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22
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 24 (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:
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 17 (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
Resolution of coloured operads and rectification of homotopy algebras
 CONTEMPORARY MATHEMATICS
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Spaces of maps into classifying spaces for equivariant crossed complexes
 Indag. Math. (N.S
, 1997
"... Abstract. The results of a previous paper on the equivariant homotopy theory of crossed complexes are generalised from the case of a discrete group to general topological groups. The principal new ingredient necessary for this is an analysis of homotopy coherence theory for crossed complexes, using ..."
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Cited by 12 (7 self)
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Abstract. The results of a previous paper on the equivariant homotopy theory of crossed complexes are generalised from the case of a discrete group to general topological groups. The principal new ingredient necessary for this is an analysis of homotopy coherence theory for crossed complexes, using detailed results on the appropriate Eilenberg–Zilber theory, and of its relation to simplicial homotopy coherence. Again, our results give information not just on the homotopy classification of certain equivariant maps, but also on the weak equivariant homotopy type of the corresponding equivariant function spaces. Mathematics Subject Classifications (2001): 55P91, 55U10, 18G55. Key words: equivariant homotopy theory, classifying space, function space, crossed complex.
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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We show that complete Segal spaces and Segal categories are Quillen equivalent to quasicategories.
The BoardmanVogt resolution of operads in monoidal model categories, in preparation
"... Abstract. We extend the Wconstruction of Boardman and Vogt to operads of an arbitrary monoidal model category with suitable interval, and show that it provides a cofibrant resolution for wellpointed Σcofibrant operads. The standard simplicial resolution of Godement as well as the cobarbar chain ..."
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Cited by 11 (9 self)
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Abstract. We extend the Wconstruction of Boardman and Vogt to operads of an arbitrary monoidal model category with suitable interval, and show that it provides a cofibrant resolution for wellpointed Σcofibrant operads. The standard simplicial resolution of Godement as well as the cobarbar chain resolution are shown to be particular instances of this generalised Wconstruction.
Higher homotopy operations
"... Abstract. We provide a general definition of higher homotopy operations, encompassing most known cases, including higher Massey and Whitehead products, and long Toda brackets. These operations are defined in terms of the Wconstruction of Boardman and Vogt, applied to the appropriate diagram categor ..."
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Cited by 6 (3 self)
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Abstract. We provide a general definition of higher homotopy operations, encompassing most known cases, including higher Massey and Whitehead products, and long Toda brackets. These operations are defined in terms of the Wconstruction of Boardman and Vogt, applied to the appropriate diagram category; we also show how some classical families of polyhedra (including simplices, cubes, associahedra, and permutahedra) arise in this way. 1.
Higher homotopy operations and cohomology
"... Abstract. We explain how higher homotopy operations, defined topologically, may be identified under mild assumptions with (the last of) the DwyerKanSmith cohomological obstructions to rectifying homotopycommutative diagrams. ..."
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Cited by 3 (0 self)
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Abstract. We explain how higher homotopy operations, defined topologically, may be identified under mild assumptions with (the last of) the DwyerKanSmith cohomological obstructions to rectifying homotopycommutative diagrams.
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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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
A SURVEY OF (∞, 1)CATEGORIES
, 2006
"... Abstract. In this paper we give a summary of the comparisons between different definitions of socalled (∞,1)categories, which are considered to be models for ∞categories whose nmorphisms are all invertible for n> 1. They are also, from the viewpoint of homotopy theory, models for the homotopy ..."
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Abstract. In this paper we give a summary of the comparisons between different definitions of socalled (∞,1)categories, which are considered to be models for ∞categories whose nmorphisms are all invertible for n> 1. They are also, from the viewpoint of homotopy theory, models for the homotopy theory of homotopy theories. The four different structures, all of which are equivalent, are simplicial categories, Segal categories, complete Segal spaces, and quasicategories. 1.