Results 1  10
of
15
Galois extensions of structured ring spectra
, 2005
"... We introduce the notion of a Galois extension of commutative Salgebras (E ∞ ring spectra), often localized with respect to a fixed homology theory. There are numerous examples, including some involving Eilenberg–MacLane spectra of commutative rings, real and complex topological Ktheory, Lubin–Tate ..."
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Cited by 19 (1 self)
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We introduce the notion of a Galois extension of commutative Salgebras (E ∞ ring spectra), often localized with respect to a fixed homology theory. There are numerous examples, including some involving Eilenberg–MacLane spectra of commutative rings, real and complex topological Ktheory, Lubin–Tate spectra and cochain Salgebras. We establish the main theorem of Galois theory in this generality. Its proof involves the notions of separable (and étale) extensions of commutative Salgebras, and the Goerss–Hopkins–Miller theory for E ∞ mapping spaces. We show that the global sphere spectrum S is separably closed (using Minkowski’s discriminant theorem), and we estimate the separable closure of its localization with respect to each of the Morava Ktheories. We also define Hopf–Galois extensions of commutative Salgebras, and study the complex cobordism spectrum MU as a common integral model for all of the local Lubin–Tate Galois extensions.
Operads and Chain Rules for the Calculus of Functors
"... Abstract. We study the structure possessed by the Goodwillie derivatives of a pointed homotopy functor of based topological spaces. These derivatives naturally form a bimodule over the operad consisting of the derivatives of the identity functor. We then use these bimodule structures to give a chain ..."
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Cited by 11 (0 self)
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Abstract. We study the structure possessed by the Goodwillie derivatives of a pointed homotopy functor of based topological spaces. These derivatives naturally form a bimodule over the operad consisting of the derivatives of the identity functor. We then use these bimodule structures to give a chain rule for higher derivatives in the calculus of functors, extending that of Klein and Rognes. This chain rule expresses the derivatives of FG as a derived composition product of the derivatives of F and G over the derivatives of the identity. There are two main ingredients in our proofs. Firstly, we construct new models for the Goodwillie derivatives of functors of spectra. These models allow for natural composition maps that yield operad and module structures. Then, we use a cosimplicial cobar construction to transfer this structure to functors of topological spaces. A form of Koszul duality for operads of spectra plays a key role in this. In a landmark series of papers, [16], [17] and [18], Goodwillie outlines his ‘calculus of homotopy functors’. Let F: C → D (where C and D are each either Top ∗, the category of pointed topological spaces, or Spec, the category of spectra) be a pointed homotopy functor. One of the things that Goodwillie does is associate with F a sequence of spectra, which are called the derivatives of F.
Postnikov extensions for ring spectra
, 2006
"... We give a functorial construction of kinvariants for ring spectra, and use these to classify extensions in the Postnikov tower of a ring spectrum. ..."
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Cited by 9 (3 self)
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We give a functorial construction of kinvariants for ring spectra, and use these to classify extensions in the Postnikov tower of a ring spectrum.
Topological Hochschild homology of Thom spectra which are . . .
, 2008
"... We identify the topological Hochschild homology (THH) of the Thom spectrum associated to an E ∞ classifying map X → BG, for G an appropriate group or monoid (e.g. U, O, and F). We deduce the comparison from the observation of McClure, Schwanzl, and Vogt that THH of a cofibrant commutative Salgebra ..."
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Cited by 8 (2 self)
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We identify the topological Hochschild homology (THH) of the Thom spectrum associated to an E ∞ classifying map X → BG, for G an appropriate group or monoid (e.g. U, O, and F). We deduce the comparison from the observation of McClure, Schwanzl, and Vogt that THH of a cofibrant commutative Salgebra (E ∞ ring spectrum) R can be described as an indexed colimit together with a verification that the LewisMay operadic Thom spectrum functor preserves indexed colimits. We prove a splitting result THH(Mf) ≃ Mf ∧BX+ which yields a convenient description of THH(MU). This splitting holds even when the classifying map f: X → BG is only a homotopy commutative A ∞ map, provided that the induced multiplication on Mf extends to an E ∞ ring structure; this permits us to recover Bokstedt’s calculation of THH(HZ).
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 3 (2 self)
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There are many interesting situations in which algebraic structure can be described
Topological AndréQuillen homology for cellular commutative
 Salgebras, Abhand. Math. Sem. Univ. Hamburg
"... Abstract. Topological AndréQuillen homology for commutative Salgebras was introduced by Basterra following work of Kriz, and has been intensively studied by several authors. In this paper we discuss it as a homology theory on CW commutative Salgebras and apply it to obtain results on minimal atom ..."
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Cited by 3 (2 self)
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Abstract. Topological AndréQuillen homology for commutative Salgebras was introduced by Basterra following work of Kriz, and has been intensively studied by several authors. In this paper we discuss it as a homology theory on CW commutative Salgebras and apply it to obtain results on minimal atomic plocal Salgebras which generalise those of Baker and May for plocal spectra and simply connected spaces. We exhibit some new examples of minimal atomic Salgebras.
Model Categories and Simplicial Methods
 CONTEMPORARY MATHEMATICS
"... There are many ways to present model categories, each with a different point of view. Here we’d like to treat model categories as a way to build and control resolutions. This an historical approach, as in his original and spectacular applications of model categories, Quillen used this technology as ..."
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Cited by 3 (0 self)
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There are many ways to present model categories, each with a different point of view. Here we’d like to treat model categories as a way to build and control resolutions. This an historical approach, as in his original and spectacular applications of model categories, Quillen used this technology as a way to construct resolutions in nonabelian settings; for example, in his work on the homology of commutative algebras [29], it was important to be very flexible with the notion of a free resolution of a commutative algebra. Similar issues arose in the paper on rational homotopy theory [31]. (This paper is the first place where the nowtraditional axioms of a model category are enunciated.) We’re going to emphasize the analog of projective resolutions, simply because these are the sort of resolutions most people see first. Of course, the theory is completely flexible and can work with injective resolutions as well. There are now any number of excellent sources for getting into the subject and since this monograph is not intended to be complete, perhaps the reader should have some of these nearby. For example, the paper of Dwyer and
Homotopy theory of modules over operads in symmetric spectra
 In preparation
, 2007
"... Abstract. We establish model category structures on algebras and modules over operads in symmetric spectra, and study when a morphism of operads induces a Quillen equivalence between corresponding categories of algebras (resp. modules) over operads. 1. ..."
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Cited by 3 (2 self)
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Abstract. We establish model category structures on algebras and modules over operads in symmetric spectra, and study when a morphism of operads induces a Quillen equivalence between corresponding categories of algebras (resp. modules) over operads. 1.
BAR CONSTRUCTIONS AND QUILLEN HOMOLOGY OF MODULES OVER OPERADS
, 802
"... There are many situations in algebraic topology, homotopy theory, and homological algebra in which operads parametrize interesting algebraic structures [10, 16, 27, 30, 35]. In many of these, there is a notion of abelianization or stabilization which provides a notion of homology [1, 2, 14, 42, 44]. ..."
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Cited by 1 (0 self)
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There are many situations in algebraic topology, homotopy theory, and homological algebra in which operads parametrize interesting algebraic structures [10, 16, 27, 30, 35]. In many of these, there is a notion of abelianization or stabilization which provides a notion of homology [1, 2, 14, 42, 44]. In these contexts,