of as a family of ob... |

105 |
Calculus of Fractions and Homotopy Theory
- Gabriel, Zisman
- 1967
(Show Context)
Citation Context ...ecome invertible in B. One has to verify the properties 6.5.3(i)–6.5.3(iv), and then apply Conjecture 6.5.4. To verify the properties (i)–(iv) use Theorem 2.5.1. This is the n-categorical analogue of =-=[14]-=-. Caution: If A is an m-category considered as an n-category then S −1 A may not be an m-category. In particular, note that by taking the group completion (see below) of 1-categories one gets all homo... |

99 | The algebra of oriented simplexes - Street - 1987 |

75 | Higher-dimensional algebra III: ncategories and the algebra of opetopes
- Baez, Dolan
- 1998
(Show Context)
Citation Context ...een the categorical notion of limit and the homotopical notion of holim. We treat Tamsamani’s notion of n-category [36], but similar arguments and results should hold for the Baez-Dolan approach [3], =-=[5]-=-, or the Batanin approach [6], [7]. We define the notions of direct and inverse limits in an arbitrary (fibrant cf [32]) ncategory C. Suppose A is an n-category, and suppose ϕ : A → C is a morphism, w... |

50 |
des notions de n-catégorie et n-groupoïde non strictes via des ensembles multi-simpliciaux. K-Theory 16
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- 1999
(Show Context)
Citation Context ... begin to develop the notion of limit for n-categories, which should be a bridge between the categorical notion of limit and the homotopical notion of holim. We treat Tamsamani’s notion of n-category =-=[36]-=-, but similar arguments and results should hold for the Baez-Dolan approach [3], [5], or the Batanin approach [6], [7]. We define the notions of direct and inverse limits in an arbitrary (fibrant cf [... |

47 |
Abstract homotopy theory and generalized sheaf cohomology
- Brown
(Show Context)
Citation Context ...ver that was not the first time that such objects were encountered—the condition of being a homotopy sheaf is the essential part of the condition of being a fibrant (or “flasque”) simplicial presheaf =-=[11]-=- [19] [21]. Suppose C is some type of category-like object (such as an n-category or ∞-category or other such thing). Suppose that we have a notion of inverse limit of a family of objects of C indexed... |

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

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25 |
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Citation Context ...s nCAT ′ . 886.7 Relative Malcev completion An example which gets more to the point of my motivation for doing all of this type of thing is the following generalization of relative Malcev completion =-=[18]-=- to higher homotopy. 6.7.1 Fix a Q-algebraic group G. Fix an n-groupoid X with base-object x (which is the same thing as an n-truncated pointed homotopy type). Fix a representation ρ : π1(X, x) → G. L... |

24 | Homotopie Rationnelle: Modeles de - Tanre - 1983 |

23 | A closed model structure for n-categories, internal Hom, n-stacks and generalized Seifert-Van
- Simpson
- 1997
(Show Context)
Citation Context ...], but similar arguments and results should hold for the Baez-Dolan approach [3], [5], or the Batanin approach [6], [7]. We define the notions of direct and inverse limits in an arbitrary (fibrant cf =-=[32]-=-) ncategory C. Suppose A is an n-category, and suppose ϕ : A → C is a morphism, which we think of as a family of objects of C indexed by A. For any object U ∈ C we can define the (n − 1)-category Hom(... |

17 |
Homotopy over the complex numbers and generalized cohomology theory, in Moduli of vector bundles (Taniguchi Symposium
- Simpson
- 1994
(Show Context)
Citation Context ... product, which does not preserve any substack of finite type. 6.4 The notion of stack, in general We give here a very general discussion of the notion of “stack”. This was called “homotopy-sheaf” in =-=[28]-=- (cf also [29] which predates [28] but which was made available much later), however that was not the first time that such objects were encountered—the condition of being a homotopy sheaf is the essen... |

16 |
Pursuing stacks, unpublished manuscript
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(Show Context)
Citation Context ...n particular, note that by taking the group completion (see below) of 1-categories one gets all homotopy types of n-groupoids. (This fact, which seems to be due to Quillen, was discussed at length in =-=[17]-=-...). 6.5.6 Group completion: The theory of n-categories which are not groupoids actually has a long history in homotopy theory, in the form of the study of topological monoids. In Adams’ book [1] the... |

16 | Mixed twistor structures - Simpson |

8 | Coherence for tricategories Memoirs A.M.S - Gordon, Power, et al. - 1995 |

8 | Algebraic (geometric) n-stacks
- Simpson
(Show Context)
Citation Context ...we should be able to construct the n + 1-stack nSTACK/X. 6.3.4 Now that we have a notion of n-stack not necessarily of groupoids, one can ask how to generalize the definition of geometricity given in =-=[31]-=-, to the case where the values may not be groupoids. If A is an n-category and X, Y are sets with maps a : X → A and b : Y → A then the pullback (a o , b) ∗ (Arr(A)) =: X × Y → (n − 1)CAT ′ may be con... |

6 |
On the definition of weak ω-category. Macquarie mathematics report number 96/207
- Batanin
(Show Context)
Citation Context ... limit and the homotopical notion of holim. We treat Tamsamani’s notion of n-category [36], but similar arguments and results should hold for the Baez-Dolan approach [3], [5], or the Batanin approach =-=[6]-=-, [7]. We define the notions of direct and inverse limits in an arbitrary (fibrant cf [32]) ncategory C. Suppose A is an n-category, and suppose ϕ : A → C is a morphism, which we think of as a family ... |

5 | Introduction to Bicategories, Lect - Bénabou - 1967 |

5 |
Homotopical algebra Springer LNM
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(Show Context)
Citation Context ...t. This remark is the starting point for [32]. There, one considers the full category of presheaves of sets on Θ (these presheaves are called n-precats) and [32] provides a closed model structure (cf =-=[25]-=- [26] [19]) on the category nPC of n-precats, corresponding to the homotopy theory of n-categories. In this section we briefly recall how this works. 2.2.1 It is more convenient for the purposes of th... |

3 |
n-Categories, sketch of a definition. Letter to
- Baez, Dolan
- 1995
(Show Context)
Citation Context ... between the categorical notion of limit and the homotopical notion of holim. We treat Tamsamani’s notion of n-category [36], but similar arguments and results should hold for the Baez-Dolan approach =-=[3]-=-, [5], or the Batanin approach [6], [7]. We define the notions of direct and inverse limits in an arbitrary (fibrant cf [32]) ncategory C. Suppose A is an n-category, and suppose ϕ : A → C is a morphi... |

3 |
Pure and Appl. Algebra 47
- Jardine, presheaves, et al.
- 1987
(Show Context)
Citation Context ...mark is the starting point for [32]. There, one considers the full category of presheaves of sets on Θ (these presheaves are called n-precats) and [32] provides a closed model structure (cf [25] [26] =-=[19]-=-) on the category nPC of n-precats, corresponding to the homotopy theory of n-categories. In this section we briefly recall how this works. 2.2.1 It is more convenient for the purposes of the closed m... |

3 | Letter to A. Grothendieck (refered - Joyal |

3 |
Homotopy everything H-spaces
- Segal
(Show Context)
Citation Context ... functor A : ∆ o → Sets) defined by setting Ap equal to the set of composable p-uples of arrows in C. This satisfies the property that the “Segal maps” (cf the discussion of Segal’s delooping machine =-=[27]-=- in [1] for the origin of this terminology) Ap → A1 ×A0 . . . ×A0 A1 are isomorphisms. To be precise this map is given by the p-uple of face maps 1 → p which take 0 to i and 1 to i + 1 for i = 0, . . ... |

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

2 |
Flexible sheaves. Preprint available on q-alg
- Simpson
(Show Context)
Citation Context ...sult. It says that a morphism which is an equivalence has an inverse which is essentially unique, if the notion of “inverse” is defined in the right way. It is an n-category version of the theorem of =-=[29]-=- which gives a canonical inverse for a homotopy equivalence of spaces. Theorem 2.5.1 For any fibrant n-category C the morphism restriction from I to I: r : Hom(I, C) → Hom(I, C) is fully faithful, so ... |

1 |
The topological realization of a simplicial presheaf. Preprint, available on q-alg
- Simpson
(Show Context)
Citation Context ...site, and when ϕ and ψ are families of n-groupoids. Then 〈ϕ, ψ〉 is an n-groupoid, and we conjecture that it corresponds to the topological space given as realization of the two functors as defined in =-=[30]-=-. 6.6.5 In the main example of [30] one took X to be the site of schemes over Spec(C) and one took ϕ to be the functor associating to each scheme the n-truncation of the homotopy type of the underlyin... |