ScalingStacks

Observation 3.5.17.

Assume \(\mathcal Z\in \mathrm{CAlg}(\mathcal S)\) and \(\mathbb{K}\in \mathrm{CAlg}(\mathrm{Sp}_{\geq 0})\). The symmetric monoidal functor \(- \otimes \mathrm{Sp}\colon \mathrm{Pr}^{\mathrm{L}, \mathrm{cp}}_{\mathrm{add}}\rightarrow\mathrm{Pr}^{\mathrm{L}, \mathrm{c}}_{\mathrm{st}}\) from construction 3.4.4 takes \(\mathrm{Mod}_{\mathbb{K}}^{\geq 0, \mathcal Z}\) with its Day convolution monoidal structure to \(\mathrm{Mod}_{\mathbb{K}}^{\mathcal Z}\) with its Day convolution monoidal structure. Indeed, we have the following sequence\[\mathrm{Mod}_{\mathbb{K}}^{\geq 0, \mathcal Z} \otimes \mathrm{Sp}\simeq \mathcal P(\mathcal Z) \otimes \mathrm{Mod}_{\mathbb{K}}^{\geq 0} \otimes \mathrm{Sp}\simeq \mathcal P(\mathcal Z) \otimes \mathrm{Mod}_{\mathbb{K}} \simeq \mathrm{Mod}_{\mathbb{K}}^{\mathcal Z}\] of symmetric monoidal equivalences. In particular, it follows that the fully faithful inclusion \(\mathrm{Mod}_{\mathbb{K}}^{\geq 0, \mathcal Z} \hookrightarrow \mathrm{Mod}_{\mathbb{K}}^{\mathcal Z}\) is symmetric monoidal and hence a morphism in \(\mathrm{CAlg}(\mathrm{Pr}^\mathrm{L})\).