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.
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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 \[\underline{\mathrm{Hom}}_{\mathbb V}(v,v') \rightarrow G \underline{\mathrm{Hom}}_{\mathbb W}(Fv, Fv').\]
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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.
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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.
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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 \(\mathrm{Cat}_{\infty}^{\mathcal K} \rightarrow\mathrm{Mod}_{\mathbb V}(\mathrm{Cat}_{\infty}^{\mathcal K})\). ◻