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.