Non-commutative ambits and equivariant compactifications

We prove that an action $ρ:A\to M(C_0(\mathbb{G})\otimes A)$ of a locally compact quantum group on a $C^*$-algebra has a universal equivariant compactification, and prove a number of other category-theoretic results on $\mathbb{G}$-equivariant compactifications: that the categories compactifications of $ρ$ and $A$ respectively are locally presentable (hence complete and cocomplete), that the forgetful functor between them is a colimit-creating left adjoint, and that epimorphisms therein are surjective and injections are regular monomorphisms. When $\mathbb{G}$ is regular coamenable we also show that the forgetful functor from unital $\mathbb{G}$-$C^*$-algebras to unital $C^*$-algebras creates finite limits and is comonadic, and that the monomorphisms in the former category are injective.

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