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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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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
"... ..."
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 10 (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.
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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Cited by 7 (1 self)
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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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Cited by 5 (4 self)
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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.
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.
SIMPLICIAL LOCALIZATION OF MONOIDAL STRUCTURES, AND A NONLINEAR VERSION OF DELIGNE’S CONJECTURE JOACHIM KOCK AND BERTRAND TO ËN
, 2003
"... is a (weak) 2monoid in sSet. This applies in particular when M is the category of Abimodules over a simplicial monoid A: the derived endomorphisms of A then form its Hochschild cohomology, which therefore becomes a simplicial 2monoid. Deligne’s conjecture. Deligne’s conjecture (stated informally ..."
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is a (weak) 2monoid in sSet. This applies in particular when M is the category of Abimodules over a simplicial monoid A: the derived endomorphisms of A then form its Hochschild cohomology, which therefore becomes a simplicial 2monoid. Deligne’s conjecture. Deligne’s conjecture (stated informally in a letter in 1993) states that the Hochschild cohomology HH(A) of an associative algebra A is a 2algebra — this means that up to homotopy it has two compatible multiplication laws.