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29
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 .
Derived Quot schemes
"... (0.1) A typical moduli problem in geometry is to construct a “space ” H parametrizing, up to isomorphism, objects of some given category Z (e.g., manifolds, vector bundles etc.). This can be seen as a kind of a nonAbelian cohomology problem and the construction usually consists of two steps of oppo ..."
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(0.1) A typical moduli problem in geometry is to construct a “space ” H parametrizing, up to isomorphism, objects of some given category Z (e.g., manifolds, vector bundles etc.). This can be seen as a kind of a nonAbelian cohomology problem and the construction usually consists of two steps of opposite nature, namely applying
Derived Hilbert schemes
 J. A.M.S
"... (0.1) The Derived Deformation Theory (DDT) program (see [Kon], [CK] for more details and historical references) seeks to avoid the difficulties related to the singular nature of the moduli spaces in geometry by “passing to the derived category”, i.e., developing an appropriate version of the (nonabe ..."
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Cited by 12 (1 self)
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(0.1) The Derived Deformation Theory (DDT) program (see [Kon], [CK] for more details and historical references) seeks to avoid the difficulties related to the singular nature of the moduli spaces in geometry by “passing to the derived category”, i.e., developing an appropriate version of the (nonabelian) derived functor of the functor
Operads and Chain Rules for the Calculus of Functors
"... Abstract. We study the structure possessed by the Goodwillie derivatives of a pointed homotopy functor of based topological spaces. These derivatives naturally form a bimodule over the operad consisting of the derivatives of the identity functor. We then use these bimodule structures to give a chain ..."
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Abstract. We study the structure possessed by the Goodwillie derivatives of a pointed homotopy functor of based topological spaces. These derivatives naturally form a bimodule over the operad consisting of the derivatives of the identity functor. We then use these bimodule structures to give a chain rule for higher derivatives in the calculus of functors, extending that of Klein and Rognes. This chain rule expresses the derivatives of FG as a derived composition product of the derivatives of F and G over the derivatives of the identity. There are two main ingredients in our proofs. Firstly, we construct new models for the Goodwillie derivatives of functors of spectra. These models allow for natural composition maps that yield operad and module structures. Then, we use a cosimplicial cobar construction to transfer this structure to functors of topological spaces. A form of Koszul duality for operads of spectra plays a key role in this. In a landmark series of papers, [16], [17] and [18], Goodwillie outlines his ‘calculus of homotopy functors’. Let F: C → D (where C and D are each either Top ∗, the category of pointed topological spaces, or Spec, the category of spectra) be a pointed homotopy functor. One of the things that Goodwillie does is associate with F a sequence of spectra, which are called the derivatives of F.
Configuration spaces with summable labels
 Proceedings of BCAT98
"... Let M be an nmanifold, and let A be a space with a partial sum behaving as an nfold loop sum. We define the space C(M; A) of configurations in M with summable labels in A via operad theory. Some examples are symmetric products, labelled configuration spaces, and spaces of rational curves. We show ..."
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Let M be an nmanifold, and let A be a space with a partial sum behaving as an nfold loop sum. We define the space C(M; A) of configurations in M with summable labels in A via operad theory. Some examples are symmetric products, labelled configuration spaces, and spaces of rational curves. We show that C(I n, ∂I n; A) is an nfold classifying space of C(I n; A), and for n = 1 it is homeomorphic to the classifying space by Stasheff. If M is compact, parallelizable, and A is path connected, then C(M; A) is homotopic to the mapping space Map(M, C(I n, ∂I n; A)).
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.
Generalized operads and their inner cohomomorphisms
, 2007
"... In this paper we introduce a notion of generalized operad containing as special cases various kinds of operad–like objects: ordinary, cyclic, modular, properads etc. We then construct inner cohomomorphism objects in their categories (and categories of algebras over them). We argue that they provid ..."
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In this paper we introduce a notion of generalized operad containing as special cases various kinds of operad–like objects: ordinary, cyclic, modular, properads etc. We then construct inner cohomomorphism objects in their categories (and categories of algebras over them). We argue that they provide an approach to symmetry and moduli objects in noncommutative geometries based upon these “ring–like ” structures. We give a unified axiomatic treatment of generalized operads as functors on categories of abstract labeled graphs. Finally, we extend inner cohomomorphism constructions to more general categorical contexts. This version differs from the previous ones by several local changes (including the title) and two extra references.
Every homotopy theory of simplicial algebras admits a proper model
 Topology Appl. 119
, 2002
"... Abstract. We show that any closed model category of simplicial algebras over an algebraic theory ..."
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Abstract. We show that any closed model category of simplicial algebras over an algebraic theory