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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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Cited by 12 (1 self)
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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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Cited by 7 (5 self)
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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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Cited by 3 (2 self)
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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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Cited by 2 (1 self)
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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.
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:
Interpolation categories for homology theories
, 2004
"... For a homological functor from a triangulated category to an abelian category satisfying some technical assumptions we construct a tower of interpolation categories. These are categories over which the functor factorizes and which capture more and more information according to the injective dimensio ..."
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For a homological functor from a triangulated category to an abelian category satisfying some technical assumptions we construct a tower of interpolation categories. These are categories over which the functor factorizes and which capture more and more information according to the injective dimension of the images of the functor. The categories are obtained by proving the existence of truncated versions of resolution or E2model structures. Examples of functors fitting in our framework are given by every generalized homology theory represented by a ring spectrum satisfying the AdamsAtiyah condition. The constructions are closely related to the modified Adams spectral sequence and give a very conceptual approach to the associated moduli problem and obstruction theory. As application we establish an isomorphism between certain E(n)local Picard groups and some Extgroups.
TRUNCATED RESOLUTION MODEL STRUCTURES
, 2006
"... Abstract. Using the dual of BousfieldFriedlander localization we colocalize resolution model structures on cosimplicial objects over a left proper model category to get truncated resolution model structures. These are useful to study realization and moduli problems in algebraic topology. 1. ..."
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Abstract. Using the dual of BousfieldFriedlander localization we colocalize resolution model structures on cosimplicial objects over a left proper model category to get truncated resolution model structures. These are useful to study realization and moduli problems in algebraic topology. 1.