THE CLASSIFYING TOPOS OF A TOPOLOGICAL BICATEGORY

被引:1
作者
Bakovic, Igor [1 ]
Jurco, Branislav [2 ]
机构
[1] Univ Split, Fac Nat Sci & Math, Split 21000, Croatia
[2] Max Planck Inst Math, D-53111 Bonn, Germany
关键词
bicategory; classifying topos; classifying space; principal bundle; EQUIVARIANT CROSSED COMPLEXES; CATEGORY; SPACES; SETS; MAPS;
D O I
10.4310/HHA.2010.v12.n1.a14
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
For any topological bicategory B, the Duskin nerve NB of B is a simplicial space. We introduce the classifying topos BB of B as the Deligne topos of sheaves Sh(NB) on the simplicial space NB. It is shown that the category of geometric morphisms Hom(Sh(X), BB) from the topos of sheaves Sh(X) on a topological space X to the Deligne classifying topos is naturally equivalent to the category of principal B-bundles. As a simple consequence, the geometric realization vertical bar NB vertical bar of the nerve NB of a locally contractible topological bicategory B is the classifying space of principal B-bundles, giving a variant of the result of Baas, Bokstedt and Kro derived in the context of bicategorical K-theory. We also define classifying topoi of a topological bicategory B using sheaves on other types of nerves of a bicategory given by Lack and Paoli, Simpson and Tamsamani by means of bisimplicial spaces, and we examine their properties.
引用
收藏
页码:279 / 300
页数:22
相关论文
共 30 条
[21]  
Moerdijk Ieke, 1995, Lecture Notes in Mathematics, V1616
[22]   CLASSIFYING SPACES RELATED TO FOLIATIONS [J].
SEGAL, G .
TOPOLOGY, 1978, 17 (04) :367-382
[23]  
SIMPSON C, 1997, ALGGEOM9704006
[24]   THE ALGEBRA OF ORIENTED SIMPLEXES [J].
STREET, R .
JOURNAL OF PURE AND APPLIED ALGEBRA, 1987, 49 (03) :283-335
[25]   On the notions of a nonstrict n-category and n-groupoid via multisimplicial sets [J].
Tamsamani, Z .
K-THEORY, 1999, 16 (01) :51-99
[27]  
VERITY D, 2005, MEMOIRS AM MATH SOC, V193
[28]   Weak complicial sets I. Basic homotopy theory [J].
Verity, D. R. B. .
ADVANCES IN MATHEMATICS, 2008, 219 (04) :1081-1149
[29]  
Verity D, 2007, CONTEMP MATH, V431, P441
[30]   What does the classifying space of a category classify? [J].
Weiss, Michael .
HOMOLOGY HOMOTOPY AND APPLICATIONS, 2005, 7 (01) :185-195