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14
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 36 (7 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:
Quasicategories vs Segal spaces
 IN CATEGORIES IN ALGEBRA, GEOMETRY AND MATHEMATICAL
, 2006
"... We show that complete Segal spaces and Segal categories are Quillen equivalent to quasicategories. ..."
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Cited by 35 (0 self)
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We show that complete Segal spaces and Segal categories are Quillen equivalent to quasicategories.
Pseudo limits, biadjoints, and pseudo algebras: categorical foundations of conformal field theory
 Mem. Amer. Math. Soc
"... The purpose of this paper is to work out the categorical basis for the foundations of Conformal Field Theory. The definition of Conformal Field Theory was outlined in Segal [45] and recently given in [24] and [25]. Concepts of 2category theory, such as versions of algebra, limit, colimit, and adjun ..."
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Cited by 28 (11 self)
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The purpose of this paper is to work out the categorical basis for the foundations of Conformal Field Theory. The definition of Conformal Field Theory was outlined in Segal [45] and recently given in [24] and [25]. Concepts of 2category theory, such as versions of algebra, limit, colimit, and adjunction, are necessary for this
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 18 (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 10 (4 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.
Weighted limits in simplicial homotopy theory
 Journal of Pure and Applied Algebra vol
, 2010
"... Abstract. We extend the theory of Quillen adjunctions by combining ideas of homotopical algebra and of enriched category theory. Our results describe how the formulas for homotopy colimits of Bousfield and Kan arise from general formulas describing the derived functor of the weighted colimit functo ..."
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Cited by 9 (0 self)
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Abstract. We extend the theory of Quillen adjunctions by combining ideas of homotopical algebra and of enriched category theory. Our results describe how the formulas for homotopy colimits of Bousfield and Kan arise from general formulas describing the derived functor of the weighted colimit functor. 1.
HOMOTOPY LIMITS FOR 2CATEGORIES
"... Abstract. We study homotopy limits for 2categories using the theory of Quillen model categories. In order to do so, we establish the existence of projective and injective model structures on diagram 2categories. Using this result, we describe the homotopical behaviour not only of conical limits bu ..."
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Cited by 3 (1 self)
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Abstract. We study homotopy limits for 2categories using the theory of Quillen model categories. In order to do so, we establish the existence of projective and injective model structures on diagram 2categories. Using this result, we describe the homotopical behaviour not only of conical limits but also of weighted limits for 2categories. Finally, homotopy limits are related to pseudolimits. 1. Quillen model structures in 2category theory The 2category of groupoids, functors, and natural transformations admits a model structure in which the weak equivalences are the equivalence of categories and the fibrations are the Grothendieck fibrations [1, 5, 13]. Similarly, the 2category of small categories, functors, and natural transformations admits a model structure in which the weak equivalences are the equivalence of categories and the fibrations are the isofibrations, which are functors satisfying a restricted version of the lifting condition for Grothendieck fibrations which involves only isomorphisms [13, 19]. Steve Lack has vastly
Reprints in Theory and Applications of Categories, No. 10, 2005. BASIC CONCEPTS OF ENRICHED CATEGORY THEORY
"... ii Acknowledgements for the Reprint: From the author: ..."
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