ScalingStacks

Example 4.1.3.

Let \(\mathbb V\in \mathrm{Alg}(\mathrm{Pr}^\mathrm{L})\) and \(\mathcal C\in \mathrm{LMod}_{\mathbb V}(\mathrm{Pr}^\mathrm{L})\), i.e. \(\mathcal C\) is a presentable \(\infty\)-category with an action \(- \otimes -\colon \mathbb V\times \mathcal C\rightarrow\mathcal C\) by a presentable monoidal \(\infty\)-category \(\mathbb V\) which is cocontinuous in both variables. It follows from the adjoint functor theorem, proposition 3.1.4, that \(\mathrm{Hom}_{\mathcal C}(- \otimes x, y)\colon \mathbb V^{\mathrm{op}} \rightarrow\mathcal S\) is representable for all \(x, y \in \mathcal C\), i.e. that the \(\mathbb V\)-module category \(\mathcal C\) is closed. In particular, any presentably monoidal \(\infty\)-category is closed monoidal.