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Homotopy Coherent Category Theory
, 1996
"... this paper we try to lay some of the foundations of such a theory of categories `up to homotopy' or more exactly `up to coherent homotopies'. The method we use is based on earlier work on: ..."
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Cited by 22 (6 self)
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this paper we try to lay some of the foundations of such a theory of categories `up to homotopy' or more exactly `up to coherent homotopies'. The method we use is based on earlier work on:
Theory and Applications of Crossed Complexes
, 1993
"... ... L are simplicial sets, then there is a strong deformation retraction of the fundamental crossed complex of the cartesian product K \Theta L onto the tensor product of the fundamental crossed complexes of K and L. This satisfies various sideconditions and associativity/interchange laws, as for t ..."
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Cited by 15 (2 self)
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... L are simplicial sets, then there is a strong deformation retraction of the fundamental crossed complex of the cartesian product K \Theta L onto the tensor product of the fundamental crossed complexes of K and L. This satisfies various sideconditions and associativity/interchange laws, as for the chain complex version. Given simplicial sets K 0 ; : : : ; K r , we discuss the rcube of homotopies induced on (K 0 \Theta : : : \Theta K r ) and show these form a coherent system. We introduce a definition of a double crossed complex, and of the associated total (or codiagonal) crossed complex. We introduce a definition of homotopy colimits of diagrams of crossed complexes. We show that the homotopy colimit of crossed complexes can be expressed as the
On the geometry of 2categories and their classifying spaces, KTheory 29
, 2003
"... Abstract. In this paper we prove that realizations of geometric nerves are classifying spaces for 2categories. This result is particularized to strict monoidal categories and it is also used to obtain a generalization of Quillen’s Theorem A. ..."
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Cited by 6 (2 self)
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Abstract. In this paper we prove that realizations of geometric nerves are classifying spaces for 2categories. This result is particularized to strict monoidal categories and it is also used to obtain a generalization of Quillen’s Theorem A.
Flexible sheaves
, 1996
"... This is an unfinished explanation of the notion of “flexible sheaf”, that is a homotopical notion of sheaf of topological spaces (homotopy types) over a site. See “Homotopy over the complex numbers and generalized de Rham cohomology ” (submitted to proceedings of the Taniguchi conference on vector b ..."
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Cited by 3 (0 self)
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This is an unfinished explanation of the notion of “flexible sheaf”, that is a homotopical notion of sheaf of topological spaces (homotopy types) over a site. See “Homotopy over the complex numbers and generalized de Rham cohomology ” (submitted to proceedings of the Taniguchi conference on vector bundles, preprint of Toulouse 3, and also available at my homepage 2) for a more detailed introduction, and also for a further development of the ideas presented here. The present paper was finished in December 1993 while I was visiting MIT. Since writing this, I have realized that the theory sketched here is essentially equivalent to JardineIllusie’s theory of presheaves of topological spaces (although they talk about presheaves of simplicial sets which is the same thing). This is the point of view adopted in “Homotopy over the complex numbers and generalized de Rham cohomology”. 1 Added in August 1996: I am finally sending this to “Duke eprints ” because it now seems that there may be several useful features of the treatment given here. First of all, the direct construction of the homotopysheafification (by doing a certain operation n+2 times) seems to be useful, for example I need
DÉCALAGE AND KAN’S SIMPLICIAL LOOP GROUP FUNCTOR
"... Abstract. Given a bisimplicial set, there are two ways to extract from it a simplicial set: the diagonal simplicial set and the less well known total simplicial set of Artin and Mazur. There is a natural comparison map between these simplicial sets, and it is a theorem due to Cegarra and Remedios an ..."
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Cited by 2 (1 self)
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Abstract. Given a bisimplicial set, there are two ways to extract from it a simplicial set: the diagonal simplicial set and the less well known total simplicial set of Artin and Mazur. There is a natural comparison map between these simplicial sets, and it is a theorem due to Cegarra and Remedios and independently Joyal and Tierney, that this comparison map is a weak homotopy equivalence for any bisimplicial set. In this paper we will give a new, elementary proof of this result. As an application, we will revisit Kan’s simplicial loop group functor G. We will give a simple formula for this functor, which is based on a factorization, due to Duskin, of Eilenberg and Mac Lane’s classifying complex functor W. We will give a new, short, proof of Kan’s result that the unit map for the adjunction G ⊣ W is a weak homotopy equivalence for reduced simplicial sets. 1.
A full and faithful nerve for 2categories
 Appl. Categ. Structures
, 2005
"... We prove that there is a full and faithful nerve functor defined on the category 2Catlax of 2categories and (normal) lax 2functors. This functor extends the usual notion of nerve of a category and it coincides on objects with the socalled geometric nerve of a 2category or of a 2groupoid. We al ..."
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We prove that there is a full and faithful nerve functor defined on the category 2Catlax of 2categories and (normal) lax 2functors. This functor extends the usual notion of nerve of a category and it coincides on objects with the socalled geometric nerve of a 2category or of a 2groupoid. We also show that (normal) lax 2natural transformations produce homotopies of a special kind, and that two lax 2functors from a 2category to a 2groupoid have homotopic nerves if and only if there is a lax 2natural transformation between them. 1
HOMOTOPY FIBRE SEQUENCES INDUCED BY 2FUNCTORS
, 909
"... Abstract. This paper contains some contributions to the study of the relationship between 2categories 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 2functors. Mathematical Subject Classif ..."
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Abstract. This paper contains some contributions to the study of the relationship between 2categories 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 2functors. Mathematical Subject Classification: 18D05, 55P15, 18F25.