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Pasting Schemes for the Monoidal Biclosed Structure on ωCat
, 1995
"... Using the theory of pasting presentations, developed in chapter 2, I give a detailed description of the tensor product on ωcategories, which extends Gray's tensor product on 2categories and which is closely related to BrownHiggins's tensor product on ωgroupoids. Immediate consequences ..."
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Using the theory of pasting presentations, developed in chapter 2, I give a detailed description of the tensor product on ωcategories, which extends Gray's tensor product on 2categories and which is closely related to BrownHiggins's tensor product on ωgroupoids. Immediate consequences are a general and uniform definition of higher dimensional lax natural transformations, and a nice and transparent description of the corresponding internal homs. Further consequences will be in the development of a theory for weak ncategories, since both tensor products and lax structures are crucial in this.
The obstruction to excision in Ktheory and cyclic homology
 Invent. Math
"... Abstract Let f: A → B be a ring homomorphism of not necessarily unital rings and I ⊳ A an ideal which is mapped by f isomorphically to an ideal of B. The obstruction to excision in Ktheory is the failure of the map between relative Kgroups K∗(A: I) → K∗(B: f(I)) to be an isomorphism; it is measur ..."
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Abstract Let f: A → B be a ring homomorphism of not necessarily unital rings and I ⊳ A an ideal which is mapped by f isomorphically to an ideal of B. The obstruction to excision in Ktheory is the failure of the map between relative Kgroups K∗(A: I) → K∗(B: f(I)) to be an isomorphism; it is measured by the birelative groups K∗(A, B: I). We show that these are rationally isomorphic to the corresponding birelative groups for cyclic homology up to a dimension shift. In the particular case when A and B are Qalgebras we obtain an integral isomorphism. Algebraic Ktheory does not satisfy excision. This means that if f: A → B is a ring homomorphism and I ⊳ A is an ideal carried isomorphically to an ideal of B, then the map of relative Kgroups K∗(A: I) → K∗(B: I): = K∗(B: f(I)) is not an isomorphism in general. The obstruction is measured by birelative groups
Quillen Closed Model Structures for Sheaves
, 1995
"... In this chapter I give a general procedure of transferring closed model structures along adjoint functor pairs. As applications I derive from a global closed model structure on the category of simplicial sheaves closed model structures on the category of sheaves of 2groupoids, the category of bisim ..."
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In this chapter I give a general procedure of transferring closed model structures along adjoint functor pairs. As applications I derive from a global closed model structure on the category of simplicial sheaves closed model structures on the category of sheaves of 2groupoids, the category of bisimplicial sheaves and the category of simplicial sheaves of groupoids. Subsequently, the homotopy theories of these categories are related to the homotopy theory of simplicial sheaves. 1 Introduction There are two ways of trying to generalize the well known closed model structure on the category of simplicial sets to the category of simplicial objects in a Grothendieck topos. One way is to concentrate on the local aspect, and to use the Kanfibrations as a starting point. In [14] Heller showed that for simplicial presheaves there is a local (there called right) closed model structure. In [2] K. Brown showed that for a topological space X the category of "locally fibrant" sheaves of spectra on ...
Pasting Diagrams in nCategories with Applications to Coherence Theorems and Categories of Paths
, 1987
"... ..."
Additivity for derivator Ktheory
, 2008
"... We prove the additivity theorem for the Ktheory of triangulated derivators. This solves one of the conjec ..."
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We prove the additivity theorem for the Ktheory of triangulated derivators. This solves one of the conjec
Iterated wreath product of the simplex category and iterated loop spaces
 Adv. Math
"... Abstract. Generalising Segal’s approach to 1fold loop spaces, the homotopy theory of nfold loop spaces is shown to be equivalent to the homotopy theory of reduced Θnspaces, where Θn is an iterated wreath product of the simplex category ∆. A sequence of functors from Θn to Γ allows for an alternat ..."
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Abstract. Generalising Segal’s approach to 1fold loop spaces, the homotopy theory of nfold loop spaces is shown to be equivalent to the homotopy theory of reduced Θnspaces, where Θn is an iterated wreath product of the simplex category ∆. A sequence of functors from Θn to Γ allows for an alternative description of the Segal spectrum associated to a Γspace. In particular, each EilenbergMacLane space has a canonical reduced Θnset model. The number of (n + d)dimensional cells of the resulting CWcomplex of type K(Z/2Z, n) is a generalised Fibonacci number.
Quantized reduction as a tensor product
 QUANTIZATION OF SINGULAR SYMPLECTIC QUOTIENTS. BASEL: BIRKHÄUSER, 2001. EPRINT MATHPH/0008004
, 2008
"... Matched bimodules for rings may be composed through the (algebraic) bimodule tensor product, the canonical bimodule R → R ← R serving as a unit for ⊗R. We describe this picture also for C ∗algebras, von Neumann algebras, Lie groupoids, Poisson manifolds, and symplectic groupoids. This hinges on th ..."
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Matched bimodules for rings may be composed through the (algebraic) bimodule tensor product, the canonical bimodule R → R ← R serving as a unit for ⊗R. We describe this picture also for C ∗algebras, von Neumann algebras, Lie groupoids, Poisson manifolds, and symplectic groupoids. This hinges on the correct notion of bimodule, tensor product, and unit: for C ∗algebras B one has Hilbert (C ∗ ) bimodules with Rieffel’s tensor product and the canonical Hilbert bimodule over B, for von Neumann algebras one uses correspondences with Connes’s tensor product and the standard form, for (symplectic) Lie groupoids G one has regular (symplectic) bibundles with the Hilsum–Skandalis tensor product and the canonical bibundle over G, and for integrable Poisson manifolds P one deals with regular symplectic bimodules (dual pairs) with Xu’s tensor product and the sconnected and ssimply connected symplectic groupoid over P. Morita theory relates socalled equivalence bimodules to equivalence of representation theories. Subsequently, we study certain interconnections between the various constructions. The relation between Hilbert bimodules and correspondences is reviewed in detail. The notion of Marsden–Weinstein reduction makes sense for Poisson manifolds, C ∗algebras, and von Neumann algebras. Poisson manifolds and Lie groupoids join in the theory of Lie algebroids and symplectic groupoids. Finally, we note that the Poisson manifolds associated to Morita equivalent sconnected and ssimply connected Lie groupoids are Morita equivalent in the sense of Xu.
Weak factorizations, fractions and homotopies
 MR2141595 (2006c:18001), Zbl 1077.18002
"... Abstract. We show that the homotopy category can be assigned to any category equipped with a weak factorization system. A classical example of this construction is the stable category of modules. We discuss a connection with the open map approach to bisimulations proposed by Joyal, Nielsen and Win ..."
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Abstract. We show that the homotopy category can be assigned to any category equipped with a weak factorization system. A classical example of this construction is the stable category of modules. We discuss a connection with the open map approach to bisimulations proposed by Joyal, Nielsen and Winskel. 1.
Quasicategories vs Segal spaces
 IN CATEGORIES IN ALGEBRA, GEOMETRY AND MATHEMATICAL
, 2006
"... We show that complete Segal spaces and Segal categories are Quillen equivalent to quasicategories. ..."
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We show that complete Segal spaces and Segal categories are Quillen equivalent to quasicategories.
New Approach to Arakelov Geometry
"... The principal aim of this work is to provide an alternative algebraic framework for Arakelov geometry, and to demonstrate its usefulness by presenting several simple applications. This framework, called theory of generalized ..."
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The principal aim of this work is to provide an alternative algebraic framework for Arakelov geometry, and to demonstrate its usefulness by presenting several simple applications. This framework, called theory of generalized