For a small set \(\mathcal K\) of simplicial sets, let \(\mathrm{Cat}_{\infty}^{\mathcal K}\) denote the subcategory of \(\mathrm{Cat}_{\infty}\) on those small \(\infty\)-categories which admit colimits of diagrams indexed by elements of \(\mathcal K\), and those functors which preserve such colimits.
3.1.4 Adjoining colimits[0039]
The \(\infty\)-categories \(\mathrm{Cat}_{\infty}^\mathrm{idem}, \mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}\) and \(\mathrm{Cat}_{\infty}^{\mathrm{rex}, \mathrm{idem}}\) from notation 3.1.1 are instances of \(\mathrm{Cat}_{\infty}^\mathcal K\) for \(\mathcal K\) consisting of the ‘walking idempotent’ of [Lur09, § 4.4.5], or the walking idempotent together with the set of finite (discrete) sets or the set of finite simplicial sets, respectively.
Just like the presheaf \(\infty\)-category \(\mathcal P(\mathcal C)\) is the free completion of a small \(\infty\)-category \(\mathcal C\) under small colimits, we may complete under other classes of colimits. The following combines [Lur17, Lem. 4.8.4.2, Rem. 4.8.1.8] and [Lur09, Cor. 5.3.6.10]:
Let \(\mathcal K\) be a small set of simplicial sets.
The \(\infty\)-category \(\mathrm{Cat}_{\infty}^{\mathcal K}\) is presentable and admits a presentably symmetric monoidal structure, which can be characterized as follows: If \(\mathcal C, \mathcal D\in \mathrm{Cat}_{\infty}^{\mathcal K}\), the tensor product \(\mathcal C\otimes \mathcal D\) is equipped with a functor \(\mathcal C\times \mathcal D\rightarrow\mathcal C\otimes \mathcal D\) which preserves \(\mathcal K\)-colimits separately in both variables and which induces for all \(\mathcal E\in \mathrm{Cat}_{\infty}^{\mathcal K}\) an equivalence \[\mathrm{Fun}^{\mathcal K}(\mathcal C\otimes \mathcal D, \mathcal E) \rightarrow\mathrm{\mathrm{Fun}^{\mathcal K\times \mathcal K}}(\mathcal C\times \mathcal D, \mathcal E),\] where \(\mathrm{Fun}^{\mathcal K}(\mathcal C\otimes \mathcal D, \mathcal E)\) denotes the full subcategory of \(\mathrm{Fun}(\mathcal C\otimes \mathcal D,\mathcal E)\) on those functors which preserve \(\mathcal K\)-colimits and where \(\mathrm{Fun}^{\mathcal K\times \mathcal K}(\mathcal C\times \mathcal D, \mathcal E)\) denotes the full subcategory of \(\mathrm{Fun}(\mathcal C\times \mathcal D, \mathcal E)\) on those functors which preserve \(\mathcal K\)-colimits separately in both variables.
Let \(\mathcal K'\) be a small set of simplicial sets with containing \(\mathcal K\). Then the subcategory inclusion \(\mathrm{Cat}_{\infty}^{\mathcal K'} \rightarrow\mathrm{Cat}_{\infty}^{\mathcal K}\) admits a symmetric monoidal left adjoint \[\mathcal P_{\mathcal K}^{\mathcal K'}\colon \mathrm{Cat}_{\infty}^{\mathcal K} \rightarrow\mathrm{Cat}_{\infty}^{\mathcal K'}\] whose unit \(\mathcal C\rightarrow\mathcal P_{\mathcal K}^{\mathcal K'}(\mathcal C)\) for \(\mathcal C\in \mathrm{Cat}_{\infty}^{\mathcal K'}\) is a fully faithful functor.
The second statement of proposition 3.1.11 implies that \(\mathcal C\hookrightarrow \mathcal P_{\mathcal K}^{\mathcal K'}(\mathcal C)\) may be thought of as a generalized Yoneda embedding: It is the free cocompletion of \(\mathcal C\) under \(\mathcal K'\)-shaped colimits subject to the relation that \(\mathcal K\)-shaped colimits in \(\mathcal C\) are preserved.
Original source: arXiv:2401.02956v2
Original source · 2401.02956v2