Let \(\mathbb V\in \mathrm{CAlg}(\mathrm{Pr}^\mathrm{L})\) and let \(\widehat{\mathrm{Cat}}[\mathbb V]\) denote the \(\infty\)-category of large \(\mathbb V\)-enriched \(\infty\)-categories equipped with the enriched tensor product. The construction of an enriched \(\infty\)-category from a presentable module category then assembles into a lax symmetric monoidal faithful functor \[\mathrm{Mod}_{\mathbb V}(\mathrm{Pr}^\mathrm{L}) \rightarrow\widehat{\mathrm{Cat}}[\mathbb V].\] In particular, this induces a functor \[ \mathrm{CAlg}(\mathrm{Mod}_{\mathbb V}(\mathrm{Pr}^\mathrm{L})) \simeq \mathrm{CAlg}(\mathrm{Pr}^\mathrm{L})_{\mathbb V/} \rightarrow\mathrm{CAlg}(\widehat{\mathrm{Cat}}[\mathbb V]).\]
4.1.2 Presentably enriched \(\infty\)-categories[007H]
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. As in example 4.1.3, it follows from the adjoint functor theorem that the action is closed, i.e. for any pair of objects \(x, y \in \mathcal C\), there exists a morphism object \(\underline{\mathrm{Hom}}_{\mathcal C}(x, y ) \in \mathbb V\). It is shown in [GH15, Cor. 7.4.13] that these morphism objects assemble \(\mathcal C\) into a \(\mathbb V\)-enriched \(\infty\)-category with space of objects \(\mathcal C^{\simeq}\), and which we will also denote by \(\mathcal C\). By [Hei23, Thm. 7.21, Thm. 1.2], this construction is functorial and multiplicative in the following sense:
The functor ([007J]) will be our meain tool to construct symmetric monoidal enriched \(\infty\)-categories and symmetric monoidal enriched functors between them. In particular, if \(\mathbb V\in \mathrm{CAlg}(\mathrm{Pr}^\mathrm{L})\), then \(\mathbb V\) itself may be considered as self-enriched, i.e. \(\mathbb V\in \mathrm{CAlg}(\widehat{\mathrm{Cat}}[\mathbb V])\).
We follow [MS21, § A.3] and call a \(\mathbb V\)-enriched \(\infty\)-category \(\mathcal C\in \widehat{\mathrm{Cat}}[\mathbb V]\) presentably \(\mathbb V\)-enriched if its underlying \(\infty\)-category is presentable, admits tensors16, and if moreover for every \(v\in \mathbb V\), the induced functor \(v\otimes -\colon \mathcal C\rightarrow\mathcal C\) between the underlying \(\infty\)-categories preserves small colimits.
Let \(\Pr^L_{\mathbb V}\) denote the (non-full) subcategory of \(\widehat{\mathrm{Cat}}[\mathbb V]\) on the presentably \(\mathbb V\)-enriched \(\infty\)-categories \(\mathcal C\) and on those \(\mathbb V\)-enriched functors that are left adjoint in the \(\mathbb V\)-enriched sense,17see [MS21, Def. A.2.12]. Then, it is shown in [MS21, Thm. A.3.8] that the functor \(\mathrm{Mod}_{\mathbb V}(\mathrm{Pr}^\mathrm{L}) \rightarrow\widehat{\mathrm{Cat}}[\mathbb V]\) factors as an equivalence through \(\Pr^L_{\mathbb V}\). In particular, \(\mathrm{Mod}_{\mathbb V}(\mathrm{Pr}^\mathrm{L}) \simeq \Pr^L_{\mathbb V}\) is a subcategory of \(\widehat{\mathrm{Cat}}[\mathbb V]\); it is merely a property of large \(\mathbb V\)-enriched categories and \(\mathbb V\)-enriched functors to be in the image of \(\mathrm{Mod}_{\mathbb V}(\mathrm{Pr}^\mathrm{L}) \rightarrow\widehat{\mathrm{Cat}}[\mathbb V]\).
Original source: arXiv:2401.02956v2
Original source · 2401.02956v2