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104
Equivalences of monoidal model categories
 Algebr. Geom. Topol
, 2002
"... Abstract: We construct Quillen equivalences between the model categories of monoids (rings), modules and algebras over two Quillen equivalent model categories under certain conditions. This is a continuation of our earlier work where we established model categories of monoids, modules and algebras [ ..."
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Cited by 55 (12 self)
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Abstract: We construct Quillen equivalences between the model categories of monoids (rings), modules and algebras over two Quillen equivalent model categories under certain conditions. This is a continuation of our earlier work where we established model categories of monoids, modules and algebras [SS00]. As an application we extend the DoldKan equivalence to show that the model categories of simplicial rings, modules and algebras are Quillen equivalent to the associated model categories of connected differential graded rings, modules and algebras. We also show that our classification results from [SS] concerning stable model categories translate to any one of the known symmetric monoidal model categories of spectra. 1.
Resolution of coloured operads and rectification of homotopy algebras
 CONTEMPORARY MATHEMATICS
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Homotopy Batalin–Vilkovisky algebras
"... This paper provides an explicit cofibrant resolution of the operad encoding BatalinVilkovisky algebras. Thus it defines the notion of homotopy BatalinVilkovisky algebras with the required homotopy properties. To define this resolution we extend the theory of Koszul duality to operads and properads ..."
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Cited by 35 (4 self)
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This paper provides an explicit cofibrant resolution of the operad encoding BatalinVilkovisky algebras. Thus it defines the notion of homotopy BatalinVilkovisky algebras with the required homotopy properties. To define this resolution we extend the theory of Koszul duality to operads and properads that are defined by quadratic and linear relations. The operad encoding Batalin– Vilkovisky algebras is shown to be Koszul in this sense. This allows us to prove a PoincaréBirkhoffWitt Theorem for such an operad and to give an explicit small quasifree resolution for it. This particular resolution enables us to describe the deformation theory and homotopy theory of BValgebras and of homotopy BValgebras. We show that any topological conformal field theory carries a homotopy BValgebra structure which lifts the BValgebra structure on homology. The same result is proved for the singular chain complex of the double loop space of a topological space endowed with an action of the circle. We also prove the cyclic Deligne conjecture with this cofibrant resolution of the operad BV. We develop the general obstruction theory for algebras over the Koszul resolution of a properad and apply it to extend a conjecture of Lian–Zuckerman, showing that certain vertex algebras have an explicit homotopy BValgebra structure.
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 33 (10 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.
DEFORMATION THEORY OF REPRESENTATIONS OF PROP(ERAD)S I
"... Abstract. In this paper and its followup [MV08], we study the deformation theory of morphisms of properads and props thereby extending Quillen’s deformation theory for commutative rings to a nonlinear framework. The associated chain complex is endowed with an L∞algebra structure. Its MaurerCarta ..."
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Cited by 32 (7 self)
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Abstract. In this paper and its followup [MV08], we study the deformation theory of morphisms of properads and props thereby extending Quillen’s deformation theory for commutative rings to a nonlinear framework. The associated chain complex is endowed with an L∞algebra structure. Its MaurerCartan elements correspond to deformed structures, which allows us to give a geometric interpretation of these results.
OPERADS AND PROPS
, 2006
"... We review definitions and basic properties of operads, PROPs and algebras over these structures. ..."
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Cited by 20 (5 self)
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We review definitions and basic properties of operads, PROPs and algebras over these structures.
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.
The symmetrisation of noperads and compactification of real configuration spaces
 Adv. Math
"... It is well known that the forgetful functor from symmetric operads to nonsymmetric operads has a left adjoint Sym1 given by product with the symmetric group operad. It is also well known that this functor does not affect the category of algebras of the operad. From the point of view of the author’s ..."
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Cited by 19 (5 self)
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It is well known that the forgetful functor from symmetric operads to nonsymmetric operads has a left adjoint Sym1 given by product with the symmetric group operad. It is also well known that this functor does not affect the category of algebras of the operad. From the point of view of the author’s theory of higher operads, the nonsymmmetric operads are 1operads and Sym1 is the first term of the infinite series of left adjoint functors Symn, called symmetrisation functors, from noperads to symmetric operads with the property that the category of one object, one arrow,..., one (n − 1)arrow algebras of an noperad A is isomorphic to the category of algebras of Symn(A). In this paper we consider some geometrical and homotopical aspects of the symmetrisation of noperads. We follow Getzler and Jones and consider their decomposition of the FultonMacpherson operad of compactified real configuration spaces. We construct an noperadic counterpart of
Homotopy theory of modules over operads and nonΣ operads in monoidal model categories
 J. Pure Appl. Algebra
"... There are many interesting situations in which algebraic structure can be described ..."
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Cited by 18 (2 self)
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There are many interesting situations in which algebraic structure can be described