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22
On Braidings, Syllepses, and Symmetries
, 1998
"... this paper is that these are the only differences between (semistrict) braided monoidal 2categories (as defined in [10]) and braided 2D teisi. The interpretation of this is that the main obstacles for proving the conjecture above will be the weakness of functoriality and the weakness of invertibili ..."
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this paper is that these are the only differences between (semistrict) braided monoidal 2categories (as defined in [10]) and braided 2D teisi. The interpretation of this is that the main obstacles for proving the conjecture above will be the weakness of functoriality and the weakness of invertibility. I should mention here that Baez and Neuchl [5, p. 242] (as corrected by me [10, p. 206]) have shown that either one of the functoriality triangles above can be made into an identity, but it is essential to the proof that the other one is not. Defining monoidal 2D teisi as 3D teisi with one object involves a shift of dimension: the arrows, 2arrows and 3arrows of the 3D tas C become the objects, arrows and 2arrows of a 2D tas which will be called the looping of
Segal topoi and . . .
, 2002
"... In [ToVe2] we began the study of higher sheaf theory (i.e. stacks theory) on higher categories endowed with a suitable notion of topology: precisely, we defined the notions of Ssite and of model site, and the associated categories of stacks on them. This led us to study a notion of model topos (or ..."
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In [ToVe2] we began the study of higher sheaf theory (i.e. stacks theory) on higher categories endowed with a suitable notion of topology: precisely, we defined the notions of Ssite and of model site, and the associated categories of stacks on them. This led us to study a notion of model topos (orginally due to C. Rezk), a model category version of the notion of Grothendieck topos. In this paper we treat the analogous theory starting from (1)Segal categories in place of Scategories and model categories. We introduce notions of Segal topologies, Segal sites and stacks over them. We define an abstract notion of Segal topos and relate it with Segal categories of stacks over Segal sites. We compare the notions of Segal topoi and of model topoi, showing that the two theories are equivalent in some sense. However, the existence of a nice Segal category of morphisms between Segal categories allows us to improve the treatment of topoi in this context. In particular we construct the 2Segal category of Segal topoi and geometric morphisms, and we provide a Giraudlike statement characterizing Segal topoi among Segal categories. As example of applications, we show how to reconstruct a topological space up to homotopy from the Segal topos of locally constant stacks on it, thus extending the main theorem of [To1] to the case of unbased spaces. We also give some hints of how to define homotopy types of Segal sites: this approach
On the Cyclic Homology of Ringed Spaces and Schemes
, 1999
"... We prove that the cyclic homology of a scheme with an ample line bundle coincides with the cyclic homology of its category of algebraic vector bundles. As a byproduct of the proof, we obtain a new construction of the Chern character of a perfect complex on a ringed space. ..."
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We prove that the cyclic homology of a scheme with an ample line bundle coincides with the cyclic homology of its category of algebraic vector bundles. As a byproduct of the proof, we obtain a new construction of the Chern character of a perfect complex on a ringed space.
A QUILLEN MODEL STRUCTURE APPROACH TO THE FINITISTIC DIMENSION CONJECTURES
, 908
"... Abstract. We explore the interlacing between model category structures attained to classes of modules of finite Xdimension, for certain classes of modules X. As an application we give a model structure approach to the Finitistic Dimension Conjectures and present a new conceptual framework in which ..."
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Abstract. We explore the interlacing between model category structures attained to classes of modules of finite Xdimension, for certain classes of modules X. As an application we give a model structure approach to the Finitistic Dimension Conjectures and present a new conceptual framework in which these conjectures can be studied. Let Λ be a finite dimensional algebra over a field k (or more generally, let Λ be an artin ring). The big finitistic dimension of Λ, Findim(Λ), is defined as the supremum of the projective dimensions of all modules having finite projective dimension. And the little finitistic dimension of Λ, findim(Λ), is defined in a similar way by restricting to the subclass of all finitely generated modules of finite projective dimension. In 1960, Bass stated the socalled Finitistic Dimension Conjectures: (I) Findim(Λ) = findim(Λ), and (II) findim(Λ) is finite. The first conjecture was proved to be false by HuisgenZimmermann in [19], but the second one still remains open. It has been proved to be true, for instance, for finitedimensional monomial algebras [15], for Artin algebras with vanishing cube radical [18], or Artin algebras
WEAK EQUIVALENCES OF SIMPLICIAL PRESHEAVES
"... Abstract. Weak equivalences of simplicial presheaves are usually defined in terms of sheaves of homotopy groups. We give another characterization using relativehomotopyliftings, and develop the tools necessary to prove that this agrees with the usual definition. From our lifting criteria we are ab ..."
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Abstract. Weak equivalences of simplicial presheaves are usually defined in terms of sheaves of homotopy groups. We give another characterization using relativehomotopyliftings, and develop the tools necessary to prove that this agrees with the usual definition. From our lifting criteria we are able to prove some foundational (but new) results about the local homotopy theory of simplicial presheaves. 1.
Generalised Sheaf Cohomology Theories
, 2003
"... This paper is an expanded version of notes for a set of lectures given at the Isaac Newton Institute for Mathematical Sciences during a NATO ASI Workshop entitled "Homotopy Theory of Geometric Categories" on September 23 and 24, 2002. This workshop was part of a program entitled New Contex ..."
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This paper is an expanded version of notes for a set of lectures given at the Isaac Newton Institute for Mathematical Sciences during a NATO ASI Workshop entitled "Homotopy Theory of Geometric Categories" on September 23 and 24, 2002. This workshop was part of a program entitled New Contexts in Stable Homotopy Theory that was held at the Institute during the fall of 2002
DOI: 10.1017/S0305004103007175 Printed in the United Kingdom Hypercovers and simplicial presheaves
, 2002
"... We use hypercovers to study the homotopy theory of simplicial presheaves. The main result says that model structures for simplicial presheaves involving local weak equivalences can be constructed by localizing at the hypercovers. One consequence is that the fibrant objects can be explicitly describe ..."
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We use hypercovers to study the homotopy theory of simplicial presheaves. The main result says that model structures for simplicial presheaves involving local weak equivalences can be constructed by localizing at the hypercovers. One consequence is that the fibrant objects can be explicitly described in terms of a hypercover descent condition, and the fibrations can be described by a relative descent condition. We give a few applications for this new description of the homotopy theory of simplicial presheaves. 1.