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23
Moduli Spaces of Commutative Ring Spectra
, 2003
"... Let E be a homotopy commutative ring spectrum, and suppose the ring of cooperations E#E is flat over E# . We wish to address the following question: given a commutative E# algebra A in E#Ecomodules, is there an E# ring spectrum X with E#X = A as comodule algebras? We will formulate this as ..."
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Let E be a homotopy commutative ring spectrum, and suppose the ring of cooperations E#E is flat over E# . We wish to address the following question: given a commutative E# algebra A in E#Ecomodules, is there an E# ring spectrum X with E#X = A as comodule algebras? We will formulate this as a moduli problem, and give a way  suggested by work of Dwyer, Kan, and Stover  of dissecting the resulting moduli space as a tower with layers governed by appropriate AndreQuillen cohomology groups. A special case is A = E#E itself. The final section applies this to discuss the LubinTate or Morava spectra En .
Moduli problems for structured ring spectra
 DANIEL DUGGER AND BROOKE
, 2005
"... In this document we make good on all the assertions we made in the previous paper “Moduli spaces of commutative ring spectra ” [20] wherein we laid out a theory a moduli spaces and problems for the existence and uniqueness of E∞ring spectra. In that paper, we discussed the the HopkinsMiller theore ..."
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Cited by 11 (0 self)
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In this document we make good on all the assertions we made in the previous paper “Moduli spaces of commutative ring spectra ” [20] wherein we laid out a theory a moduli spaces and problems for the existence and uniqueness of E∞ring spectra. In that paper, we discussed the the HopkinsMiller theorem on the LubinTate or Morava spectra En; in particular, we showed how to prove that the moduli space of all E ∞ ring spectra X so that (En)∗X ∼ = (En)∗En as commutative (En) ∗ algebras had the homotopy type of BG, where G was an appropriate variant of the Morava stabilizer group. This is but one point of view on these results, and the reader should also consult [3], [38], and [41], among others. A point worth reiterating is that the moduli problems here begin with algebra: we have a homology theory E ∗ and a commutative ring A in E∗E comodules and we wish to discuss the homotopy type of the space T M(A) of all E∞ring spectra so that E∗X ∼ = A. We do not, a priori, assume that T M(A) is nonempty, or even that there is a spectrum X so that E∗X ∼ = A as comodules.
The fiber of functors between categories of algebras
"... Abstract. We investigate the fiber of a functor F: C → D between sketchable categories of algebras over an object D ∈ D from two points of view: characterizing its classifying space as a universal Aut(D)space, and parametrizing its objects in cohomological terms. ..."
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Abstract. We investigate the fiber of a functor F: C → D between sketchable categories of algebras over an object D ∈ D from two points of view: characterizing its classifying space as a universal Aut(D)space, and parametrizing its objects in cohomological terms.
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 4 (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
Higher homotopy operations and cohomology
"... Abstract. We explain how higher homotopy operations, defined topologically, may be identified under mild assumptions with (the last of) the DwyerKanSmith cohomological obstructions to rectifying homotopycommutative diagrams. ..."
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Abstract. We explain how higher homotopy operations, defined topologically, may be identified under mild assumptions with (the last of) the DwyerKanSmith cohomological obstructions to rectifying homotopycommutative diagrams.
A∞–monads and completion
, 2007
"... ABSTRACT. Given an operad A of topological spaces, we consider Amonads in a topological category C. When A is an A∞operad, any Amonad K: C → C can be thought of as a monad up to coherent homotopies. We define the completion functor with respect to an A∞monad and prove that it is an A∞monad itse ..."
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ABSTRACT. Given an operad A of topological spaces, we consider Amonads in a topological category C. When A is an A∞operad, any Amonad K: C → C can be thought of as a monad up to coherent homotopies. We define the completion functor with respect to an A∞monad and prove that it is an A∞monad itself. In a second part, we construct a combinatorial model of an A∞operad which acts simplicially on the cobar resolution (not just its total space) of a simplicial set with respect to a ring R. 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.
Moduli spaces of homotopy theory
 Contemp. Math
"... Abstract. The moduli spaces refered to are topological spaces whose path components parametrize homotopy types. Such objects have been studied in two separate contexts: rational homotopy types, in the work of several authors in the late 1970’s; and general homotopy types, in the work of DwyerKan an ..."
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Abstract. The moduli spaces refered to are topological spaces whose path components parametrize homotopy types. Such objects have been studied in two separate contexts: rational homotopy types, in the work of several authors in the late 1970’s; and general homotopy types, in the work of DwyerKan and their collaborators. We here explain the two approaches, and show how they may be related to each other. 1.
COMPARING HOMOTOPY CATEGORIES
, 2006
"... Abstract. Given a suitable functor T: C → D between model categories, we define a long exact sequence relating the homotopy groups of any X ∈ C with those of TX, and use this to describe an obstruction theory for lifting an object G ∈ D to C. Examples include finding spaces with given homology or ho ..."
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Abstract. Given a suitable functor T: C → D between model categories, we define a long exact sequence relating the homotopy groups of any X ∈ C with those of TX, and use this to describe an obstruction theory for lifting an object G ∈ D to C. Examples include finding spaces with given homology or homotopy groups. A number of fundamental problems in algebraic topology can be described as measuring the extent to which a given functor T: C → D between model categories induces an equivalence of homotopy categories: more specifically, which objects (or maps) from D are in the image of T, and in how many different ways. For example:
A SIMPLICIAL A∞OPERAD ACTING ON RRESOLUTIONS
, 901
"... ABSTRACT. We construct a combinatorial model of an A∞operad which acts simplicially on the cobar resolution (not just its total space) of a simplicial set with respect to a ring R. 1. ..."
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ABSTRACT. We construct a combinatorial model of an A∞operad which acts simplicially on the cobar resolution (not just its total space) of a simplicial set with respect to a ring R. 1.