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18
Local projective model structures on simplicial presheaves
 Ktheory
"... Abstract. We give a model structure on the category of simplicial presheaves on some essentially small Grothendieck site T. When T is the Nisnevich site it specializes to a proper simplicial model category with the same weak equivalences as in [MV], but with fewer cofibrations and consequently more ..."
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Abstract. We give a model structure on the category of simplicial presheaves on some essentially small Grothendieck site T. When T is the Nisnevich site it specializes to a proper simplicial model category with the same weak equivalences as in [MV], but with fewer cofibrations and consequently more fibrations. This allows a simpler proof of the comparison theorem of [V2], one which makes no use of ∆closed classes. The purpose of this note is to introduce different model structures on the categories of simplicial presheaves and simplicial sheaves on some essentially small Grothendieck site T and to give some applications of these simplified model categories. In particular, we prove that the stable homotopy categories SH((Sm/k)Nis, A 1) and SH((Sch/k)cdh, A 1) are equivalent. This result was first proven by Voevodsky in [V2] and our proof uses many of his techniques, but it does not use his theory of ∆closed classes developed in [V3]. 1. The local projective model structure on presheaves We first recall some of the other wellknown model structures on simplicial presheaves. Definition 1.1. A map f: X → Y of simplicial presheaves (or sheaves) is a local weak equivalence if f ∗ : π0(X) → π0(Y) induces an isomorphism of associated sheaves and, for all U ∈ T, f ∗ : πn(X, x) → πn(Y, f(x)) induces an isomorphism of associated sheaves on T/U for any choice of basepoint x ∈ X(U). The map f is a sectionwise weak equivalence (respectively sectionwise fibration) if for all U ∈ T, the map f(U) : X(U) → Y (U) is a weak equivalence (respectively Kan fibration) of simplicial sets. Heller [He] discovered a model structure on simplicial presheaves whose weak equivalences are the sectionwise weak equivalences. We will refer to his model structure as the injective model structure. Date: January 11, 2001. I would like to thank Dan Isaksen for his many helpful suggestions, and I thank my adviser Peter May for his encouragement and careful reading of many drafts. I am also grateful to Vladimir Voevodsky for noticing an error in an earlier version and for his work that inspired this note. 1 2 BENJAMIN BLANDER
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
On the BreenBaezDolan stabilization hypothesis for Tamsamani’s weak ncategories
, 1998
"... In [2] Baez and Dolan established their stabilization hypothesis as one of a list of the key properties that a good theory of higher categories should have. It is the analogue for ncategories of the wellknown stabilization theorems in homotopy theory. To explain the statement, recall that BaezDol ..."
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In [2] Baez and Dolan established their stabilization hypothesis as one of a list of the key properties that a good theory of higher categories should have. It is the analogue for ncategories of the wellknown stabilization theorems in homotopy theory. To explain the statement, recall that BaezDolan introduce the notion of kuply monoidal ncategory which is an n + kcategory having only one imorphism for all i < k. This includes the notions previously defined and examined by many authors, of monoidal (resp. braided monoidal, symmetric monoidal) category (resp. 2category) and so forth, as is explained in [2] [4]. See the bibliographies of those preprints as well as that of the the recent preprint [9] for many references concerning these types of objects. In the case where the ncategory in question is an ngroupoid, this notion is—except for truncation at n—the same thing as the notion of kfold iterated loop space, or “Ekspace ” which appears in Dunn [10] (see also some anterior references from there). The fully stabilized notion of kuply monoidal ncategories for k ≫ n is what Grothendieck calls Picard ncategories in [12]. The stabilization hypothesis [2] states that for n + 2 ≤ k ≤ k ′ , the kuply monoidal
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
PERVERSE BUNDLES AND CALOGEROMOSER SPACES
, 2007
"... Abstract. We present a simple description of moduli spaces of torsionfree Dmodules (Dbundles) on general smooth complex curves, generalizing the identification of the space of ideals in the Weyl algebra with CalogeroMoser quiver varieties. Namely, we show that the moduli of Dbundles form twiste ..."
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Abstract. We present a simple description of moduli spaces of torsionfree Dmodules (Dbundles) on general smooth complex curves, generalizing the identification of the space of ideals in the Weyl algebra with CalogeroMoser quiver varieties. Namely, we show that the moduli of Dbundles form twisted cotangent bundles to moduli of torsion sheaves on X, answering a question of Ginzburg. The corresponding (untwisted) cotangent bundles are identified with moduli of perverse vector bundles on T ∗ X, which contain as open subsets the moduli of framed torsionfree sheaves (the Hilbert schemes T ∗ X [n] in the rank one case). The proof is based on the description of the derived category of Dmodules on X by a noncommutative version of the Beilinson transform on P 1. 1.
Flexible sheaves
, 1996
"... This is an unfinished explanation of the notion of “flexible sheaf”, that is a homotopical notion of sheaf of topological spaces (homotopy types) over a site. See “Homotopy over the complex numbers and generalized de Rham cohomology ” (submitted to proceedings of the Taniguchi conference on vector b ..."
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This is an unfinished explanation of the notion of “flexible sheaf”, that is a homotopical notion of sheaf of topological spaces (homotopy types) over a site. See “Homotopy over the complex numbers and generalized de Rham cohomology ” (submitted to proceedings of the Taniguchi conference on vector bundles, preprint of Toulouse 3, and also available at my homepage 2) for a more detailed introduction, and also for a further development of the ideas presented here. The present paper was finished in December 1993 while I was visiting MIT. Since writing this, I have realized that the theory sketched here is essentially equivalent to JardineIllusie’s theory of presheaves of topological spaces (although they talk about presheaves of simplicial sets which is the same thing). This is the point of view adopted in “Homotopy over the complex numbers and generalized de Rham cohomology”. 1 Added in August 1996: I am finally sending this to “Duke eprints ” because it now seems that there may be several useful features of the treatment given here. First of all, the direct construction of the homotopysheafification (by doing a certain operation n+2 times) seems to be useful, for example I need
RIMS1748 BAR CONSTRUCTION AND TANNAKIZATION By
, 2012
"... Abstract. We continue our study of tannakizations of symmetric monoidal stable ∞categories, begun in [17]. The issue treated in this paper is the calculation of tannakizations of examples of symmetric monoidal stable ∞categories with fiber functors. We consider the case of symmetric monoidal ∞cat ..."
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Abstract. We continue our study of tannakizations of symmetric monoidal stable ∞categories, begun in [17]. The issue treated in this paper is the calculation of tannakizations of examples of symmetric monoidal stable ∞categories with fiber functors. We consider the case of symmetric monoidal ∞categories of perfect complexes on perfect derived stacks. The first main result especially says that our tannakization includes the bar construction for an augmented commutative ring spectrum and its equivariant version as a special case. We apply it to the study of the tannakization of the stable infinitycategory of mixed Tate motives over a perfect field. We prove that its tannakization can be obtained from the Gmequivariant bar construction of a commutative differential graded algebra equipped with Gmaction. Moreover, under BeilinsonSoulé vanishing conjecture, we prove that the underlying group scheme of the tannakization is the motivic Galois group for mixed Tate motives, constructed in [4], [21], [22]. 1.