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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.
ΓHomology of Commutative Rings and of E∞ Ring Spectra
, 1996
"... this paper we introduce the natural homology and cohomology theories for E1 ring spectra, which arise in problems of classification and extension of these generalized rings. The theory can be specialized to commutative rings in the ordinary algebraic sense by applying it to the corresponding Eilenbe ..."
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this paper we introduce the natural homology and cohomology theories for E1 ring spectra, which arise in problems of classification and extension of these generalized rings. The theory can be specialized to commutative rings in the ordinary algebraic sense by applying it to the corresponding EilenbergMac Lane spectra. This gives a completely new homology theory for commutative rings, which has applications in homotopy theory and probably elsewhere. It is analogous to, but distinct from, the homology theory for commutative rings constructed by Andr'e [1] and by Quillen [10] (after work by Grothendieck, Harrison, Lichtenbaum and Schlessinger). One would not expect our theory to specialize to Andr'e's, because polynomial algebras (which are acyclic in Andr'e/Quillen homology) do not give free E1 ring spectra, and there is no reason why they should be acyclic in the homology which is appropriate there. The first author's work in this field began in the late 1980's, and was announced at the Adams Memorial Conference in 1990. The second author has contributed in her Ph.D. thesis, written during 199294 [14], and since. There is also unpublished work of Waldhausen which, starting from different motivation and using an entirely different construction, obtains a homology theory for E1 algebras by using methods of stable homotopy theory. This theory has been developed for discrete commutative algebras by Schwede [13]. Their work and ours are far apart in concept and in detail, but it is likely that they will eventually be proved equivalent where they overlap. We have furthermore very recently received a draft paper by I. Kriz which discusses E1 cohomology. 1. The \Gammacotangent spectrum
Dedicated to the memory of Jon Beck
"... During the academic year 1966/67 a seminar on various aspects of category theory and its 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 ..."
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During the academic year 1966/67 a seminar on various aspects of category theory and its 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
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.
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
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
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
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.