For \(\mathbb{K}\in \mathrm{CAlg}(\mathrm{Sp}_{\geq 0})\), we define
the \(\infty\)-category \({\mathrm{Pr}^{\mathrm{L}}_{{\mathrm{add}_{\mathbb{K}}}}}\) of additive presentable \(\mathbb{K}\)-linear \(\infty\)-categories as \[{\mathrm{Pr}^{\mathrm{L}}_{{\mathrm{add}_{\mathbb{K}}}}}:= \mathrm{Mod}_{\mathrm{Mod}^{\geq 0}_{\mathbb{K}}}(\mathrm{Pr}^\mathrm{L});\]
the \(\infty\)-category \(\mathrm{add}_{\mathbb{K}}\) of small additive, idempotent-complete \(\mathbb{K}\)-linear \(\infty\)-categories as \[\mathrm{add}_{\mathbb{K}}:=\mathrm{Mod}_{\mathrm{CProj}_{\mathbb{K}}}(\mathrm{add}).\]
For \(\mathbb{K}\in \mathrm{CAlg}(\mathrm{Sp})\), we define
the \(\infty\)-category \({\mathrm{Pr}^{\mathrm{L}}_{{\mathrm{st}_{\mathbb{K}}}}}\) of stable presentable \(\mathbb{K}\)-linear \(\infty\)-categories as \[{\mathrm{Pr}^{\mathrm{L}}_{{\mathrm{st}_{\mathbb{K}}}}}:= \mathrm{Mod}_{\mathrm{Mod}_{\mathbb{K}}}(\mathrm{Pr}^\mathrm{L});\]
the \(\infty\)-category \(\mathrm{st}_{\mathbb{K}}\) of small stable, idempotent-complete \(\mathbb{K}\)-linear \(\infty\)-categories to be \[\mathrm{st}_{\mathbb{K}}:= \mathrm{Mod}_{\mathrm{Perf}_{\mathbb{K}}}(\mathrm{st}).\]