Let |$T$| be a torus, |$X$| a smooth separated scheme of finite type equipped with a |$T$| -action, and |$[X/T]$| the associated quotient stack. Given any localizing |${\mathbb {A}}^{1}$| -homotopy invariant of dg categories |$E$| (homotopy |$K$| -theory, algebraic |$K$| -theory with coefficients, étale |$K$| -theory with coefficients, |$l$| -adic algebraic |$K$| -theory, |$l$| -adic étale |$K$| -theory, semi-topological |$K$| -theory, topological |$K$| -theory, periodic cyclic homology, etc), we prove that the derived completion of |$E([X/T])$| at the augmentation ideal |$I$| of the representation ring |$R(T)$| of |$T$| agrees with the classical Borel construction associated to the |$T$| -action on |$X$|. Moreover, for certain localizing |${\mathbb {A}}^{1}$| -homotopy invariants, we extend this result to the case of a linearly reductive group scheme |$G$|. As a first application, we obtain an alternative proof of Krishna's completion theorem in algebraic |$K$| -theory, of Thomason's completion theorem in étale |$K$| -theory with coefficients, and also of Atiyah-Segal's completion theorem in topological |$K$| -theory (for those topological |$M$| -spaces |$X^{\textrm {an}}$| arising from analytification; |$M$| is a(ny) maximal compact Lie subgroup of |$G^{\textrm {an}}$|). These alternative proofs lead to a spectral enrichment of the corresponding completion theorems and also to the following improvements: in the case of Thomason's completion theorem the base field |$k$| no longer needs to be separably closed, and in the case of Atiyah-Segal's completion theorem the topological |$M$| -space |$X^{\textrm {an}}$| no longer needs to be compact and the |$M$| -equivariant topological |$K$| -theory groups of |$X^{\textrm {an}}$| no longer need to be finitely generated over the representation ring |$R(M)$|. As a second application, we obtain new completion theorems in |$l$| -adic étale |$K$| -theory, in semi-topological |$K$| -theory and also in periodic cyclic homology. As a third application, we obtain a description of the different equivariant cohomology groups in the literature (motivic, |$l$| -adic, morphic, Betti, de Rham, etc) in terms of derived completion. Finally, in two appendixes of independent interest, we extend a result of Weibel on homotopy |$K$| -theory from the realm of schemes to the broad setting of quotient stacks and establish some useful properties of semi-topological |$K$| -theory. [ABSTRACT FROM AUTHOR]