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Two kinds of derived categories, Koszul duality, and comodulecontramodule correspondence
, 2009
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Equivariant cohomology and the MaurerCartan equation
, 2005
"... Let G be a compact, connected Lie group, acting smoothly on a manifold M. In their 1998 paper, GoreskyKottwitzMacPherson described a small Cartan model for the equivariant cohomology of M, quasiisomorphic to the standard (large) Cartan complex of equivariant differential forms. In this paper, w ..."
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Let G be a compact, connected Lie group, acting smoothly on a manifold M. In their 1998 paper, GoreskyKottwitzMacPherson described a small Cartan model for the equivariant cohomology of M, quasiisomorphic to the standard (large) Cartan complex of equivariant differential forms. In this paper, we construct an explicit cochain map from the small Cartan model into the large Cartan model, intertwining the (Sg ∗)invmodule structures and inducing an isomorphism in cohomology. The construction involves the solution of a remarkable inhomogeneous MaurerCartan equation. This solution has further applications to the theory of transgression in the Weil algebra, and to the ChevalleyKoszul theory of the
Origins and breadth of the theory of higher homotopies
, 2007
"... Higher homotopies are nowadays playing a prominent role in mathematics as well as in certain branches of theoretical physics. The purpose of the talk is to recall some of the connections between the past and the present developments. Higher homotopies were isolated within algebraic topology at least ..."
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Higher homotopies are nowadays playing a prominent role in mathematics as well as in certain branches of theoretical physics. The purpose of the talk is to recall some of the connections between the past and the present developments. Higher homotopies were isolated within algebraic topology at least as far back as the 1940’s. Prompted by the failure of the AlexanderWhitney multiplication of cocycles to be commutative, Steenrod developed certain operations which measure this failure in a coherent manner. Dold and Lashof extended Milnor’s classifying space construction to associative Hspaces, and a careful examination of this extension led Stasheff to the discovery of Anspaces and A∞spaces as notions which control the failure of associativity in a coherent way so that the classifying space construction can still be pushed through. Algebraic versions of higher homotopies have, as we all know, led Kontsevich eventually to the proof of the formality conjecture. Homological perturbation theory (HPT), in a simple form first isolated by Eilenberg and Mac Lane in the early 1950’s, has nowadays become a standard tool to handle algebraic incarnations of higher homotopies. A basic
Documenta Math. 243 Koszul Duality and Equivariant Cohomology
, 2005
"... Abstract. Let G be a topological group such that its homology H(G) with coefficients in a principal ideal domain R is an exterior algebra, generated in odd degrees. We show that the singular cochain functor carries the duality between Gspaces and spaces over BG to the Koszul duality between modules ..."
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Abstract. Let G be a topological group such that its homology H(G) with coefficients in a principal ideal domain R is an exterior algebra, generated in odd degrees. We show that the singular cochain functor carries the duality between Gspaces and spaces over BG to the Koszul duality between modules up to homotopy over H(G) and H ∗ (BG). This gives in particular a Cartantype model for the equivariant cohomology of a Gspace with coefficients in R. As another corollary, we obtain a multiplicative quasiisomorphism C ∗ (BG) → H ∗ (BG). A key step in the proof is to show that a differential Hopf algebra is formal in the category of A ∞ algebras provided that it is free over R and its homology an exterior algebra.
Relative . . . Equivariant De Rham COHOMOLOGY, AND KOSZUL DUALITY
, 2004
"... Equivariant de Rham theory for a general (not necessarily finite dimensional compact) Lie group is described in terms of a relative differential graded Extfunctor. Appropriate models for this functor, constructed via homological perturbation theory, yield small models for equivariant de Rham cohomo ..."
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Equivariant de Rham theory for a general (not necessarily finite dimensional compact) Lie group is described in terms of a relative differential graded Extfunctor. Appropriate models for this functor, constructed via homological perturbation theory, yield small models for equivariant de Rham cohomology. Suitably defined functors in equivariant de Rham theory yield a notion of duality which includes a version of Koszul duality. The latter is shown to rely on the extended functoriality of the differential graded Ext and Torfunctors.
Contents
, 2008
"... We relate a construction of Kadeishvili’s establishing an A∞structure on the homology of a differential graded algebra or more generally of an A ∞ algebra with certain constructions of Chen and Gugenheim. Thereafter we establish the links of ..."
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We relate a construction of Kadeishvili’s establishing an A∞structure on the homology of a differential graded algebra or more generally of an A ∞ algebra with certain constructions of Chen and Gugenheim. Thereafter we establish the links of
Equivariant cohomology over Lie groupoids and LieRinehart algebras
, 2009
"... Using the language and terminology of relative homological algebra, in particular that of derived functors, we introduce equivariant cohomology over a general LieRinehart algebra and equivariant de Rham cohomology over a locally trivial Lie groupoid in terms of suitably defined monads (also known a ..."
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Using the language and terminology of relative homological algebra, in particular that of derived functors, we introduce equivariant cohomology over a general LieRinehart algebra and equivariant de Rham cohomology over a locally trivial Lie groupoid in terms of suitably defined monads (also known as triples) and the associated standard constructions. This extends a characterization of equivariant de Rham cohomology in terms of derived functors developed earlier for the special case where the Lie groupoid is an ordinary Lie group, viewed as a Lie groupoid with a single object; in that theory over a Lie group, the ordinary BottDupontShulmanStasheff complex arises as an a posteriori object. We prove that, given a locally trivial Lie groupoid Ω and a smooth Ωmanifold f: M → BΩ over the space BΩ of objects of Ω, the resulting Ωequivariant de Rham theory of f boils down to the ordinary equivariant de Rham theory of a vertex manifold f −1 (q) relative to the vertex group Ω q q, for any vertex q in the space BΩ of objects of Ω; this implies that the equivariant de Rham cohomology introduced here coincides with the stack de Rham cohomology of the associated transformation groupoid whence this stack de Rham cohomology can be characterized as a relative derived functor. We introduce a notion of cone on a LieRinehart algebra and in particular that of cone on a Lie algebroid. This cone is an indispensable tool for the description of the requisite monads.
Equivariant cohomology via relative homological algebra
, 2008
"... We will explain how the appropriate categorical framework involving (co)monads and standard constructions provides categorical definitions of various relative derived functors including equivariant de Rham cohomology, LieRinehart cohomology, Poisson cohomology, etc. This leads, in particular, to a ..."
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We will explain how the appropriate categorical framework involving (co)monads and standard constructions provides categorical definitions of various relative derived functors including equivariant de Rham cohomology, LieRinehart cohomology, Poisson cohomology, etc. This leads, in particular, to a description of equivariant de Rham theory as a suitable differential Ext, in the sense of Eilenberg and Moore, over a category of modules relative to the group and the cone on its Lie algebra. Extending a decomposition lemma of Bott’s, we obtain a decomposition of the functor defining equivariant de Rham cohomology into two constituents, one constituent being a kind of Lie algebra cohomology functor and the other one the differentiable cohomology functor. For the case of a compact group, standard comparison arguments then lead to the familiar Weil and Cartan models and in particular explain why these models calculate the equivariant cohomology initially defined via a Borel construction. Pushing further the approach we arrive at a construction defining equivariant Lie algebroid or, somewhat more generally, equivariant LieRinehart cohomology. This kind of construction provides, perhaps, a framework to explore constrained hamiltonian systems, the variational bicomplex, the Noether theorems and related topics. Interesting issues arise, e.g. how to define the cone in the category of LieRinehart algebras or Lie algebroids. The question whether and how these constructions extend to Lie groupoids is momentarily open as is the question of existence of injective modules over Lie groupoids. The talk will illustrate the formal approach, with an emphasis on the development of these ideas (Cartan, Weil, Cartan