most recent 30 from http://mathoverflow.net 2013-05-18T06:32:52Z http://mathoverflow.net/feeds/question/123680 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/123680/k-theory-of-a-certain-category-of-groups Werner Thumann 2013-03-05T23:09:50Z 2013-03-05T23:09:50Z

<p>Consider the category of groups together with group homomorphisms. Define a homomorphism to be a weak equivalence if it has a finite kernel and an image of finite index. One can compute that this category is a <a href="http://nlab.mathforge.org/nlab/show/homotopical+category" rel="nofollow">homotopical</a> category satisfying a right <a href="http://nlab.mathforge.org/nlab/show/calculus+of+fractions" rel="nofollow">calculus of fractions</a>. Two groups are isomorphic in the homotopy category iff they are virtually isomorphic. The category has even more structure, it is a <a href="http://nlab.mathforge.org/nlab/show/Waldhausen+category" rel="nofollow">Waldhausen</a> category (with fibrations) when defining the surjections as fibrations.</p> <p>What can be said about the K-groups or even the spectrum of this category?</p>