## HOMOTOPY FIBRE SEQUENCES INDUCED BY 2-FUNCTORS (909)

### BibTeX

@MISC{Cegarra909homotopyfibre,

author = {A. M. Cegarra},

title = {HOMOTOPY FIBRE SEQUENCES INDUCED BY 2-FUNCTORS},

year = {909}

}

### OpenURL

### Abstract

Abstract. This paper contains some contributions to the study of the relationship between 2-categories and the homotopy types of their classifying spaces. Mainly, generalizations are given of both Quillen’s Theorem B and Thomason’s Homotopy Colimit Theorem to 2-functors. Mathematical Subject Classification: 18D05, 55P15, 18F25.

### Citations

984 |
Categories for the Working Mathematician
- Lane
- 1997
(Show Context)
Citation Context ...he study of 2-diagrams of 2-categories by means of the higher Grothendieck construction. 2. Preliminaries and notations For the general background on 2-categories used in this paper, we refer to [2], =-=[18]-=- and [24], and on simplicial sets to [19], [17] and, mainly, to [10]. The simplicial category is denoted by ∆. It has as objects the ordered sets [n] = {0, . . .,n}, n ≥ 0, and as arrows the (weakly) ... |

367 |
Homotopical algebra
- Quillen
- 1967
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Citation Context ...C diagSF, where −→ ∆C into a trivial cofibration (= ∆[0] u → Y q → ∆C is a (any) factorization of ∆[0] z0 injective weak equivalence) followed by a Kan fibration. By using the “small object argument” =-=[16, 10]-=- to find such a factorization, the trivial cofibration u is a colimit of a sequence of pushouts of coproducts of simplicial inclusions uk : Λk [n] ֒→ ∆[n], of kth-horn subcomplexes of standard n-simpl... |

360 | Homotopy limits, completions and localizations - Kan |

183 | Simplicial homotopy theory
- Goerss, Jardine
- 1999
(Show Context)
Citation Context ...ndieck construction. 2. Preliminaries and notations For the general background on 2-categories used in this paper, we refer to [2], [18] and [24], and on simplicial sets to [19], [17] and, mainly, to =-=[10]-=-. The simplicial category is denoted by ∆. It has as objects the ordered sets [n] = {0, . . .,n}, n ≥ 0, and as arrows the (weakly) monotone maps α : [n] → [m]. This category is generated by the direc... |

183 | Complexe cotangent et deformations - Illusie |

106 |
Handbook of Categorical Algebra 1: Basic Category Theory
- Borceux
- 1994
(Show Context)
Citation Context ... to the study of 2-diagrams of 2-categories by means of the higher Grothendieck construction. 2. Preliminaries and notations For the general background on 2-categories used in this paper, we refer to =-=[2]-=-, [18] and [24], and on simplicial sets to [19], [17] and, mainly, to [10]. The simplicial category is denoted by ∆. It has as objects the ordered sets [n] = {0, . . .,n}, n ≥ 0, and as arrows the (we... |

40 |
A splitting for the stable mapping class group
- Tillmann
- 1999
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Citation Context ... also shown its relevance in several other mathematical contexts such as in algebraic K-theory [13], conformal fields theory [27], or in the study of geometric structures on low-dimensional manifolds =-=[26]-=-. This work deals with questions such as, when do 2-functors induce homotopy equivalences or homotopy cartesian squares of classifying spaces? In this paper we mainly state and prove an extension of t... |

39 |
Homotopy colimits in the category of small categories
- Thomason
- 1979
(Show Context)
Citation Context ...o-called Grothendieck construction on a lax functor F : Co � Cat, for C a category, underlies our 2-categorical construction ∫ F, and recall that the C well-known Homotopy Colimit Theorem by Thomason =-=[25]-=- establishes that the C� � �� HOMOTOPY FIBRE SEQUENCES INDUCED BY 2-FUNCTORS 3 Grothendieck construction on a diagram of categories is actually a categorical model for the homotopy type of the homoto... |

32 |
Classifying spaces and spectral sequences”, Pub
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- 1968
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Citation Context ... K-groups are defined by taking homotopy groups of classifying spaces of certain categories, or that every CW-complex is homotopy equivalent to the classifying space of a small category. Later, Segal =-=[22]-=- extended the realization process to simplicial topological spaces. He observed that, if C is a topological category, then NC is, in a natural way, a simplicial space and he defines the classifying sp... |

26 | Simplicial matrices and the nerves of weak n-categories I: nerves of bicategories, Theory and Applications of Categories, 9
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Citation Context ...u ◦ v) ◦ Fw 1 ◦ Fv,w �� Fu ◦ F(v ◦ w) F u◦v,w F u,v◦w � F(u ◦ v ◦ w). A normal lax functor in which all constraints Fu,v are identities is precisely a 2functor. The geometric nerve of a 2-category C, =-=[5, 23]-=-, is defined to be the simplicial set (2) ∆C = laxFunc(−, C) : ∆ o → Set, with ∆nC = laxFunc([n], C) the set of normal lax functors x : [n] � C. Thus, for a 2-category C, the vertices of its geometric... |

23 |
Algebraic classification of equivariant homotopy 2-types
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Citation Context ...esponding homotopy category of 2-categories and the ordinary homotopy category of CW-complexes. By this correspondence, 2-groupoids correspond to spaces whose homotopy groups πn are trivial for n > 2 =-=[21]-=-, and from this point of view the use of 2-categories and their classifying spaces in homotopy theory goes back to Whitehead (1949) and Mac Lane-Whitehead (1950), since 2-groupoids with only one objec... |

19 |
Categorical structures’, in Handbook of Algebra Volume
- Street
- 1996
(Show Context)
Citation Context ...of 2-diagrams of 2-categories by means of the higher Grothendieck construction. 2. Preliminaries and notations For the general background on 2-categories used in this paper, we refer to [2], [18] and =-=[24]-=-, and on simplicial sets to [19], [17] and, mainly, to [10]. The simplicial category is denoted by ∆. It has as objects the ordered sets [n] = {0, . . .,n}, n ≥ 0, and as arrows the (weakly) monotone ... |

16 | Bisimplicial sets and the group-completion theorem. In Algebraic K-theory: connections with geometry and topology - Moerdijk - 1987 |

12 |
Categories ´ Fibrees ´ et Descente
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Citation Context ...ifying spaces of 2-categories and, in particular, of monoidal categories. To help motivate the reader, and to establish the setting for our discussions, let us briefly recall that it was Grothendieck =-=[12]-=- who first associated a simplicial set NC to a small category C, calling it its nerve. The set of n-simplices ⊔ NnC = C(x1, x0) × C(x2, x1) × · · · × C(xn, xn−1) (x0,...,xn)∈ObC n+1 consists of length... |

11 |
Discrete models for the category of Riemann surfaces
- Tillmann
- 1997
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Citation Context ...ll as in homotopy theory, the use of classifying spaces of 2-categories has also shown its relevance in several other mathematical contexts such as in algebraic K-theory [13], conformal fields theory =-=[27]-=-, or in the study of geometric structures on low-dimensional manifolds [26]. This work deals with questions such as, when do 2-functors induce homotopy equivalences or homotopy cartesian squares of cl... |

10 |
The algebra of oriented simplices
- Street
- 1987
(Show Context)
Citation Context ...u ◦ v) ◦ Fw 1 ◦ Fv,w �� Fu ◦ F(v ◦ w) F u◦v,w F u,v◦w � F(u ◦ v ◦ w). A normal lax functor in which all constraints Fu,v are identities is precisely a 2functor. The geometric nerve of a 2-category C, =-=[5, 23]-=-, is defined to be the simplicial set (2) ∆C = laxFunc(−, C) : ∆ o → Set, with ∆nC = laxFunc([n], C) the set of normal lax functors x : [n] � C. Thus, for a 2-category C, the vertices of its geometric... |

9 |
Closed categories, lax limits, and homotopy limits
- Gray
- 1980
(Show Context)
Citation Context ...S INDUCED BY 2-FUNCTORS 9 3. Homotopy cartesian squares induced by 2-functors Suppose F : B → C any given 2-functor between 2-categories B and C. For each object z of C, the homotopy fibre 2-category =-=[11]-=-, denoted by z//F, has objects the pairs (x, v), where x is an object of B and v : z → Fx is a morphism in C. A morphism (u, β) : (x, v) → (x ′ , v ′ ) consists of a morphism u : x → x ′ in B together... |

8 |
On the van Kampen theorem, Topology 5
- Artin, Mazur
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(Show Context)
Citation Context ...� �� �� �� �� �� � �� �� �� �� �� �� �� �� �� �� �� �� �� �� ���� �� �� �� �� � �� �� 8 A. M. CEGARRA Proof. Suppose α : F ⇒ F ′ : B � C is a lax transformation. There is a normal lax functor H : B × =-=[1]-=- � C making the diagram commutative (3) B × [0] ∼ = B 1×δ0 B × [1] 1×δ1 B × [0] ∼ = B �� �� �� F �� �� � H C, �� �� that carries a morphism in B × [1] of the form (x u → y, 1 → 0) : (x, 1) → (y, 0) to... |

8 |
The relationship between the diagonal and the bar constructions on a bisimplicial set, Topology and its applications, vol 153, number 1 (2005) D.-C. Cisinski, Invariance de la K-théorie par équivalences dérivées
- Cegarra, Remedios
(Show Context)
Citation Context ...h i ti+1,p−i−1, . . .,d h i tp,0), si(t0,p . . . , tp,0) = (s v i t0,p, . . .,s v 0 ti,p−i, s h i ti,p−i, . . . , s h i tp,0). For any bisimplicial set S, there is a natural weak homotopy equivalence =-=[7, 8]-=- (23) η : diag S → WS, the so-called Zisman comparison map (see [4]), which carries a p-simplex tp,p ∈ diag S to ( ηtp,p = (d h 1) p tp,p, (d h 2) p−1 d v 0tp,p, . . . , (d h m+1) p−m (d v 0) m tp,p, ... |

8 |
Higher Algebraic K-Theory:I
- Quillen
- 1971
(Show Context)
Citation Context ...do 2-functors induce homotopy equivalences or homotopy cartesian squares of classifying spaces? In this paper we mainly state and prove an extension of the relevant and well-known Quillen’s Theorem B =-=[17]-=- to 2-categories by showing, under reasonable necessary conditions, a category-theoretical interpretation of the homotopy fibres of the realization map BF : BB → BC, of a 2-functor F : B → C. More pre... |

7 |
les limites homotopiques de diagrammes homotopiquement cohérents
- Cordier, Sur
- 1987
(Show Context)
Citation Context ...d by composing F with the geometric nerve functor ∆ : 2Cat → Simpl.Set given by (2). Proof. We shall use the bar construction on a bisimplicial set WS, also called its “codiagonal” or “total complex” =-=[1, 4]-=-. Let us recall that the functor W : Bisimpl.Set → Simpl.Set is the right Kan extension along the ordinal sum functor or : ∆ × ∆ → ∆, ([p], [q]) ↦→ [p + 1 + q]. For any given bisimplicial set S, WS ca... |

7 |
Geometry of category of complexes and algebraic
- Hinich, Schechtman
- 1985
(Show Context)
Citation Context ...as crossed modules. But, as well as in homotopy theory, the use of classifying spaces of 2-categories has also shown its relevance in several other mathematical contexts such as in algebraic K-theory =-=[13]-=-, conformal fields theory [27], or in the study of geometric structures on low-dimensional manifolds [26]. This work deals with questions such as, when do 2-functors induce homotopy equivalences or ho... |

6 | On the geometry of 2-categories and their classifying spaces
- Bullejos, Cegarra
(Show Context)
Citation Context ...ollowing normalization equations hold: xi,i = 1xi, xi,j,j = 1xi,j = xi,i,j. Note that, if C is a category, regarded as a 2-category with all deformations identities, then ∆C = NC. As a main result in =-=[6]-=-, the following is proved: Theorem 2.2. For any 2-category C there is a natural homotopy equivalence |∆C| ≃ BC. Example 2.3. Let (M, ⊗) be a strict monoidal category, regarded as a 2-category with onl... |

5 | A model structure à la Thomason on 2-Cat
- Worytkiewicz, Hess, et al.
(Show Context)
Citation Context ... ⊗) is just the classifying space of the one object 2-category, with category of endomorphisms M, that it defines. The category 2Cat of small 2-categories and 2-functors has a Quillen model structure =-=[28]-=-, such that the functor C ↦→ BC induces an equivalence between the corresponding homotopy category of 2-categories and the ordinary homotopy category of CW-complexes. By this correspondence, 2-groupoi... |

3 |
Homotopy classification of graded Picard categories
- Cegarra, Khmaladze
(Show Context)
Citation Context ... ∆ ∫ ∫ F is a normal lax functor [p] � C CF, which can be described as a pair (y ′ ,x), where x : [p] → C is a functor, that is, a p-simplex of NC, and y ′ : [p] � F is a normal x-crossed lax functor =-=[9]-=-, that is, a family (27) y ′ = {y ′ i , y′ i,j , y′ i,j,k }0≤i≤j≤k≤p in which each y ′ i is an object of the 2-category Fxi, each y ′ i,j : y′ j → x∗ i,j y′ i is a morphism in Fxj, and the y ′ i,j,k :... |

2 |
The behaviour of the W-construction on the homotopy theory of bisimplicial sets
- Cegarra, Remedios
- 2007
(Show Context)
Citation Context ...h i ti+1,p−i−1, . . .,d h i tp,0), si(t0,p . . . , tp,0) = (s v i t0,p, . . .,s v 0 ti,p−i, s h i ti,p−i, . . . , s h i tp,0). For any bisimplicial set S, there is a natural weak homotopy equivalence =-=[7, 8]-=- (23) η : diag S → WS, the so-called Zisman comparison map (see [4]), which carries a p-simplex tp,p ∈ diag S to ( ηtp,p = (d h 1) p tp,p, (d h 2) p−1 d v 0tp,p, . . . , (d h m+1) p−m (d v 0) m tp,p, ... |

2 |
Simplicial objects in algebraic topology, viii+161 pp
- May
- 1992
(Show Context)
Citation Context ... means of the higher Grothendieck construction. 2. Preliminaries and notations For the general background on 2-categories used in this paper, we refer to [2], [18] and [24], and on simplicial sets to =-=[19]-=-, [17] and, mainly, to [10]. The simplicial category is denoted by ∆. It has as objects the ordered sets [n] = {0, . . .,n}, n ≥ 0, and as arrows the (weakly) monotone maps α : [n] → [m]. This categor... |