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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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Cited by 35 (0 self)
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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
ADHESIVE AND QUASIADHESIVE CATEGORIES
 THEORETICAL INFORMATICS AND APPLICATIONS
, 1999
"... We introduce adhesive categories, which are categories with structure ensuring that pushouts along monomorphisms are wellbehaved, as well as quasiadhesive categories which restrict attention to regular monomorphisms. Many examples of graphical structures used in computer science are shown to be ex ..."
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Cited by 34 (3 self)
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We introduce adhesive categories, which are categories with structure ensuring that pushouts along monomorphisms are wellbehaved, as well as quasiadhesive categories which restrict attention to regular monomorphisms. Many examples of graphical structures used in computer science are shown to be examples of adhesive and quasiadhesive categories. Doublepushout graph rewriting generalizes well to rewriting on arbitrary adhesive and quasiadhesive categories.
Motivic Functors
 DOCUMENTA MATH.
, 2003
"... The notion of motivic functors refers to a motivic homotopy theoretic analog of continuous functors. In this paper we lay the foundations for a homotopical study of these functors. Of particular interest is a model structure suitable for studying motivic functors which preserve motivic weak equivale ..."
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Cited by 9 (7 self)
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The notion of motivic functors refers to a motivic homotopy theoretic analog of continuous functors. In this paper we lay the foundations for a homotopical study of these functors. Of particular interest is a model structure suitable for studying motivic functors which preserve motivic weak equivalences and a model structure suitable for motivic stable homotopy theory. The latter model is Quillen equivalent to the category of motivic symmetric spectra. There is a symmetric monoidal smash product of motivic functors, and all model structures constructed are compatible with the smash product in the sense that we can do homotopical algebra on the various categories of modules and algebras. In particular, motivic cohomology
Techniques, computations, and conjectures for semitopological Ktheory
 MATH. ANN
, 2004
"... We establish the existence of an “AtiyahHirzebruchlike” spectral sequence relating the morphic cohomology groups of a smooth, quasiprojective complex variety to its semitopological Kgroups. This spectral sequence is compatible with (and, indeed, is built from) the motivic spectral sequence tha ..."
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Cited by 6 (1 self)
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We establish the existence of an “AtiyahHirzebruchlike” spectral sequence relating the morphic cohomology groups of a smooth, quasiprojective complex variety to its semitopological Kgroups. This spectral sequence is compatible with (and, indeed, is built from) the motivic spectral sequence that connects the motivic cohomology and algebraic Ktheory of varieties, and it is also compatible with the classical AtiyahHirzebruch spectral sequence in algebraic topology. In the second part of this paper, we use this spectral sequence in conjunction with another computational tool that we introduce — namely, a variation on the integral weight filtration of the BorelMoore (singular) cohomology of complex varieties introduced by H. Gillet and C. Soulé — to compute the semitopological Ktheory of a large class of varieties. In particular, we prove that for curves, surfaces, toric varieties, projective rational threefolds, and related varieties, the semitopological Kgroups and topological Kgroups are isomorphic in all degrees permitted by cohomological considerations. We also
Toric Varieties, Monoid Schemes and cdh descent
"... Abstract. We give conditions for the MayerVietoris property to hold for the algebraic Ktheory of blowup squares of toric varieties and schemes, using the theory of monoid schemes. These conditions are used to relate algebraic Ktheory to topological cyclic homology in characteristic p. To achieve ..."
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Abstract. We give conditions for the MayerVietoris property to hold for the algebraic Ktheory of blowup squares of toric varieties and schemes, using the theory of monoid schemes. These conditions are used to relate algebraic Ktheory to topological cyclic homology in characteristic p. To achieve our goals, we develop many notions for monoid schemes based on classical algebraic geometry, such as separated and proper maps and resolution of singularities. The goal of this paper is to prove Haesemeyer’s Theorem [18, 3.12] for toric schemes in any characteristic. It is proven below as Corollary 14.4. Theorem 0.1. Assume k is a commutative regular noetherian ring containing an infinite field and let G be a presheaf of spectra defined on the category of schemes of finite type over k. If G satisfies the MayerVietoris property for Zariski covers, finite abstract blowup squares, and blowups along regularly embedded closed subschemes, then G satisfies the MayerVietoris property for all abstract blowup squares of toric kschemes obtained from subdividing a fan. The application we have in mind is to understand the relationship between the
CONTENTS
, 2006
"... ABSTRACT. A kind of unstable homotopy theory on the category of associative rings (without unit) is developed. There are the notions of fibrations, homotopy (in the sense of Karoubi), path spaces, Puppe sequences, etc. One introduces the notion of a quasiisomorphism (or weak equivalence) for rings ..."
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ABSTRACT. A kind of unstable homotopy theory on the category of associative rings (without unit) is developed. There are the notions of fibrations, homotopy (in the sense of Karoubi), path spaces, Puppe sequences, etc. One introduces the notion of a quasiisomorphism (or weak equivalence) for rings and shows that similar to spaces the derived category obtained by inverting the quasiisomorphisms is naturally left triangulated. Also, homology theories on rings are studied. These must be homotopy invariant in the algebraic sense, meet the MayerVietoris property and plus some minor natural axioms. To any functor X from rings to pointed simplicial sets a homology theory is associated in a natural way. If X = GL and fibrations are the GLfibrations, one recovers KaroubiVillamayor’s functors KVi,i> 0. If X is Quillen’s Ktheory functor and fibrations are the surjective homomorphisms, one recovers the (nonnegative) homotopy Ktheory in the sense of Weibel. Technical tools we use are the homotopy information for the category of simplicial functors on rings and the Bousfield localization theory for model categories. The machinery developed in the paper