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13
A Cellular Nerve for Higher Categories
, 2002
"... ... categories. The associated cellular nerve of an ocategory extends the wellknown simplicial nerve of a small category. Cellular sets (like simplicial sets) carry a closed model structure in Quillen’s sense with weak equivalences induced by a geometric realisation functor. More generally, there ..."
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... categories. The associated cellular nerve of an ocategory extends the wellknown simplicial nerve of a small category. Cellular sets (like simplicial sets) carry a closed model structure in Quillen’s sense with weak equivalences induced by a geometric realisation functor. More generally, there exists a dense subcategory YA of the category of Aalgebras for each ooperad A in Batanin’s sense. Whenever A is contractible, the resulting homotopy category of Aalgebras (i.e. weak ocategories) is
Descente pour les nchamps
"... We develop the theory of nstacks (or more generally Segal nstacks which are ∞stacks such that the morphisms are invertible above degree n). This is done by systematically using the theory of closed model categories (cmc). Our main results are: a definition of nstacks in terms of limits, which sh ..."
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We develop the theory of nstacks (or more generally Segal nstacks which are ∞stacks such that the morphisms are invertible above degree n). This is done by systematically using the theory of closed model categories (cmc). Our main results are: a definition of nstacks in terms of limits, which should be perfectly general for stacks of any type of objects; several other characterizations of nstacks in terms of “effectivity of descent data”; construction of the stack associated to an nprestack; a strictification result saying that any “weak ” nstack is equivalent to a (strict) nstack; and a descent result saying that the (n + 1)prestack of nstacks (on a site) is an (n + 1)stack. As for other examples, we start from a “left Quillen presheaf ” of cmc’s and introduce the associated Segal 1prestack. For this situation, we prove a general descent result, giving sufficient conditions for this prestack to be a stack. This applies to the case of complexes, saying how complexes of sheaves of Omodules can be glued together via quasiisomorphisms. This was the problem that originally motivated us. Résumé
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 ...
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
Symplectic groups are Ndetermined 2compact groups
"... Abstract. We show that for n ≥ 3 the symplectic group Sp(n) is as a 2compact group determined up to isomorphism by the isomorphism type of its maximal torus normalizer. This allows us to determine the integral homotopy type of Sp(n) among connected finite loop spaces with maximal torus. 1. ..."
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Abstract. We show that for n ≥ 3 the symplectic group Sp(n) is as a 2compact group determined up to isomorphism by the isomorphism type of its maximal torus normalizer. This allows us to determine the integral homotopy type of Sp(n) among connected finite loop spaces with maximal torus. 1.
ON THE MOD p COHOMOLOGY OF BPU(p)
, 2003
"... Abstract. We study the mod p cohomology of the classifying space of the projective unitary group PU(p). We first proof that old conjectures due to J.F. Adams, and Kono and Yagita [16] about the structure of the mod p cohomology of classifying space of connected compact Lie groups held in the case of ..."
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Abstract. We study the mod p cohomology of the classifying space of the projective unitary group PU(p). We first proof that old conjectures due to J.F. Adams, and Kono and Yagita [16] about the structure of the mod p cohomology of classifying space of connected compact Lie groups held in the case of PU(p). Finally, we proof that the classifying space of the projective unitary group PU(p) is determined by its mod p cohomology as an unstable algebra over the Steenrod algebra for p> 3, completing previous works [10] and [6] for the cases p = 2, 3. 1.
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
A homotopy construction of the adjoint representation for Lie groups
, 2000
"... Let G be a compact, simplyconnected, simple Lie group and T ⊂ G a maximal torus. The purpose of this paper is to study the connection between various fibrations over BG (where G is a compact, simplyconnected, simple Lie group) associated to the adjoint representation and homotopy colimits over pos ..."
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Let G be a compact, simplyconnected, simple Lie group and T ⊂ G a maximal torus. The purpose of this paper is to study the connection between various fibrations over BG (where G is a compact, simplyconnected, simple Lie group) associated to the adjoint representation and homotopy colimits over poset categories �, hocolim�BGI where GI are certain connected maximal rank subgroups of G. 1.