What does the classifying space of a category classify? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T07:40:18Z http://mathoverflow.net/feeds/question/23857 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/23857/what-does-the-classifying-space-of-a-category-classify What does the classifying space of a category classify? Don Stanley 2010-05-07T14:22:01Z 2010-05-07T18:27:43Z <p>A finite group \$G\$ can be considered as a category with one object. Taking its nerve \$NG\$, and then geometrically realizing we get \$BG\$ the classifying space of \$G\$, which classifies principle \$G\$ bundles.</p> <p>Instead starting with any category \$C\$, what does \$NC\$ classify? (Either before or after taking realization.) Does it classify something reasonable?</p> http://mathoverflow.net/questions/23857/what-does-the-classifying-space-of-a-category-classify/23863#23863 Answer by Andrew Stacey for What does the classifying space of a category classify? Andrew Stacey 2010-05-07T14:39:14Z 2010-05-07T14:39:14Z <p>It's one level up the categorical ladder, but you may find this paper interesting:</p> <p><a href="http://arxiv.org/abs/math/0612549" rel="nofollow">http://arxiv.org/abs/math/0612549</a><br> Two-Categorical Bundles and Their Classifying Spaces<br> Authors: Nils. A. Baas, Marcel Bokstedt, Tore August Kro</p> http://mathoverflow.net/questions/23857/what-does-the-classifying-space-of-a-category-classify/23865#23865 Answer by Peter Arndt for What does the classifying space of a category classify? Peter Arndt 2010-05-07T14:43:06Z 2010-05-07T18:27:43Z <p>Ieke Moerdijk has written a small Springer Lecture Notes tome addressing this question:<a href="http://openlibrary.org/books/OL800793M/Classifying_spaces_and_classifying_topoi" rel="nofollow">"Classifying Spaces and Classifying Topoi" SLNM 1616</a>.</p> <p>Roughly the answer is: A G-bundle is a map whose fibers have a G-action, i.e. are G-sets (if they are discrete), i.e. they are functors from G seen as a category to Sets. Likewise a C-bundle for a category C is a map whose fibers are functors from C to sets, or, if you want, a disjoint union of sets (one for each object of C) and an action by the morphisms of C - a morphism A-->B in C takes elements of the set corresponding to A to elements of the set corresponding to B.</p> <p>There is a completely analogous version for topological categories also.</p>