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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 38 (5 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
Homotopy theory of modules over operads in symmetric spectra
, 2009
"... We establish model category structures on algebras and modules over operads in symmetric spectra, and study when a morphism of operads induces a Quillen equivalence between corresponding categories of algebras (resp. modules) over operads. ..."
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Cited by 16 (2 self)
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We establish model category structures on algebras and modules over operads in symmetric spectra, and study when a morphism of operads induces a Quillen equivalence between corresponding categories of algebras (resp. modules) over operads.
doi:10.1112/S0024611505015492 EQUIVARIANT UNIVERSAL COEFFICIENT AND KÜNNETH SPECTRAL SEQUENCES
"... In the nonequivariant context, universal coefficient and Künneth spectral sequences provide important tools for computing generalized homology and cohomology. In [5] Elmendorf, Kriz, Mandell and May (EKMM) construct examples of these types of spectral sequences for theories that come from ‘Salgeb ..."
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In the nonequivariant context, universal coefficient and Künneth spectral sequences provide important tools for computing generalized homology and cohomology. In [5] Elmendorf, Kriz, Mandell and May (EKMM) construct examples of these types of spectral sequences for theories that come from ‘Salgebras ’ (or, equivalently,