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29
DG quotients of DG categories
 J. Algebra
"... Abstract. Keller introduced a notion of quotient of a differential graded category modulo a full differential graded subcategory which agrees with Verdier’s notion of quotient of a triangulated category modulo a triangulated subcategory. This work is an attempt to further develop his theory. ..."
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Abstract. Keller introduced a notion of quotient of a differential graded category modulo a full differential graded subcategory which agrees with Verdier’s notion of quotient of a triangulated category modulo a triangulated subcategory. This work is an attempt to further develop his theory.
ON LEFT AND RIGHT MODEL CATEGORIES AND LEFT AND RIGHT BOUSFIELD LOCALIZATIONS
"... We verify the existence of left Bousfield localizations and of enriched left Bousfield localizations, and we prove a collection of useful technical results characterizing certain fibrations of (enriched) left Bousfield localizations. We also use such Bousfield localizations to construct a number of ..."
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We verify the existence of left Bousfield localizations and of enriched left Bousfield localizations, and we prove a collection of useful technical results characterizing certain fibrations of (enriched) left Bousfield localizations. We also use such Bousfield localizations to construct a number of new model categories, including models for the homotopy limit of right Quillen presheaves, for Postnikov towers in model categories, and for presheaves valued in a symmetric monoidal model category satisfying a homotopycoherent descent condition. We then verify the existence of right Bousfield localizations of right model categories, and we apply this to construct a model of the homotopy limit of a left Quillen presheaf as a right model category.
Homotopy limits and colimits and enriched homotopy theory
, 2006
"... Abstract. Homotopy limits and colimits are homotopical replacements for the usual limits and colimits of category theory, which can be approached either using classical explicit constructions or the modern abstract machinery of derived functors. Our first goal is to explain both and show their equiv ..."
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Abstract. Homotopy limits and colimits are homotopical replacements for the usual limits and colimits of category theory, which can be approached either using classical explicit constructions or the modern abstract machinery of derived functors. Our first goal is to explain both and show their equivalence. Our second goal is to generalize this result to enriched categories and homotopy weighted limits, showing that the classical explicit constructions still give the right answer in the abstract sense, thus partially bridging the gap between classical homotopy theory and modern abstract homotopy theory. To do this we introduce a notion of “enriched homotopical categories”, which are more general than enriched model categories, but are still a good place to do enriched homotopy theory. This demonstrates that the presence of enrichment often simplifies rather than complicates matters, and goes some way toward achieving a better understanding of “the role of homotopy in homotopy theory.” Contents
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.
Weighted limits in simplicial homotopy theory
 Journal of Pure and Applied Algebra vol
, 2010
"... Abstract. We extend the theory of Quillen adjunctions by combining ideas of homotopical algebra and of enriched category theory. Our results describe how the formulas for homotopy colimits of Bousfield and Kan arise from general formulas describing the derived functor of the weighted colimit functo ..."
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Abstract. We extend the theory of Quillen adjunctions by combining ideas of homotopical algebra and of enriched category theory. Our results describe how the formulas for homotopy colimits of Bousfield and Kan arise from general formulas describing the derived functor of the weighted colimit functor. 1.
Derived Koszul Duality and Involutions in the Algebraic KTheory of Spaces
 J. Topology
"... ar ..."
Localization with respect to a class of maps I – Equivariant localization of diagrams of spaces
"... Abstract. Homotopical localizations with respect to a set of maps are known to exist in cofibrantly generated model categories (satisfying additional assumptions) [3, 12, 20, 29]. In this paper we expand the existing framework, so that it will apply to not necessarily cofibrantly generated model cat ..."
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Abstract. Homotopical localizations with respect to a set of maps are known to exist in cofibrantly generated model categories (satisfying additional assumptions) [3, 12, 20, 29]. In this paper we expand the existing framework, so that it will apply to not necessarily cofibrantly generated model categories and, more important, will allow for a localization with respect to a class of maps (satisfying some restrictive conditions). We illustrate our technique by applying it to the equivariant model category of diagrams of spaces [11]. This model category is not cofibrantly generated [7]. We give conditions on a class of maps which ensure the existence of the localization functor; these conditions are satisfied by any set of maps and by the class of maps which induces ordinary localizations on the generalized fixedpoints sets.
Deformations of sheaves of algebras
 Adv. Math
, 2005
"... Abstract. A construction of the tangent dg Lie algebra of a sheaf of operad algebras on a site is presented. The requirements on the site are very mild; the requirements on the algebra are more substantial. A few applications including the description of deformations of a scheme and equivariant defo ..."
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Cited by 7 (2 self)
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Abstract. A construction of the tangent dg Lie algebra of a sheaf of operad algebras on a site is presented. The requirements on the site are very mild; the requirements on the algebra are more substantial. A few applications including the description of deformations of a scheme and equivariant deformations are considered. The construction is based upon a model structure on the category of presheaves which should be of an independent interest. 0.1. In this paper we study formal deformations of sheaves of algebras. The most obvious (and very important) example is that of deformations of a scheme X over a field k of characteristic zero. In two different cases, the first when X is smooth, and the second when X is affine, the description is wellknown. In both cases there is a differential graded