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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:
Homotopy limits and colimits and enriched homotopy theory
, 2006
"... Abstract. Homotopy limits and colimits are homotopical replacements for the usual limits and colimits of category theory, which can be approached either using classical explicit constructions or the modern abstract machinery of derived functors. Our first goal is to explain both and show their equiv ..."
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Cited by 15 (2 self)
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Abstract. Homotopy limits and colimits are homotopical replacements for the usual limits and colimits of category theory, which can be approached either using classical explicit constructions or the modern abstract machinery of derived functors. Our first goal is to explain both and show their equivalence. Our second goal is to generalize this result to enriched categories and homotopy weighted limits, showing that the classical explicit constructions still give the right answer in the abstract sense, thus partially bridging the gap between classical homotopy theory and modern abstract homotopy theory. To do this we introduce a notion of “enriched homotopical categories”, which are more general than enriched model categories, but are still a good place to do enriched homotopy theory. This demonstrates that the presence of enrichment often simplifies rather than complicates matters, and goes some way toward achieving a better understanding of “the role of homotopy in homotopy theory.” Contents
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
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 10 (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.
Categorical Aspects of Equivariant Homotopy
 Applied Categorical Structures
, 1996
"... this paper we will use equivariant homotopy theory as a case study for how our new methods relate to old ones, and in the process will extend results of Dwyer and Kan to an enriched setting. We will try to illustrate the uses and interpretation of the categorical results that we have proved, as we b ..."
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Cited by 4 (4 self)
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this paper we will use equivariant homotopy theory as a case study for how our new methods relate to old ones, and in the process will extend results of Dwyer and Kan to an enriched setting. We will try to illustrate the uses and interpretation of the categorical results that we have proved, as we believe that these results will have significant applications in other parts of the subject. The second author would like to thank Ronnie Brown, Marek Golasi'nski and Andy Tonks for conversations on the area. This work was, in part, prepared for them in order to extend the use of crossed complexes to the equivariant case and in particular to extend the theory of classifying spaces to this setting. This work on equivariant crossed complexes will be the subject of a joint publication later on. The diagrams in this article were produced using Paul Taylor's "Diagrams" package. The authors would like to thank him for the use of these macros. 2 GSets and OrGdiagrams.
ON HOMOTOPY VARIETIES
, 2005
"... Abstract. Given an algebraic theory T, a homotopy Talgebra is a simplicial set where all equations from T hold up to homotopy. All homotopy Talgebras form a homotopy variety. We will give a characterization of homotopy varieties analogous to the characterization of varieties. We will also study ho ..."
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Abstract. Given an algebraic theory T, a homotopy Talgebra is a simplicial set where all equations from T hold up to homotopy. All homotopy Talgebras form a homotopy variety. We will give a characterization of homotopy varieties analogous to the characterization of varieties. We will also study homotopy models of limit theories which leads to homotopy locally presentable categories. These were recently considered by Simpson, Lurie, Toën and Vezzosi. 1.
Accessible categories and . . .
, 2007
"... Accessible categories have recently turned out to be useful in homotopy theory. This text is prepared as notes for a series of lectures at ..."
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Accessible categories have recently turned out to be useful in homotopy theory. This text is prepared as notes for a series of lectures at