Results 1 
9 of
9
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: ..."
Abstract

Cited by 22 (6 self)
 Add to MetaCart
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:
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 ..."
Abstract

Cited by 18 (8 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 15 (2 self)
 Add to MetaCart
... 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
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. ..."
Abstract

Cited by 11 (0 self)
 Add to MetaCart
We show that complete Segal spaces and Segal categories are Quillen equivalent to quasicategories.
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. ..."
Abstract

Cited by 6 (2 self)
 Add to MetaCart
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.
An Australian conspectus of higher categories

, 2004
"... Much Australian work on categories is part of, or relevant to, the development of higher categories and their theory. In this note, I hope to describe some of the origins and achievements of our efforts that they might perchance serve as a guide to the development of aspects of higherdimensional wo ..."
Abstract

Cited by 6 (0 self)
 Add to MetaCart
Much Australian work on categories is part of, or relevant to, the development of higher categories and their theory. In this note, I hope to describe some of the origins and achievements of our efforts that they might perchance serve as a guide to the development of aspects of higherdimensional work. I trust that the somewhat autobiographical style will add interest rather than be a distraction. For so long I have felt rather apologetic when describing how categories might be helpful to other mathematicians; I have often felt even worse when mentioning enriched and higher categories to category theorists. This is not to say that I have doubted the value of our work, rather that I have felt slowed down by the continual pressure to defend it. At last, at this meeting, I feel justified in speaking freely amongst motivated researchers who know the need for the subject is well established. Australian Category Theory has its roots in homology theory: more precisely, in the treatment of the cohomology ring and the Künneth formulas in the book by Hilton and Wylie [HW]. The first edition of the book had a mistake concerning the cohomology ring of a product. The Künneth formulas arise from splittings of the natural short exact sequences
Reprints in Theory and Applications of Categories, No. 13, 2005, pp. 1–13. ON THE OPERADS OF J.P. MAY
, 1972
"... asked that it be expanded to study the relation of operads to clubs. The author found this too daunting a task at a busy time and the manuscript was never published. Reading through the manuscript now, more than thirty years later, elicits two strong impressions. First, the treatment is very complet ..."
Abstract
 Add to MetaCart
asked that it be expanded to study the relation of operads to clubs. The author found this too daunting a task at a busy time and the manuscript was never published. Reading through the manuscript now, more than thirty years later, elicits two strong impressions. First, the treatment is very complete: the only item not discussed in detail is the coherence of the monoidal structure given by the functor T ◦ S on [P, V]. Secondly, it was done—for instance in proving the associativity (R ◦ T) ◦ S ∼ = R ◦ (T ◦ S)—with bare hands. Today one could argue as follows, using universal properties; the author learned this approach from Aurelio Carboni. P op, which is in fact isomorphic to P, is the free symmetric monoidal category on 1. So to give an object of [P, V], or a functor T:1 → [P, V], is equally to give a strong monoidal functor P op → [P, V], where the latter has the convolution monoidal structure ⊗; this is the strong monoidal functor sending m to the tensor power T m = T ⊗T ⊗...⊗T. By Theorem 5.1 of [12], this is equally to give a cocontinuous strong monoidal functor T ′:[P, V] → [P, V]; this is the left Kan extension −◦T,andT is recovered from T ′ as T ′ (J) =J ◦ T. Now the desired associativity ( − ◦T) ◦ S ∼ = −◦(T ◦ S) isjustthe associativity of these cocontinuous strong monoidal functors. I am grateful to my colleagues Lack, Street, and Wood for suggesting this article for the TAC Reprint series, and to Flora Armaghanian for producing the LaTeX version. 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 ..."
Abstract
 Add to MetaCart
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
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 ..."
Abstract
 Add to MetaCart
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.