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ANDRÉQUILLEN COHOMOLOGY AND RATIONAL HOMOTOPY OF FUNCTION SPACES
, 2003
"... Abstract. We develop a simple theory of AndréQuillen cohomology for commutative differential graded algebras over a field of characteristic zero. We then relate it to the homotopy groups of function spaces and spaces of homotopy selfequivalences of rational nilpotent CWcomplexes. This puts certai ..."
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Abstract. We develop a simple theory of AndréQuillen cohomology for commutative differential graded algebras over a field of characteristic zero. We then relate it to the homotopy groups of function spaces and spaces of homotopy selfequivalences of rational nilpotent CWcomplexes. This puts certain results of Sullivan in a more conceptual framework. 1.
Generalized AndréQuillen cohomology
 J. Homotopy Relat. Struct
"... Abstract. We explain how the approach of André and Quillen to defining cohomology and homology as suitable derived functors extends to generalized (co)homology theories, and how this identification may be used to study the relationship between them. ..."
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Abstract. We explain how the approach of André and Quillen to defining cohomology and homology as suitable derived functors extends to generalized (co)homology theories, and how this identification may be used to study the relationship between them.
Cohomology of diagrams of algebras
, 2008
"... We consider cohomology of diagrams of algebras by Beck’s approach, using comonads. We then apply this theory to computing the cohomology of Ψrings. Our main result is that there is a spectral sequence connecting the cohomology of the diagram of an algebra to the cohomology of the underlying algebra ..."
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We consider cohomology of diagrams of algebras by Beck’s approach, using comonads. We then apply this theory to computing the cohomology of Ψrings. Our main result is that there is a spectral sequence connecting the cohomology of the diagram of an algebra to the cohomology of the underlying algebra. 1
Locally complete intersection
"... homomorphisms and a conjecture of Quillen on the vanishing of cotangent homology ..."
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homomorphisms and a conjecture of Quillen on the vanishing of cotangent homology
Contents
, 810
"... and it is closed under taking arbitrary intersections in a manifold. A derived manifold is a space together with a sheaf of local C ∞rings that is obtained by patching together homotopy zerosets of smooth functions on Euclidean spaces. We show that derived manifolds come equipped with a stable nor ..."
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and it is closed under taking arbitrary intersections in a manifold. A derived manifold is a space together with a sheaf of local C ∞rings that is obtained by patching together homotopy zerosets of smooth functions on Euclidean spaces. We show that derived manifolds come equipped with a stable normal bundle and can be imbedded into Euclidean space. We define a cohomology theory called derived cobordism, and use a PontrjaginThom argument to show that the derived cobordism theory is isomorphic to the classical cobordism theory. This allows us to define fundamental classes in cobordism for all derived manifolds. In particular, the intersection A ∩ B of submanifolds A, B ⊂ X exists on the categorical level in our theory, and a cup product formula [A] ⌣ [B] = [A ∩ B] holds, even if the submanifolds are not transverse. One can thus consider the theory of derived manifolds as a categorification of intersection theory.
Deformations via Simplicial Deformation Complexes
, 2008
"... There has long been a philosophy that every deformation problem in characteristic 0 should give rise to a differential graded Lie algebra (DGLA). This DGLA should not ..."
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There has long been a philosophy that every deformation problem in characteristic 0 should give rise to a differential graded Lie algebra (DGLA). This DGLA should not
Deformations via Simplicial Deformation Complexes
, 2008
"... There has long been a philosophy that every deformation problem in characteristic 0 should give rise to a differential graded Lie algebra (DGLA). This DGLA should not ..."
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There has long been a philosophy that every deformation problem in characteristic 0 should give rise to a differential graded Lie algebra (DGLA). This DGLA should not
SEMINAR ON TRIPLES AND CATEGORICAL HOMOLOGY THEORY
"... applications was held at the Forschungsinstitut für Mathemtik, ETH, Zürich. This volume is a report on those lectures and discussions which concentrated on two closely related topics of special interest: namely a) on the concept of “triple ” or standard construction with special reference to the ass ..."
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applications was held at the Forschungsinstitut für Mathemtik, ETH, Zürich. This volume is a report on those lectures and discussions which concentrated on two closely related topics of special interest: namely a) on the concept of “triple ” or standard construction with special reference to the associated “algebras”, and b) on homology theories in general categories, based upon triples and simplicial methods. In some respects this report is unfinished and to be continued in later volumes; thus in particular the interpretation of the general homology concept on the functor level (as satellites of Kan extensions), is only sketched in a short survey. I wish to thank all those who have contributed to the seminar; the authors for their lectures and papers, and the many participants for their active part in the discussions. Special thanks are due to Myles Tierney and Jon Beck for their efforts in collecting the material for this volume. B. Eckmann 2 Preface to the reprint This volume was the culmination of a very exciting year at the Forsch (as we called it)
AndréQuillen Homology for Simplicial Algebras and Ring Spectra
, 2008
"... We discuss AndréQuillen homology for simplicial algebras and algebras over simplicial algebras, extending the classical notion for rings. This extension is also discussed by Goerss and Hopkins [22], however our statements are proven in a more explicit way. We are then further able to construct spec ..."
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We discuss AndréQuillen homology for simplicial algebras and algebras over simplicial algebras, extending the classical notion for rings. This extension is also discussed by Goerss and Hopkins [22], however our statements are proven in a more explicit way. We are then further able to construct spectral sequences for AndréQuillen homology like the spectral sequence for the indecomposables or the Fundamental spectral sequence according to Quillen. The AndréQuillen homology for algebras constructed as cellular complexes is calculated and we apply this homology theory to obtain notions of atomic and nuclear algebras, thus extending results from Baker and May. We define the notion of istable algebras and are able to give a comparison theorem between AndréQuillen homology, stabilisation and Γhomology for istable algebras up to degree i. In the second part of the thesis we discuss topological AndréQuillen homology and extend certain results by Gilmour about cellular complexes
AN EQUIVARIANT SMASH SPECTRAL SEQUENCE AND AN UNSTABLE BOX PRODUCT
"... Abstract. Let G be a finite group. We construct a first quadrant spectral sequence which converges to the equivariant homotopy groups of the smash product X ∧ Y for suitably connected, based GCW complexes X and Y.The E 2 term is described in terms of a tensor product functor of equivariant Πalgebr ..."
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Abstract. Let G be a finite group. We construct a first quadrant spectral sequence which converges to the equivariant homotopy groups of the smash product X ∧ Y for suitably connected, based GCW complexes X and Y.The E 2 term is described in terms of a tensor product functor of equivariant Πalgebras. A homotopy version of the nonequivariant Künneth theorem and the equivariant suspension theorem of Lewis are both shown to be special cases of the corner of the spectral sequence. We also give a categorical description of this tensor product functor which is analogous to the description in equivariant stable homotopy theory of the box product of Mackey functors. For this reason, the tensor product functor deserves to be called an “unstable box product”. 1.