Given \(\mathbb V\in \mathrm{Alg}(\mathrm{Pr}^{\mathrm{L},\mathrm{cp}})\) and \(A, B \in \mathrm{Alg}(\mathbb V)\), we denote by \[{}_{A}\mathrm{BMod}^{\mathrm{cp}}_{B}(\mathbb V)\subseteq {}_{A}\mathrm{BMod}_{B}(\mathbb V)\]the full subcategory on those \(A\)–\(B\)-bimodules which are compact-projective as right \(B\)-modules.
Similarly, given \(\mathbb V\in \mathrm{Alg}(\mathrm{Pr}^{\mathrm{L},\mathrm{c}})\) and \(A, B \in \mathrm{Alg}(\mathbb V)\), we denote by \[{}_{A}\mathrm{BMod}^{\mathrm{c}}_{B}(\mathbb V)\subseteq {}_{A}\mathrm{BMod}_{B}(\mathbb V)\] the full subcategory on those \(A\)–\(B\)-bimodules which are compact as right \(B\)-modules.