A categorical perspective on the Atiyah-Segal completion theorem in KK-theory

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We investigate the homological ideal JGʰ, the kernel of the restriction functors in compact Lie group equivariant Kasparov categories. Applying the relative homological algebra developed by Meyer and Nest, we relate the Atiyah-Segal completion theorem with the comparison of JGʰ with the augmentation ideal of the representation ring. In relation to it, we study on the Atiyah-Segal completion theorem for groupoid equivariant KK-theory, McClure's restriction map theorem, permanence property of the Baum-Connes conjecture under extensions of groups and a class of JG-injective objects coming from C*-dynamical systems, continuous Rokhlin property.