ScalingStacks

The forgetful functor \(\mathrm{add}_k \rightarrow\mathrm{Cat}_{\infty}\) has a symmetric monoidal left adjoint ‘linearization functor’ \[\mathrm{Lin}_k\colon \mathrm{Cat}_{\infty}\rightarrow\mathrm{add}_k,\] which is the composite of symmetric monoidal left adjoints \[ \mathrm{Cat}_{\infty}\xrightarrow{(-)^{\sqcup, \mathrm{idem}}}\mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}\xrightarrow{\mathrm{CProj}_k \otimes -} \mathrm{Mod}_{\mathrm{CProj}_k}(\mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}) \simeq \mathrm{add}_k,\] where \((-)^{\sqcup, \mathrm{idem}}\) freely adjoints finite coproducts and splittings of idempotents (see proposition 3.1.11) and \(\mathrm{CProj}_k\otimes -\) constructs free \(\mathrm{CProj}_k\)-modules (see proposition 3.1.8). This induces a symmetric monoidal left adjoint of the forgetful functor \(\mathrm{add}_{k}^{B\mathbb{Z}}\rightarrow\mathrm{Cat}_{\infty}^{B\mathbb{Z}}\) \[\mathrm{Fun}(B\mathbb{Z}, \mathrm{Lin}_k(-))\colon \mathrm{Cat}_{\infty}^{B\mathbb{Z}}\rightarrow\mathrm{add}_{k}^{B\mathbb{Z}},\] which we will also denote by \(\mathrm{Lin}_k(-) \colon \mathrm{Cat}_{\infty}^{B\mathbb{Z}}\rightarrow\mathrm{add}_{k}^{B\mathbb{Z}}\). Unpacking observation 3.5.13, the forgetful functor \(\mathrm{Cat}_{\infty}^{B \mathbb{Z}} \rightarrow\mathrm{Cat}_{\infty}\) (i.e. the functor \(\mathrm{ev}_*\colon \mathrm{Fun}(B\mathbb{Z}, \mathrm{Cat}_{\infty}) \rightarrow\mathrm{Cat}_{\infty}\)) has a left adjoint \[- \times \mathbb{Z}: \mathrm{Cat}_{\infty}\rightarrow\mathrm{Cat}_{\infty}^{B\mathbb{Z}}\] which sends an \(\infty\)-category \(\mathcal C\) to the \(\infty\)-category \(\mathcal C\times \mathbb{Z}\) with free \(\mathbb{Z}\)-action, and which is symmetric monoidal with respect to the Day convolution structure on \(\mathrm{Cat}_{\infty}^{B\mathbb{Z}}\).