The localization theorem allows one to study the equivariant
cohomology of a smooth variety with a torus action in terms of certain data
at the fixed locus of that torus action, but only after formally inverting
certain elements in the equivariant cohomology ring of a point. Similar
results apply in equivariant topological and algebraic K-theory. It is
natural to ask, especially from the perspective of modern categorical
methods in geometric representation theory, if there is a categorical
analog: can one describe the category of equivariant coherent sheaves in
terms of certain data which localizes around the fixed points? At the heart
of this is the question of what the analog of "inverting elements in the
equivariant cohomology of a point" should mean. I will discuss one solution
to these questions, and describe how it is related to yet another
"localization" theorem in geometric representation theory.