ScalingStacks

Recall from Sections 3.4.2 and 3.4.3 that the forgetful functor \(\mathrm{st}\rightarrow\mathrm{add}\) from the presentably symmetric monoidal \(\infty\)-category of small stable \(\infty\)-categories to that of small additive \(\infty\)-categories has a symmetric monoidal left adjoint \[{\mathbf K}^b\colon \mathrm{add}\rightarrow\mathrm{st},\] which sends an ordinary additive \(1\)-category \(\mathcal A\) to the \(\infty\)-category of bounded chain complexes in \(\mathcal A\) with chain maps and (higher) chain homotopies between them. By proposition 4.3.2, this is compatible with linearity and \(\mathbb{Z}\)-action and induces a symmetric monoidal left adjoint \[{\mathbf K}^b\colon \mathrm{add}_{k}^{B\mathbb{Z}}\rightarrow\mathrm{st}^{B\mathbb{Z}}_{k}\] of the forgetful functor \(\mathrm{st}^{B\mathbb{Z}}_{k}\rightarrow\mathrm{add}_{k}^{B\mathbb{Z}}\). Applying \({\mathbf K}^b\) homwise, it follows from subsection A.10 and subsection A.8.11 that we obtain symmetric monoidal left-adjoints of the respective forgetful functors: \[{\mathbf K}^b_{\mathrm{loc}}:= \mathrm{Cat}[{\mathbf K}^b] \colon \mathrm{Cat}[\mathrm{add}_{k}^{B\mathbb{Z}}] \rightarrow\mathrm{Cat}[\mathrm{st}^{B\mathbb{Z}}_{k}]\] \[ {\mathbf K}^b_{\mathrm{loc}}\colon \mathrm{Alg}_{\mathbb E_1}\left(\mathrm{Cat}[\mathrm{add}_{k}^{B\mathbb{Z}}]\right) \rightarrow\mathrm{Alg}_{\mathbb E_1}\left(\mathrm{Cat}[\mathrm{st}^{B\mathbb{Z}}_{k}] \right).\]