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Homotopical algebraic geometry. II. Geometric stacks and applications
, 2006
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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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Cited by 50 (19 self)
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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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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 20 (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.
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.
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
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
A Model Structure for Enriched Coloured Operads
"... Abstract. We prove that, under certain conditions, the model structure on a monoidal model category V can be transferred to a model structure on the category of Venriched coloured (symmetric) operads. As a particular case we recover the known model structure on simplicial operads. ..."
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Abstract. We prove that, under certain conditions, the model structure on a monoidal model category V can be transferred to a model structure on the category of Venriched coloured (symmetric) operads. As a particular case we recover the known model structure on simplicial operads.
Cycle complex over P1 minus 3 points : toward multiple zeta values cycles
, 2012
"... Abstract. In this paper, the author constructs a family of algebraic cycles in Bloch’s cycle complex over P1 minus three points which are expected to correspond to multiple polylogarithms in one variable. Elements in this family of weight p are in the cubical cycle group of codimension p in (P1 \{0, ..."
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Abstract. In this paper, the author constructs a family of algebraic cycles in Bloch’s cycle complex over P1 minus three points which are expected to correspond to multiple polylogarithms in one variable. Elements in this family of weight p are in the cubical cycle group of codimension p in (P1 \{0, 1,∞})× (P1 \ {1})2p−1 and are, in weight greater or equal to 2, naturaly extended as equidimensional cycles over over A1. This allows to consider their fibers at the point 1 and this is one of the main differences with the work of Gangl, Goncharov and Levin. Considering the fiber at 1 makes it possible to think of these cycles as corresponding to weight n multiple zeta values. After the introduction, the author recalls some properties of Bloch’s cycle complex and enlightens the difficulties on a few examples. Then a large section is devoted to the combinatorial situation involving the combinatoric of trivalent trees. In the last section, two families of cycles are constructed as solutions to a “differential system ” in Bloch’s cycle complex. One of this