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18
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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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.
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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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.
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:
Bousfield's E 2 model theory for simplicial objects
"... this paper: Theorem 9. Suppose that is a right proper closed simplicial model category having an object which is initial and terminal, and let be a set of cogroup objects of which is closed under suspension. Then with the definitions given above the category sM of simplicial objects in ha ..."
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this paper: Theorem 9. Suppose that is a right proper closed simplicial model category having an object which is initial and terminal, and let be a set of cogroup objects of which is closed under suspension. Then with the definitions given above the category sM of simplicial objects in has the structure of a right proper closed simplicial model category
Contemporary Mathematics Model Categories and Simplicial Methods
"... Abstract. 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 tech ..."
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Abstract. 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
Proalgebraic homotopy types
, 2008
"... The purpose of this paper is to generalise Sullivan’s rational homotopy theory to nonnilpotent spaces, providing an alternative approach to defining Toën’s schematic homotopy types over any field k of characteristic zero. New features include an explicit description of homotopy groups using the Mau ..."
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The purpose of this paper is to generalise Sullivan’s rational homotopy theory to nonnilpotent spaces, providing an alternative approach to defining Toën’s schematic homotopy types over any field k of characteristic zero. New features include an explicit description of homotopy groups using the MaurerCartan equations and convergent spectral sequences comparing schematic homotopy groups with cohomology of the universal semisimple local system. For compact Kähler manifolds, the schematic homotopy groups can be described explicitly in terms of this cohomology ring, giving them canonical weight decompositions. There are also notions of minimal models, unpointed homotopy types and algebraic automorphism groups. For a space with algebraically good fundamental group and higher homotopy groups of finite rank, the schematic homotopy groups are shown to be πn(X) ⊗Z k.
Contents
, 2008
"... The purpose of this paper is to generalise Sullivan’s rational homotopy theory to nonnilpotent spaces, providing an alternative approach to defining Toën’s schematic homotopy types over any field k of characteristic zero. New features include an explicit description of homotopy groups using the Mau ..."
Abstract
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The purpose of this paper is to generalise Sullivan’s rational homotopy theory to nonnilpotent spaces, providing an alternative approach to defining Toën’s schematic homotopy types over any field k of characteristic zero. New features include an explicit description of homotopy groups using the MaurerCartan equations, convergent spectral sequences comparing schematic homotopy groups with cohomology of the universal semisimple local system, and a generalisation of the BauesLemaire conjecture. For compact Kähler manifolds, the schematic homotopy groups can be described explicitly in terms of this cohomology ring, giving them canonical weight decompositions. There are also notions of minimal models, unpointed homotopy types and algebraic automorphism groups. For a space with algebraically good fundamental group and higher homotopy groups of finite rank,
Contents
, 2008
"... The purpose of this paper is to generalise Sullivan’s rational homotopy theory to nonnilpotent spaces, providing an alternative approach to defining Toën’s schematic homotopy types over any field k of characteristic zero. New features include an explicit description of homotopy groups using the Mau ..."
Abstract
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The purpose of this paper is to generalise Sullivan’s rational homotopy theory to nonnilpotent spaces, providing an alternative approach to defining Toën’s schematic homotopy types over any field k of characteristic zero. New features include an explicit description of homotopy groups using the MaurerCartan equations, convergent spectral sequences comparing schematic homotopy groups with cohomology of the universal semisimple local system, and a generalisation of the BauesLemaire conjecture. For compact Kähler manifolds, the schematic homotopy groups can be described explicitly in terms of this cohomology ring, giving them canonical weight decompositions. There are also notions of minimal models, unpointed homotopy types and algebraic automorphism groups. For a space with algebraically good fundamental group and higher homotopy groups of finite rank,