Author: Peter Webb email: webb@math.umn.edu Address: School of Mathematics, University of Minnesota, Minneapolis MN 55455 USA Title: The Low-Dimensional Cohomology of Categories Abstract: This is a report of a talk given at Oberwolfach during the week on Group Representations March 27-31, 2006. It is not refereed. The following was given as an abstract of the talk: Many things which are usually done with the cohomology of groups can also be done with the cohomology of small categories. Some of these things have been present in the mathematical literature for a long time, including a formulation of extension theory with a correspondence between equivalence classes of extensions and second cohomology classes, and the role of derivations in describing the first cohomology. Other things appear not to have been described so far. These include 5-term sequences associated to an extension of categories, a description of the properties of the Schur multiplier of a category (defined as a second homology group), and the construction of a relation module associated to an epimorphism from a free category. The development of this theory is timely in view of the interest in representations of categories and cohomology of categories currently shown in a broad range of topics in group representation theory and group cohomology. These topics include the theories of $p$-local finite groups, homology decompositions of classifying spaces, $p$-locally determined properties of group representations including conjectures of Alperin and Brou\'e and generalizations of these conjectures, as well as presentations of groups. Journal: Oberwolfach Reports.