3.1.1 The Yoneda embedding[002P]
Any small \(\infty\)-category \(\mathcal C\) has a Yoneda embedding into its \(\infty\)-category of (\(\mathcal S\)-valued) presheaves \(\mathcal P(\mathcal C)=\mathrm{Fun}(\mathcal C^\mathrm{op}, \mathcal S)\), [Lur09, Prop. 5.1.3.1], see also [Cis19]. It is characterized by the universal property that \(\mathcal P(\mathcal C)\) has all small colimits (i.e. is cocomplete) [Lur09, Cor. 5.1.2.4], and that for any cocomplete \(\infty\)-category \(\mathcal D\) the restriction along the Yoneda embedding induces an equivalence \[
\mathrm{Fun^L}(\mathcal P(\mathcal C),\mathcal D) \rightarrow\mathrm{Fun}(\mathcal C,\mathcal D)\] where \(\mathrm{Fun^L}\) denotes the full subcategory of the \(\infty\)-category of functors on those functors which preserve all small colimits (i.e. the cocontinuous functors) [Lur09, Thm. 5.1.5.6], see also [Cis19, Thm. 6.3.13]. An \(\infty\)-category \(\mathcal C\) is called idempotent complete if its image under the Yoneda embedding \(\mathcal C\rightarrow\mathcal P(\mathcal C)\) is closed under retracts (see [Lur09, Proof of Prop. 5.1.4.2]). We refer to [Lur09, § 4.4.5] for a discussion of retracts and idempotents in \(\infty\)-categories.
[002R]
Notation 3.1.1.
We write
\(\mathrm{Cat}_{\infty}^\mathrm{idem}\) for the full subcategory of \(\mathrm{Cat}_{\infty}\) on the idempotent complete small \(\infty\)-categories,
\(\mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}\) for the subcategory of \(\mathrm{Cat}_{\infty}^\mathrm{idem}\) on the idempotent complete small \(\infty\)-categories that admit finite coproducts and functors which preserve finite coproducts, and
\(\mathrm{Cat}_{\infty}^{\mathrm{rex}, \mathrm{idem}}\) for the subcategory of \(\mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}\) on the idempotent complete small \(\infty\)-categories that admit finite colimits and functors which preserve finite colimits.