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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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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
MANIN PRODUCTS, KOSZUL DUALITY, LODAY ALGEBRAS AND DELIGNE CONJECTURE
"... Dedicated to JeanLouis Loday, on the occasion of his sixtieth birthday 1 Abstract. In this article we give a conceptual definition of Manin products in any category endowed with two coherent monoidal products. This construction can be applied to associative algebras, nonsymmetric operads, operads, ..."
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Dedicated to JeanLouis Loday, on the occasion of his sixtieth birthday 1 Abstract. In this article we give a conceptual definition of Manin products in any category endowed with two coherent monoidal products. This construction can be applied to associative algebras, nonsymmetric operads, operads, colored operads, and properads presented by generators and relations. These two products, called black and white, are dual to each other under Koszul duality functor. We study their properties and compute several examples of black and white products for operads. These products allow us to define natural operations on the chain complex defining cohomology theories. With these operations, we are able to prove that Deligne’s conjecture holds for a general class of operads and is not specific to the case of associative algebras. Finally, we prove generalized versions of a few conjectures raised by M. Aguiar and J.L. Loday related to the Koszul property of operads defined by black products. These operads provide infinitely many examples for this generalized Deligne’s conjecture.
OPERADS AND PROPS
, 2006
"... We review definitions and basic properties of operads, PROPs and algebras over these structures. ..."
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We review definitions and basic properties of operads, PROPs and algebras over these structures.
OPERADS OF NATURAL OPERATIONS I: LATTICE PATHS, BRACES AND HOCHSCHILD COCHAINS
"... Abstract. In this first paper of a series we study various operads of natural operations on Hochschild cochains and relationships between them. Contents ..."
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Abstract. In this first paper of a series we study various operads of natural operations on Hochschild cochains and relationships between them. Contents
The lattice path operad and Hochschild cochains
 CONTEMPORARY MATHEMATICS
"... We introduce two coloured operads in sets – the lattice path operad and a cyclic extension of it – closely related to iterated loop spaces and to universal operations on cochains. As main application we present a formal construction of an E2action (resp. framed E2action) on the Hochschild cochai ..."
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We introduce two coloured operads in sets – the lattice path operad and a cyclic extension of it – closely related to iterated loop spaces and to universal operations on cochains. As main application we present a formal construction of an E2action (resp. framed E2action) on the Hochschild cochain complex of an associative (resp. symmetric Frobenius) algebra.
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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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.
Contents
"... Abstract. We give an explicit construction of the free monoid in monoidal abelian categories when the monoidal product does not necessarily preserve coproducts. Then we apply it to several new monoidal categories that appeared recently in the theory of Koszul duality for operads and props. This give ..."
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Abstract. We give an explicit construction of the free monoid in monoidal abelian categories when the monoidal product does not necessarily preserve coproducts. Then we apply it to several new monoidal categories that appeared recently in the theory of Koszul duality for operads and props. This gives a conceptual explanation of the form of the free operad, free dioperad and free
LOCALIZATION OF ALGEBRAS OVER COLOURED OPERADS
"... We give sufficient conditions for homotopical localization functors to preserve algebras over coloured operads in monoidal model categories. Our approach encompasses a number of previous results about preservation of structures under localizations, such as loop spaces or infinite loop spaces, and p ..."
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We give sufficient conditions for homotopical localization functors to preserve algebras over coloured operads in monoidal model categories. Our approach encompasses a number of previous results about preservation of structures under localizations, such as loop spaces or infinite loop spaces, and provides new results of the same kind. For instance, under suitable assumptions, homotopical localizations preserve ring spectra (in the strict sense, not only up to homotopy), modules over ring spectra, and algebras over commutative ring spectra, as well as ring maps, module maps, and algebra maps. It is principally the treatment of module spectra and their maps that led us to the use of coloured operads (also called enriched multicategories) in this context.
OPERAD OF FORMAL HOMOGENEOUS SPACES AND BERNOULLI NUMBERS
, 708
"... Abstract. It is shown that for any morphism, φ: g → h, of Lie algebras the vector space underlying the Lie algebra h is canonically a ghomogeneous formal manifold with the action of g being highly nonlinear and twisted by Bernoulli numbers. This fact is obtained from a study of the 2coloured opera ..."
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Abstract. It is shown that for any morphism, φ: g → h, of Lie algebras the vector space underlying the Lie algebra h is canonically a ghomogeneous formal manifold with the action of g being highly nonlinear and twisted by Bernoulli numbers. This fact is obtained from a study of the 2coloured operad of formal homogeneous spaces whose minimal resolution gives a new conceptual explanation of both Ziv Ran’s JacobiBernoulli complex and FiorenzaManetti’s L∞algebra structure on the mapping cone of a morphism of two Lie algebras. All these constructions are iteratively extended to the case of a morphism of arbitrary L∞algebras. 1.