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31
A Homotopy Theory for Stacks
 Israel J. of Math
"... Abstract. We give a homotopy theoretic characterization of stacks on a site C as the homotopy sheaves of groupoids on C. We use this characterization to construct a model category in which stacks are the fibrant objects. We compare different definitions of stacks and show that they lead to Quillen e ..."
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Abstract. We give a homotopy theoretic characterization of stacks on a site C as the homotopy sheaves of groupoids on C. We use this characterization to construct a model category in which stacks are the fibrant objects. We compare different definitions of stacks and show that they lead to Quillen equivalent model categories. In addition, we show that these model structures are Quillen equivalent to the S 2nullification of Jardine’s model structure on sheaves of simplicial sets on C. 1.
The Center Of A pCompact Group
"... this paper we continue the study by looking at the idea of the "center" of a pcompact group and showing that two very different ways of defining the center are equivalent. This leads for instance to a reproof and generalization of a theorem from [15] about the identity component of the space of sel ..."
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Cited by 20 (1 self)
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this paper we continue the study by looking at the idea of the "center" of a pcompact group and showing that two very different ways of defining the center are equivalent. This leads for instance to a reproof and generalization of a theorem from [15] about the identity component of the space of self homotopy equivalences of BG (G compact Lie). Along the way we find various familiarlooking elements of internal structure in a pcompact group X , enumerate the X 's which are abelian in the appropriate sense, and construct what might be called the "adjoint form" of X . Before describing in more detail the main results we are aiming at, we have to introduce some ideas from [12]. A loop space X is by definition a triple (X ; BX ; e) in which X is a space, BX is a connected pointed space (called the classifying space of X ), and e : X !
Higher limits via subgroup complexes
 Ann. of Math
"... Abelian groupvalued functors on the orbit category of a finite group G are ubiquitous both in group theory and in much of homotopy theory. In this paper we give a new finite model for lim ∗ D F, the higher derived functors of the inverse limit functor, for a general functor F: Dop → Z (p) mod, whe ..."
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Cited by 18 (3 self)
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Abelian groupvalued functors on the orbit category of a finite group G are ubiquitous both in group theory and in much of homotopy theory. In this paper we give a new finite model for lim ∗ D F, the higher derived functors of the inverse limit functor, for a general functor F: Dop → Z (p) mod, where
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 .
Centric Maps and Realization of Diagrams in the Homotopy Category
 Proc. Amer. Math. Soc
, 1992
"... Introduction Let D be a small category. Suppose that ¯ X is a Ddiagram in the homotopy category (in other words, a functor from D to the homotopy category of simplicial sets). The question of whether or not ¯ X can be realized by a Ddiagram of simplicial sets has been treated by [5]. The purpose ..."
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Cited by 11 (2 self)
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Introduction Let D be a small category. Suppose that ¯ X is a Ddiagram in the homotopy category (in other words, a functor from D to the homotopy category of simplicial sets). The question of whether or not ¯ X can be realized by a Ddiagram of simplicial sets has been treated by [5]. The purpose of this note is to study a special situation in which the treatment can be simplified quite a bit. We look at two examples to which this simplified treatment is applicable; both examples involve homotopy decomposition diagrams for compact Lie groups. Our results show that in at least one of these examples ([13]) the decomposition diagram is completely determined by its underlying diagram in the homotopy category. It is possible that this "rigidity" result will eventually contribute to a general homotopy theoretic characterization theorem for
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.
Self Homotopy Equivalences Of Classifying Spaces Of Compact Connected Lie Groups
"... this paper, we extend those results to the case where G is any compact connected Lie group, but only considering self maps of BG which are rational equivalences. Most of the paper deals with self maps of the padic completions BG p ; and the results are extended to global maps only at the end. The f ..."
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Cited by 10 (1 self)
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this paper, we extend those results to the case where G is any compact connected Lie group, but only considering self maps of BG which are rational equivalences. Most of the paper deals with self maps of the padic completions BG p ; and the results are extended to global maps only at the end. The first complete description of [BG; BG] for any nonabelian connected Lie group
Homotopy Theory Of Classifying Spaces Of Compact Lie Groups
 Algebraic Topology and its Applications, MSRI Publications 27, SpringerVerlag
, 1994
"... this paper is to describe these results and the methods used to prove them. ..."
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Cited by 8 (2 self)
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this paper is to describe these results and the methods used to prove them.
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.
A spectral sequence for string cohomology
"... Let X be a 1connected space with free loop space ΛX. We introduce two spectral sequences converging towards H ∗ (ΛX; Z/p) and H ∗ ((ΛX)hT; Z/p). The E2terms are certain non Abelian derived functors applied to H ∗ (X; Z/p). When H ∗ (X; Z/p) is a polynomial algebra, the spectral sequences collapse ..."
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Cited by 5 (2 self)
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Let X be a 1connected space with free loop space ΛX. We introduce two spectral sequences converging towards H ∗ (ΛX; Z/p) and H ∗ ((ΛX)hT; Z/p). The E2terms are certain non Abelian derived functors applied to H ∗ (X; Z/p). When H ∗ (X; Z/p) is a polynomial algebra, the spectral sequences collapse for more or less trivial reasons. If X is a sphere it is a surprising fact that the spectral sequences collapse for p = 2. AMS subject classification (2000): 55N91, 55P35, 18G50 1