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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 .
(Pre)sheaves of Ring Spectra over the Moduli Stack of Formal Group Laws
, 2004
"... In the first part of this article, I will state a realization problem for diagrams of structured ring spectra, and in the second, I will discuss the moduli space which parametrizes the problem. ..."
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In the first part of this article, I will state a realization problem for diagrams of structured ring spectra, and in the second, I will discuss the moduli space which parametrizes the problem.
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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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.
CW simplicial resolutions of spaces, with an application
"... Abstract. We show how a certain type of CW simplicial resolutions of spaces by wedges of spheres may be constructed, and how such resolutions yield an obstruction theory for a given space to be a loop space. 1. ..."
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Abstract. We show how a certain type of CW simplicial resolutions of spaces by wedges of spheres may be constructed, and how such resolutions yield an obstruction theory for a given space to be a loop space. 1.
Realizing coalgebras over the Steenrod algebra
 Topology
"... (co)algebra K ∗ over the mod p Steenrod algebra as the (co)homology of a topological space, and for distinguishing between the phomotopy types of different realizations. The theories are expressed in terms of the Quillen cohomology of K∗. 1. ..."
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(co)algebra K ∗ over the mod p Steenrod algebra as the (co)homology of a topological space, and for distinguishing between the phomotopy types of different realizations. The theories are expressed in terms of the Quillen cohomology of K∗. 1.
Suspension and loop objects in theories and cohomology
 Georgian Math. J
"... Abstract. We introduce theories in which suspension, resp. loop objects, are defined and we describe the cohomology of such theories. Cohomology classes in degree 3 classify track theories. ..."
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Abstract. We introduce theories in which suspension, resp. loop objects, are defined and we describe the cohomology of such theories. Cohomology classes in degree 3 classify track theories.
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:
ON REALIZING DIAGRAMS OF ΠALGEBRAS
, 2006
"... Abstract. Given a diagram of Πalgebras (graded groups equipped with an action of the primary homotopy operations), we ask whether it can be realized as the homotopy groups of a diagram of spaces. The answer given here is in the form of an obstruction theory, of somewhat wider application, formulate ..."
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Abstract. Given a diagram of Πalgebras (graded groups equipped with an action of the primary homotopy operations), we ask whether it can be realized as the homotopy groups of a diagram of spaces. The answer given here is in the form of an obstruction theory, of somewhat wider application, formulated in terms of generalized Πalgebras. This extends a program begun in [DKS1, BDG] to study the realization of a single Πalgebra. In particular, we explicitly analyze the simple case of a single map, and provide a detailed example, illustrating the connections to higher homotopy operations. A recurring problem in algebraic topology is the rectification of homotopycommutative diagrams: given a diagram F: D → ho T ∗ (i.e., a functor from a small category to the homotopy category of topological spaces), we ask whether F lifts to ˆ F: D → T∗, and if so, in how many ways.