ScalingStacks

Applying \(\mathrm{Lin}_k(-\times \mathbb{Z})\) homwise, this induces by subsection A.10 and subsection A.8.11 symmetric monoidal left adjoints \[ \mathrm{Lin}_{k, \mathrm{loc}}^{\mathbb{Z}}\coloneqq \mathrm{Cat}[\mathrm{Lin}_k(-\times \mathbb{Z})]\colon \mathrm{Cat}_{(\infty, {2})} = \mathrm{Cat}[\mathrm{Cat}_{\infty}] \rightarrow\mathrm{Cat}[\mathrm{add}_{k}^{B\mathbb{Z}}]\] and (abusing notation) \[\mathrm{Lin}_{k, \mathrm{loc}}^{\mathbb{Z}} \coloneqq \mathrm{Alg}_{\mathbb E_1}(\mathrm{Cat}[\mathrm{Lin}_k(-\times \mathbb{Z})])\colon \mathrm{Alg}_{\mathbb E_1}(\mathrm{Cat}_{(\infty, {2})}) \rightarrow\mathrm{Alg}_{\mathbb E_1}(\mathrm{Cat}[\mathrm{add}_{k}^{B\mathbb{Z}}])\] of the respective forgetful functors.