and so on. For example, we can prove certain “approximate cohomology vanishing" type theorems, which play an essential role in the study of actions on von Neumann algebras.

Our main tool is Kasparov’s ${\mathrm{KK}}$-theory [@MR582160]. Although it is a kind of (co)homology theory and hence does not distinguish two homotopic ${\mathrm{C}^*}$-algebras, ${\mathrm{KK}}$-theory is also a powerful tool for classifications of ${\mathrm{C}^*}$-algebras up to isomorphism. For example, the Kirchberg-Phillips classification asserts that two Kirchberg algebras are isomorphic if and only if there is a ${\mathrm{KK}}$-equivalence preserving the unit classes in $K_0$-groups. Here, it is essential in the proof of [@MR1745197] that the ${\mathrm{KK}}$-group has a presentation as the set of homotopy classes of asymptotic morphisms (cf. $E$-theory [@MR1065438]).

In [@mathOA150806815], the authors study the relative homological algebra of compact group equivariant ${\mathrm{KK}}$-theory in connection with the Atiyah-Segal completion theorem. Here, we introduce a categorical counterpart of freeness of group actions on ${\mathrm{C}^*}$-algebras, ${\mathfrak{J}}_G^n$-injectivity. It directly follows from Thomsen’s picture of equivariant ${\mathrm{KK}}$-theory [@MR1680467] that $G$-${\mathrm{C}^*}$-algebras with continuous Rokhlin property [@mathOA14071277], a variation of Rokhlin property, is ${\mathfrak{J}}_G$-injective. More generally, if a ${\mathrm{C}^*}$-dynamical system has continuous Rokhlin dimension with commuting tower less than $d$, then it is ${\mathfrak{J}}_G^{d+1}$-injective (Theorem \[thm:Rokinj\]). This theorem 