## A full and faithful nerve for 2-categories (2005)

Venue: | Appl. Categ. Structures |

Citations: | 2 - 0 self |

### BibTeX

@ARTICLE{Bullejos05afull,

author = {M. Bullejos and E. Faro and V. Blanco},

title = {A full and faithful nerve for 2-categories},

journal = {Appl. Categ. Structures},

year = {2005}

}

### OpenURL

### Abstract

We prove that there is a full and faithful nerve functor defined on the category 2-Catlax of 2-categories and (normal) lax 2-functors. This functor extends the usual notion of nerve of a category and it coincides on objects with the so-called geometric nerve of a 2-category or of a 2-groupoid. We also show that (normal) lax 2-natural transformations produce homotopies of a special kind, and that two lax 2-functors from a 2-category to a 2-groupoid have homotopic nerves if and only if there is a lax 2-natural transformation between them. 1

### Citations

251 | Braided tensor categories - Joyal, Street - 1993 |

134 | Review of the elements of 2-categories - KELLY, STREET - 1974 |

97 |
The algebra of oriented simplexes
- Street
- 1987
(Show Context)
Citation Context ... natural transformation for identities. 2�� �� �� �� �� �� between them (forgetting for the most part the 3- and 2-category structures that 2-Catlax has). 2 Nerve functors for 2-categories Street in =-=[11]-=- defines a functor O : ∆ → ω-Cat to the category of (strict) ωcategories and strict functors. Then, he defines the nerve of an ω-category by means of this embedding. On the other hand, we can consider... |

26 | Simplicial matrices and the nerves of weak n-categories I: nerves of bicategories, Theory and Applications of Categories, 9
- Duskin
- 2001
(Show Context)
Citation Context ...f nerve of a 2-groupoid given by Moerdijk-Svensson in [9]. It should be noted also that the definition of the geometric nerve agrees also with the definition of nerve of a bicategory given by Duskin (=-=[4]-=-, p. 238). There is another nerve of 2-categories that is very useful. This can be deduced by an embedding of ∆ to BSset = Set ∆op ×∆ op , the category of bisimplicial sets. The ordinal sum functor +o... |

19 | Categorical structures’, in Handbook of Algebra Volume - Street - 1996 |

11 |
Adequate subcategories
- Isbell
- 1960
(Show Context)
Citation Context ...on objects as Ner(A) = A ( i(−), A ) and on arrows f : A → B in A via composition: Ner(f) = ¯ f is the simplicial map ¯fn Ner(A)n −−−→ Ner(B)n, fn(α) ¯ = f ◦ α. Using the term introduced by Isbell in =-=[5]-=- we call i : ∆ → A adequate (or simply call ∆ an adequate subcategory of A, if i is clear) if the above induced functor Ner : A → Sset is full and faithful (assuming i is itself full and faithful). Th... |

9 | adequacy, completeness and categories of algebras - Isbell, Subobjects - 1964 |

9 |
The equivariant Serre spectral sequence
- Moerdijk, Svensson
- 1993
(Show Context)
Citation Context ...ed for 2-groupoids and that it agrees with the . 3���� � � ���� � �� �� �� �� �� �� ����� �� �� �� �� ���� �� �� �� � �� �� �� �� �� definition of nerve of a 2-groupoid given by Moerdijk-Svensson in =-=[9]-=-. It should be noted also that the definition of the geometric nerve agrees also with the definition of nerve of a bicategory given by Duskin ([4], p. 238). There is another nerve of 2-categories that... |

7 |
les limites homotopiques de diagrammes homotopiquement cohérents
- Cordier, Sur
- 1987
(Show Context)
Citation Context ...th Yoneda, an embedding Tdec ◦y : ∆ ֒→ BSset. Let us write Ner ∗ : BSset → Sset for the corresponding nerve functor. The adjunction isomorphism implies that Ner ∗ = W, the Artin-Mazur codiagonal (see =-=[3]-=-). We can now use the usual nerve of categories to see a 2-category as a bisimplicial set in the following way, given a 2-category A let us write A0, A1 and A2 for the sets of zero-, one- and two-cell... |

6 | On the geometry of 2-categories and their classifying spaces
- Bullejos, Cegarra
(Show Context)
Citation Context ...ine the nerve of a category, one finds oneself using the fact that ∆, the simplicial category, whose objects are those particular posets that are the finite non-empty linear orders, 1 = [0] = {0},2 = =-=[1]-=- = {0 ≤ 1}, . . . (and whose arrows are functors or monotonic maps), can be regarded as a full subcategory of Cat. Then, given that the n-simplices of a category C are just the functors [n] → C, the f... |

5 | Categorical non-abelian cohomology and the Schreier theory of groupoids
- Blanco, Bullejos, et al.
(Show Context)
Citation Context ...uld be used to prove a representation theorem for the non abelian cohomology of groupoids. This cohomology, which generalizes the non abelian cohomology of groups given by Dedecker [7], is defined in =-=[2]-=- so as to classify non abelian extensions of groupoids by “Schreier invariants” in a way that generalizes the classical Schreier theory of groups, (see [16] and [1]). It was clear that in order to obt... |

5 |
On the equivariant 2-type of a G-space
- Bullejos, Cabello, et al.
(Show Context)
Citation Context ...sociating with each 2-category a simplicial set which captures the essential aspects of the 2-dimensionality of the structure of the 2-category, becomes useful and necessary in a variety of contexts (=-=[3]-=-, [8], [14], [18]). In this case, however, several notions of nerve have been proposed, each with its own applications, but none of them providing a full and faithful functor. In this paper we show th... |

3 | Review of the elements of 2-categories. volume 420 - Kelly, Street |

1 |
Théorie de Schereier supérieure
- Breen
- 1992
(Show Context)
Citation Context ...iven by Dedecker [7], is defined in [2] so as to classify non abelian extensions of groupoids by “Schreier invariants” in a way that generalizes the classical Schreier theory of groups, (see [16] and =-=[1]-=-). It was clear that in order to obtain the desired representation in terms of homotopy clases of simplicial maps, a nerve functor which was full and faithful was needed. Unfortunately this property w... |

1 |
Les foncteurs ExtII, H2 II et H2 II non abeliens
- Dedecker
- 1964
(Show Context)
Citation Context ... 2category which could be used to prove a representation theorem for the non abelian cohomology of groupoids. This cohomology, which generalizes the non abelian cohomology of groups given by Dedecker =-=[7]-=-, is defined in [2] so as to classify non abelian extensions of groupoids by “Schreier invariants” in a way that generalizes the classical Schreier theory of groups, (see [16] and [1]). It was clear t... |