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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
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
Calculating Maps between nCategories
, 2000
"... This short note addresses the problem of how to calculate in a reasonable way the homotopy classes of maps between two ncategories (by which we mean Tamsamani's nnerves [10]). The closed model structure of [9] gives an abstract way of calculating this but it isn't very concrete so we would like a ..."
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This short note addresses the problem of how to calculate in a reasonable way the homotopy classes of maps between two ncategories (by which we mean Tamsamani's nnerves [10]). The closed model structure of [9] gives an abstract way of calculating this but it isn't very concrete so we would like a more downtoearth calculation. For the purposes of the present note we shall use the closed model structure of [9] to prove that our method gives the right answer. It might be possible to develop the theory entirely using the present calculation as the de nition of \map", but that is left as an open problem. Denote by n Cat the category of ncategories introduced by Tamsamani [10], and by L(n Cat) the DwyerKan simplicial localization [2] where we divide out by the equivalences between ncategories (Tamsamani calls these \external equivalences"). Note that the 1category obtained from L(n Cat) by applying 0 to the simplicial Hom sets is just the GabrielZisman [4] localiza
Stacks and nonabelian cohomoloy
, 2002
"... The purpose of this first lecture is to present a general formalism of higher stacks based on the theory of simplicial presheaves introduced by A. Joyal and developped by many authors after him. My main purpose will be to explain throught examples why the homotopy theory of simplicial presheaf is ac ..."
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The purpose of this first lecture is to present a general formalism of higher stacks based on the theory of simplicial presheaves introduced by A. Joyal and developped by many authors after him. My main purpose will be to explain throught examples why the homotopy theory of simplicial presheaf is actually a very good model for a theory of higher stacks. For this, I will present the homotopy category of stacks and investigate its relations with the usual theory of (1)stacks, sheaf cohomology and nonabelian cohomology. Some references on the subjects are [Ja1], [S2], [HS], [Ja2], [Hol], [Du]. Terminology remarks: • The stacks of this talk will probably have a different flavor than usual. Indeed, instead of considering stacks from a geometrical point of view (e.g. algebraic stacks) they will be considered as coefficients for cohomology and will not be endowed with geometrical structure (algebraic, topological...). In the future they will serve to study other geometrical objects (varieties, spaces, possiblyotherstacks...) exactlyassheavesareusedtostudyschemes. The theory of higher stacks should actually be understood as part of higher topos theory. • The expression nonabelian cohomology can be quite ambiguous. One could try to make it clearer by the following observation. Abelian cohomology is certainely dual to (abelian) homology. On the other side, homology is nothing else than the abelianization of homotopy, or abelian homotopy. Therefore, nonabelian cohomology should really be understood as dual to homotopy. This is why homotopy theory will play an imporant role in this lecture. In the following I will be using the homotopy theory of simplicial sets. If one is not familliar with it, one can simply replace simplicial sets by topological spaces, andequivalences of simplicial sets by weak equivalences of topological spaces. The category of sets will be denoted by Set, and the category of simplicial sets by SSet. As any set can be considered as a discrete simplicial set one can considered Set as embedded in SSet. To fix the ideas we will work over the big site T: = (Top) of all topological spaces. However, everything will be valid over any Grothendieck site.
Some properties of the theory of ncategories
, 2001
"... Much interest has recently focused on the problem of comparing different definitions of ncategories. 1 Leinster has made a useful compendium of 10 definitions [8]; May has proposed another definition destined among other things to make comparison easier [9], and he has also led the creation of an u ..."
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Much interest has recently focused on the problem of comparing different definitions of ncategories. 1 Leinster has made a useful compendium of 10 definitions [8]; May has proposed another definition destined among other things to make comparison easier [9], and he has also led the creation of an umbrella research group including comparison as one of the main research topics; Batanin has started a comparison of his definition with Penon’s [2]; and Berger has made a comparison between Batanin’s theory and the homotopy theory of spaces, introducing techniques which should allow for other comparisons starting with Batanin’s theory [3]. The comparison question was first explicitly mentionned by Grothendieck in [7], in a prescient prediction that many different people would come up with different definitions of ncategory; and this theme was again brought up by Baez and Dolan in [1]. The purpose of this short note is to make some observations about properties which one can expect any theory of ncategories to have, and to conjecture that these properties characterize the theory of ncategories. As a small amount of evidence for this conjecture, we show how to go from these properties to the composition law between mapping objects in an ncategory. While we don’t give the proofs here, it is not too hard to see that Tamsamani’s definition of ncategory satisfies the properties listed below. We conjecture that the other definitions satisfy these properties too. This conjecture plus the conjecture of the previous paragraph would give an answer to the comparison question. Rather than giving all of the references for the various different definitions of ncategory, we refer the reader to Leinster’s excellent bibliography [8]. The fundamental tool which we use is the DwyerKan localization [5]. This is a generalization of the classical GabrielZisman localization, which keeps higher homotopy data. Dwyer and Kan obtain a mapping space between two objects, where Gabriel and Zisman obtain only the set of homotopy classes of maps i.e. the π0 of the mapping space. The fundamental observation of Dwyer and Kan is that the mapping spaces (plus their composition and higher homotopy coherence information) are determined by the data of
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.