4.1 Presentably enriched \(\infty\)-categories[0073]
4.1.1 Closed monoidal \(\infty\)-categories and closed module \(\infty\)-categories[0074]
[0075]
Definition 4.1.1. ([Lur17, Def. 4.2.1.28]).
Let \(\mathbb V\) be a (possibly large) monoidal \(\infty\)-category, and \(\mathcal C\) a left \(\mathbb V\)-module \(\infty\)-category. A morphism object between objects \(x,y \in \mathcal C\) is an object \(\underline{\mathrm{Hom}}_{\mathcal C}(x,y) \in \mathbb V\) representing the presheaf \(\mathrm{Hom}_{\mathcal C}(- \otimes x, y) \colon \mathbb V^{\mathrm{op}} \rightarrow\mathcal S\), i.e. equipped with isomorphisms natural in \(v\in \mathbb V\) \[\mathrm{Hom}_{\mathbb V}(v, \underline{\mathrm{Hom}}_{\mathcal C}(x,y)) \simeq \mathrm{Hom}_{\mathcal C}(v \otimes x, y).\] A \(\mathbb V\)-module category \(\mathcal C\) is closed if a morphism object exists between every pair of objects \(x, y \in \mathcal C\). A closed monoidal \(\infty\)-category is a monoidal \(\infty\)-category whose left action on itself is closed.
[0076]
Observation 4.1.2.
Let \(F \colon \mathbb V\rightarrow\mathbb W\) be a monoidal functor from a monoidal \(\infty\)-category to a closed monoidal \(\infty\)-category \(\mathbb W\) which is left adjoint to a functor \(G\). Then, the induced \(\mathbb V\)-action on \(\mathbb W\) is closed with morphism object \(G \underline{\mathrm{Hom}}_{\mathbb W}(w, w') \in \mathbb V\) for \(w, w' \in \mathbb W\). If \(\mathbb V\) is also closed monoidal, then for \(v, v' \in \mathbb V\), the map of spaces \(\mathrm{Hom}_{\mathbb V}(v,v') \rightarrow\mathrm{Hom}_{\mathbb W}(Fv, Fv')\) lifts along \(\mathrm{Hom}_{\mathbb V}(I, -)\colon \mathbb V\rightarrow\mathcal S\) to a \(\mathbb V\)-morphism \[\underline{\mathrm{Hom}}_{\mathbb V}(v,v') \rightarrow G \underline{\mathrm{Hom}}_{\mathbb W}(Fv, Fv').\]
[0077]
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.
[0078]
Example 4.1.4.
Let \(\mathcal K\) be a set of simplicial sets and recall from \(\mathrm{Cat}_{\infty}^{\mathcal K}\) the presentably symmetric monoidal \(\infty\)-category of \(\infty\)-categories with \(\mathcal K\)-colimits and \(\mathcal K\)-colimit preserving functors. For \(\mathcal C, \mathcal D\in \mathrm{Cat}_{\infty}^{\mathcal K}\), the full subcategory \(\mathrm{Fun}^{\mathcal K}(\mathcal C, \mathcal D)\) of \(\mathrm{Fun}(\mathcal C, \mathcal D)\) on the \(\mathcal K\)-colimit preserving functors is closed under \(\mathcal K\)-colimits [Lur17, Rem. 4.8.4.14] and hence is an object of \(\mathrm{Cat}_{\infty}^{\mathcal K}\). It follows directly from the characterization of the tensor product in \(\mathrm{Cat}_{\infty}^{\mathcal K}\), see proposition 3.1.11, that \(\mathrm{Fun}^{\mathcal K}(\mathcal C, \mathcal D) \in \mathrm{Cat}_{\infty}^{\mathcal K}\) is the morphism object between \(\mathcal C\) and \(\mathcal D\) in \(\mathrm{Cat}_{\infty}^{\mathcal K}\) (cf. proof of [Lur17, Lem. 4.8.4.2]).
We generalize example 4.1.4 to module categories using the following terminology.
[0079]
Notation 4.1.5.
Let \(\mathcal K\) be a small set of simplicial sets, \(\mathbb V\in \mathrm{Alg}(\mathrm{Cat}_{\infty}^{\mathcal K})\) and \(\mathcal C, \mathcal D\in \mathrm{LMod}_{\mathbb V}(\mathrm{Cat}_{\infty}^{\mathcal K})\). Let \(\mathrm{Fun}_{\mathbb V}(\mathcal C, \mathcal D)\) be the \(\infty\)-category of \(\mathbb V\)-module functors [Lur17, Def. 4.6.2.7] and \(\mathrm{Fun}^{\mathcal K}_{\mathbb V}(\mathcal C, \mathcal D) \subset \mathrm{Fun}_{\mathbb V}(\mathcal C, \mathcal D)\) the full subcategory on those module functors whose underlying functors preserve \(\mathcal K\)-colimits.
By [Lur17, Rem. 4.8.4.14], \(\mathrm{Fun}^{\mathcal K}_{\mathbb V}(\mathcal C, \mathcal D)\) is closed under \(\mathcal K\)-colimits, thus an object of \(\mathrm{Cat}_{\infty}^{\mathcal K}\).
[007A]
Lemma 4.1.6.
Let \(\mathcal K\) be a small set of simplicial sets and let \(\mathbb V\in \mathrm{Alg}(\mathrm{Cat}_{\infty}^{\mathcal K})\). Cconsider the right action of \(\mathrm{Cat}_{\infty}^{\mathcal K}\) on \(\mathrm{LMod}_{\mathbb V}(\mathrm{Cat}_{\infty}^{\mathcal K})\). Let \(\mathcal C, \mathcal D\in \mathrm{LMod}_{\mathbb V}(\mathrm{Cat}_{\infty}^{\mathcal K})\). Then, the following hold.
\(\mathrm{Fun}^{\mathcal K}_{\mathbb V}(\mathcal C, \mathcal D)\) is a morphism object in \(\mathrm{Cat}_{\infty}^{\mathcal K}\) between \(\mathcal C, \mathcal D\in \mathrm{LMod}_{\mathbb V}(\mathrm{Cat}_{\infty}^{\mathcal K})\).
If \(\mathbb V\) is furthermore symmetric monoidal, then \(\mathrm{Fun}^{\mathcal K}_{\mathbb V}(\mathcal C, \mathcal D)\) admits a \(\mathbb V\)-action which makes it into a morphism object in \(\mathrm{Mod}_{\mathbb V}(\mathrm{Cat}_{\infty}^{\mathcal K})\).
[007D]
Proof.
We first prove statement ([007B]) for \(\mathcal K= \emptyset\). Consider the locally coCartesian fibration \(\mathcal C^{\circledast} \rightarrow\mathbb V^{\circledast}\) from [Lur17, Not. 4.2.2.17, Lem. 4.2.2.20] associated to a \(\mathbb V\)-module category \(\mathcal C\). It follows from [Lur17, Lem. 4.8.4.12] that \(\mathrm{Fun}_{\mathbb V}(\mathcal C, \mathcal D) \subset \mathrm{Fun}_{/\mathbb V^{\circledast}}(\mathcal C^{\circledast}, \mathcal D^{\circledast})\) is the full subcategory on those functors which preserve locally coCartesian morphisms, where for given functors \(F\colon \mathcal A\rightarrow\mathcal B\leftarrow \mathcal C\colon G\) of \(\infty\)-categories, we let \(\mathrm{Fun}_{/ \mathcal B}(\mathcal A, \mathcal C) \coloneqq \mathrm{Fun}(\mathcal A, \mathcal C) \times_{\mathrm{Fun}(\mathcal A, \mathcal B)} \{F\}\) denote the over-functor category. If \(\mathcal C, \mathcal D\in \mathrm{LMod}_{\mathbb V}(\mathrm{Cat}_{\infty})\) and \(\mathcal A\in \mathrm{Cat}_{\infty}\), the evident equivalence \[\mathrm{Fun}(\mathcal A, \mathrm{Fun}_{/\mathbb V^\circledast}(\mathcal C^{\circledast}, \mathcal D^{\circledast})) \simeq \mathrm{Fun}_{/\mathbb V^{\circledast}}(\mathcal A\times \mathcal C^{\circledast} , \mathcal D^{\circledast}) \simeq \mathrm{Fun}_{/\mathbb V^{\circledast}}((\mathcal A\times \mathcal C)^{\circledast} , \mathcal D^{\circledast})\] restricts to an equivalence \[
\mathrm{Fun}(\mathcal A, \mathrm{Fun}_{\mathbb V}(\mathcal C, \mathcal D)) \simeq \mathrm{Fun}_{\mathbb V}(\mathcal A\times \mathcal C, \mathcal D)\] which upon passing to maximal \(\infty\)-subgroupoids shows that \(\mathrm{Fun}_{\mathbb V}(\mathcal C, \mathcal D)\) is the morphism object for the action of \(\mathrm{Cat}_{\infty}\) on \(\mathrm{LMod}_{\mathbb V}(\mathrm{Cat}_{\infty})\).
Now let \(\mathcal K\) be general. Let \(\mathcal A\in \mathrm{Cat}_{\infty}^{\mathcal K}\) and \(\mathcal C, \mathcal D\in \mathrm{LMod}_{\mathbb V}(\mathrm{Cat}_{\infty}^{\mathcal K})\), and let \(\otimes\) denote the action of \(\mathrm{Cat}_{\infty}^{\mathcal K}\) on \(\mathrm{LMod}_{\mathbb V}(\mathrm{Cat}_{\infty}^{\mathcal K})\). By definition of the action, it induces an equivalence \[
\mathrm{Fun}^{\mathcal K}_{\mathbb V}(\mathcal A\otimes \mathcal C, \mathcal D) \simeq \mathrm{Fun}^{\mathcal K\times\mathcal K}_{\mathbb V}(\mathcal A\times \mathcal C, \mathcal D),\] where \(\mathrm{Fun}^{\mathcal K\times \mathcal K}_{\mathbb V}(\mathcal A\times \mathcal C, \mathcal D) \subset \mathrm{Fun}_{\mathbb V}(\mathcal A\times \mathcal C, \mathcal D)\) denotes the full subcategory of \(\mathbb V\)-linear functors whose underlying functor \(\mathcal A\times \mathcal C\rightarrow\mathcal D\) preserves \(\mathcal K\)-index colimits separately in each variable. On the other hand, by the description of \(\mathcal K\)-indexed colimits in \(\mathrm{Fun}^{\mathcal K}_{\mathbb V}(\mathcal C, \mathcal D)\) [Lur17, Lem. 4.8.4.13], the equivalence ([007E]) restricts to an equivalence of full subcategories \[
\mathrm{Fun}^{\mathcal K}(\mathcal A, \mathrm{Fun}^{\mathcal K}_{\mathbb V}(\mathcal C, \mathcal D)) \simeq \mathrm{Fun}^{\mathcal K, \mathcal K}_{\mathbb V}(\mathcal A\times \mathcal C, \mathcal D).\] Composing ([007F]) and ([007G]) exhibits \(\mathrm{Fun}^{\mathcal K}_{\mathbb V}(\mathcal C, \mathcal D)\) as the morphism object of \(\mathcal C, \mathcal D\) in \(\mathrm{Cat}_{\infty}^{\mathcal K}\). This proves part ([007B]). Part ([007C]) follows now with observation 4.1.2 applied to the (symmetric) monoidal left adjoint \(\mathrm{Cat}_{\infty}^{\mathcal K} \rightarrow\mathrm{Mod}_{\mathbb V}(\mathrm{Cat}_{\infty}^{\mathcal K})\). ◻
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:
[007I]
Proposition 4.1.7.
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 \[ \mathrm{CAlg}(\mathrm{Mod}_{\mathbb V}(\mathrm{Pr}^\mathrm{L})) \simeq \mathrm{CAlg}(\mathrm{Pr}^\mathrm{L})_{\mathbb V/} \rightarrow\mathrm{CAlg}(\widehat{\mathrm{Cat}}[\mathbb V]).\]
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. \(\mathbb V\in \mathrm{CAlg}(\widehat{\mathrm{Cat}}[\mathbb V])\).
[007K]
Notation 4.1.8.
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 tensors, 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.