We introduce a new perspective on the Atiyah-Segal completion theorem
and generalize it for equivariant KK-theory of operator algebras. The
homological algebra of the Kasparov category, which is developed by Ralf
Meyer and Ryzard Nest, plays a central role. As corollaries, we
introduce Atiyah-Segal type completion theorems for K-homology and
twisted K-theory. This is a joint work with Yuki Arano.