K-theory of large categories

Abstract

We give a short overview of a new notion of continuous K-theory, which is defined for a certain class of large (enhanced) triangulated categories. For compactly generated triangulated categories, this continuous K-theory gives the usual nonconnective K-theory of the category of compact objects. We formulate a general theorem about the computation of continuous K-theory for the category of sheaves (of modules) on a locally compact Hausdorff space. This result already gives a surprising answer for the category of sheaves on the real line.