## On the geometry of 2-categories and their classifying spaces, K-Theory 29 (2003)

Citations: | 6 - 2 self |

### BibTeX

@MISC{Bullejos03onthe,

author = {M. Bullejos and A. M. Cegarra},

title = {On the geometry of 2-categories and their classifying spaces, K-Theory 29},

year = {2003}

}

### OpenURL

### Abstract

Abstract. In this paper we prove that realizations of geometric nerves are classifying spaces for 2-categories. This result is particularized to strict monoidal categories and it is also used to obtain a generalization of Quillen’s Theorem A.

### Citations

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Citation Context ...ion 2 is the main one and it includes the proof of Theorem 1. In Section 3 we prove Theorem 2. 1. Preliminaries and Notations For the general background on 2-categories used in this paper we refer to =-=[2, 15]-=-, and on simplicial sets to [16]. Throughout this paper all categories are assumed to be small. 2-Categories. A 2-category C is just a category enriched in the category of small categories. Then, C is... |

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Citation Context ...y of ℓFunc(I, C)relΘ whose objects are the functors from I to C. Observe that the coherence condition (2) for a deformation f between functors x, y : I → C reduces to the equality fστ = yσfτ ◦ fσxτ . =-=(3)-=- Let us now replace category I by a (directed and reflexive multi) graph G. By a morphism x : G → C we mean a (directed and reflexive multi-) graph morphism from G to the underlying graph of the under... |

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Citation Context ... of certain categories and also recall that classifying spaces of symmetric monoidal categories provide the most noteworthy examples of spaces with the extra structure required to define a Ω-spectrum =-=[18, 22]-=-. It was Grothendieck [10] who first associated a simplicial set NC to a small category C, calling this simplicial set the nerve of C. The psimplices of NC are diagrams in C of the form x0 → x1 → . . ... |

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Citation Context ...3 is the main one and it includes the proof of Theorem 1.1. In Section 4 we prove Theorem 1.2. 2. Preliminaries and Notations For the general background on 2-categories used in this paper we refer to =-=[1, 13]-=-, and on simplicial sets to [14]. Throughout this paper all categories are assumed to be small.214 M. BULLEJOS AND A. M. CEGARRA 2.1. 2-CATEGORIES A2-category C is just a category enriched in the cat... |

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Citation Context ...ategory C, calling this simplicial set the nerve of C. The psimplices of NC are diagrams in C of the form x0 → x1 → . . . xp−1 → xp. The classifying space of the category is the geometric realization =-=[19]-=- of its nerve, BC = |NC|. Later, Segal [21] 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 simplic... |

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Citation Context ...nerve of C. The psimplices of NC are diagrams in C of the form x0 → x1 → . . . xp−1 → xp. The classifying space of the category is the geometric realization [19] of its nerve, BC = |NC|. Later, Segal =-=[21]-=- 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... |

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Citation Context ...o our knowledge, the only comparison that has been given is in the case in which all arrows and deformations of C are invertible, that is, when C is a 2-groupoid. For this case, Moerdijk and Svensson =-=[20]-=- prove that the two classifying spaces associated as above to a 2-groupoid C are homotopically equivalent. In their proof, the restriction on C is essential, so that it can not be translated to genera... |

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Citation Context ... modules are defined as the classifying spaces of the categorical groups they define ([4, 14]). We finally note that geometric nerves of non-necessarily strict categorical groups have been treated in =-=[6, 7]-=- and, more generally, geometric nerves of arbitrary monoidal categories (even of bicategories) have been treated in [5] geometry.tex; 19/02/2002; 9:46; p.2122 BULLEJOS and CEGARRA 3. A sufficient con... |

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Citation Context ...] to 2-categories. In our theorem we replace the notion of homotopy fiber category of a functor by Gray’s notion of homotopy fiber 2-category y ′ //F of a 2-functor F : C → C ′ at an object y ′ ∈ C ′ =-=[9]-=- (see Section 1 for details). geometry.tex; 19/02/2002; 9:46; p.34 BULLEJOS and CEGARRA THEOREM 2. Let F : C → C ′ be a 2-functor. If the classifying space B(y ′ //F ) is contractible for every objec... |

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Citation Context ...ructure information. This process of taking classifying spaces leads to the homotopy theory of those categorical structures, whose interest is well recognized; for example, let us recall that Quillen =-=[13]-=- defines a higher algebraic K-theory by taking homotopy groups of the classifying spaces of certain categories and also recall that classifying spaces of symmetric monoidal categories provide the most... |

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Citation Context ...itions: • for any object i ∈ I, xIdi = Idxi , • for any arrow τ : i → j in I, xτ,Idi = xτ = xIdj,τ , • for any triple of composable arrows i γ −→j τ −→k σ −→ℓ in I, xσ,τ xγ ◦ xστ,γ = xσxτ,γ ◦ xσ,τγ . =-=(1)-=- By a deformation f : x ⇒ y, between lax-functors x, y : I → C, we mean a (normalized) lax-natural transformation. Then, f consists of a pair of maps which assign to each object i ∈ I an arrow fi : xi... |

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Citation Context ...( Z(x) = (d h 1) n x, (d h 2) n−1 d v 0x, . . . , (d h i+1) n−i (d v 0) i x, . . . , (d v 0) n ) x . This simplicial map induces a homotopy equivalence on realizations |Z| : |diag S| ∼ −→|T (S)| (cf. =-=[8]-=-). The simplicial set T N C is defined as T N C = T(NNC). We then have the canonical map which induces a homotopy equivalence Z = ZNNC : diagNNC −→ T N C, (9) |Z| : BC ∼ �� N |T C|, therefore the simp... |

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Citation Context ...tions in C, x0 x0,1 ⇓ f01 x1 x1,2 ⇓ f12 x2 xn−1 xn−1,n ⇓ fn−1n xn . x ′ 0,1 x ′ 1,2 x ′ n−1,n geometry.tex; 19/02/2002; 9:46; p.910 BULLEJOS and CEGARRA Therefore we can make the identification (cf. =-=[12]-=-) NnC = ∐ C(x0, x1) × C(x1, x2) × · · · × C(xn−1, xn) x0,...,xn∈C and N0C = Ob(C), as a discrete category. After the above identification, the face and degeneracy functors are defined in the standard ... |

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Citation Context ...e BC of the 2-category C is defined as that of the topological category BC. However, there is another convincing way of associating a space to a 2-category C. This way goes through what Duskin called =-=[5]-=- the geometric nerve ∆C of the 2-category and it was developed, among others, by Street [23] and Duskin himself. This geometric nerve ∆C is a simplicial set, which encodes the entire 2-categorical str... |

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R.: Homotopy classification of categorical torsors
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Citation Context ... modules are defined as the classifying spaces of the categorical groups they define ([4, 14]). We finally note that geometric nerves of non-necessarily strict categorical groups have been treated in =-=[6, 7]-=- and, more generally, geometric nerves of arbitrary monoidal categories (even of bicategories) have been treated in [5] geometry.tex; 19/02/2002; 9:46; p.2122 BULLEJOS and CEGARRA 3. A sufficient con... |

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Sur les complexes crois´s d’homotopie associés a quelques espaces filtrés
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Citation Context ...categorical groups are equivalent to crossed modules (see [15] for example) and that classifying spaces of crossed modules are defined as the classifying spaces of the categorical groups they define (=-=[4, 14]-=-). We finally note that geometric nerves of non-necessarily strict categorical groups have been treated in [6, 7] and, more generally, geometric nerves of arbitrary monoidal categories (even of bicate... |

1 |
The algebra of oriented simplices, J. of Pure and Appl. Algebra 49(3
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Citation Context ...is another convincing way of associating a space to a 2-category C. This way goes through what Duskin called [5] the geometric nerve ∆C of the 2-category and it was developed, among others, by Street =-=[23]-=- and Duskin himself. This geometric nerve ∆C is a simplicial set, which encodes the entire 2-categorical structure of C, whose simplices have the following pleasing geometrical description. The vertic... |