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20
The realization space of a Πalgebra: a moduli problem in algebraic topology
 Topology
"... 2. Moduli spaces 7 3. Postnikov systems for spaces 10 4. Πalgebras and their modules 14 ..."
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Cited by 18 (11 self)
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2. Moduli spaces 7 3. Postnikov systems for spaces 10 4. Πalgebras and their modules 14
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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Cited by 17 (0 self)
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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 .
Simplicial Structures on Model Categories and Functors
 Amer.J.Math.123
, 2001
"... We produce a highly structured way of associating a simplicial category to a model category which improves on work of Dwyer and Kan and answers a question of Hovey. We show that model categories satisfying a certain axiom are Quillen equivalent to simplicial model categories. A simplicial model cate ..."
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Cited by 15 (3 self)
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We produce a highly structured way of associating a simplicial category to a model category which improves on work of Dwyer and Kan and answers a question of Hovey. We show that model categories satisfying a certain axiom are Quillen equivalent to simplicial model categories. A simplicial model category provides higher order structure such as composable mapping spaces and homotopy colimits. We also show that certain homotopy invariant functors can be replaced by weakly equivalent simplicial, or "continuous," functors. This is used to show that if a simplicial model category structure exists on a model category then it is unique up to simplicial Quillen equivalence.
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 10 (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.
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 Commutative Ring Spectra as E∞ Ring Spectra
, 1999
"... We outline an obstruction theory for deciding when a homotopy commutative and associative ring spectrum is actually an E∞ ring spectrum. The obstruction groups are AndréQuillen cohomology groups of an algebra over an E∞ operad. The same cohomology theory is part of a spectral sequence for comput ..."
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Cited by 7 (2 self)
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We outline an obstruction theory for deciding when a homotopy commutative and associative ring spectrum is actually an E∞ ring spectrum. The obstruction groups are AndréQuillen cohomology groups of an algebra over an E∞ operad. The same cohomology theory is part of a spectral sequence for computing the homotopy type of mapping spaces between E∞ ring spectrum. The obstruction theory arises out of techniques of Dwyer, Kan, and Stover, and the main application here is to prove an analog of a theorem of Haynes Miller and the second author: the LubinTate spectra En are E∞ and the space of E∞ selfmaps has weakly contractible components.
Homotopy operations and rational homotopy type”, in Algebraic Topology: Categorical decomposition techniques
 Prog. in Math. 215, Birkhäuser, BostonBasel
"... In [HS] and [F1] Halperin, Stasheff, and Félix showed how an inductivelydefined sequence of elements in the cohomology of a graded commutative algebra over the rationals can be used to distinguish among the homotopy types of all possible realizations, thus providing a collection ..."
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Cited by 4 (4 self)
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In [HS] and [F1] Halperin, Stasheff, and Félix showed how an inductivelydefined sequence of elements in the cohomology of a graded commutative algebra over the rationals can be used to distinguish among the homotopy types of all possible realizations, thus providing a collection
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.
Resolutions in Model Categories
, 1997
"... Let C be a closed model category. Following work of Dwyer, Kan, and Stover, we develop a technique for building simplicial resolutions of objects in C which are free, or homotopically free, in a precise sense. Specifically, we specify in advance a set A of small, co brant, coH objects in C and the ..."
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Cited by 2 (2 self)
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Let C be a closed model category. Following work of Dwyer, Kan, and Stover, we develop a technique for building simplicial resolutions of objects in C which are free, or homotopically free, in a precise sense. Specifically, we specify in advance a set A of small, co brant, coH objects in C and the simplicial resolutions X will have the property that at each level n, the object Xn is homotopy equivalent to a coproduct of objects of A. These Stovertype resolutions are modeled on the free resolutions developed by Quillen, and are meant to be a replacement for projective resolutions in this context. In the end we develop some computational tools for spectra and structured ring spectra.