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18
Homotopical Algebraic Geometry I: Topos theory
, 2002
"... This is the first of a series of papers devoted to lay the foundations of Algebraic Geometry in homotopical and higher categorical contexts. In this first part we investigate a notion of higher topos. For this, we use Scategories (i.e. simplicially enriched categories) as models for certain kind of ..."
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This is the first of a series of papers devoted to lay the foundations of Algebraic Geometry in homotopical and higher categorical contexts. In this first part we investigate a notion of higher topos. For this, we use Scategories (i.e. simplicially enriched categories) as models for certain kind of ∞categories, and we develop the notions of Stopologies, Ssites and stacks over them. We prove in particular, that for an Scategory T endowed with an Stopology, there exists a model
Algebraic geometry over model categories  A general approach to derived algebraic geometry
, 2001
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Brave New Algebraic Geometry and global derived moduli spaces of ring spectra
, 2008
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Homotopical algebraic geometry. II. Geometric stacks and applications
, 2006
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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.
Toën, Simplicial localization of monoidal structures and a nonlinear version of Deligne’s conjecture
 Compos. Math
"... Abstract. We show that if (M, ⊗, I) is a monoidal model category then REnd ..."
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Abstract. We show that if (M, ⊗, I) is a monoidal model category then REnd
Relations between slices and quotients of the algebraic cobordism spectrum
"... Abstract. We prove a relative statement about the slices of the algebraic cobordism spectrum. If the map from MGL to a certain quotient of MGL introduced by Hopkins and Morel is the map to the zeroslice then a relative version of Voevodsky’s conjecture on the slices of MGL holds true. We outline th ..."
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Abstract. We prove a relative statement about the slices of the algebraic cobordism spectrum. If the map from MGL to a certain quotient of MGL introduced by Hopkins and Morel is the map to the zeroslice then a relative version of Voevodsky’s conjecture on the slices of MGL holds true. We outline the picture for Ktheory and rational slices.
Algebraic Ktheory, Algebraic Cycles and Arithmetic Geometry
 Handbook of Ktheory
, 2005
"... Warning: This paper is full of conjectures. If you are allergic to them it may be harmful to your health. Parts of them are proven, though. In algebraic geometry, one encounters two important kinds of objects: vector bundles and algebraic cycles. The first lead to algebraic Ktheory while the ..."
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Warning: This paper is full of conjectures. If you are allergic to them it may be harmful to your health. Parts of them are proven, though. In algebraic geometry, one encounters two important kinds of objects: vector bundles and algebraic cycles. The first lead to algebraic Ktheory while the
COFREE COALGEBRAS OVER OPERADS II.
, 2004
"... ABSTRACT. This paper develops closedform expressions for cofree coalgebras over free operads. This leads to a proof that homology equivalences of cogenerating complexes induce homology equivalences of the cofree coalgebras. We show that the underlying chain complex of any cofree coalgebra is natura ..."
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ABSTRACT. This paper develops closedform expressions for cofree coalgebras over free operads. This leads to a proof that homology equivalences of cogenerating complexes induce homology equivalences of the cofree coalgebras. We show that the underlying chain complex of any cofree coalgebra is naturally a direct summand of the underlying chaincomplex of a cofree coalgebra over a free operad. This is combined with the previous result to prove the homology invariance of all cofree coalgebras.