Definition 8.5.3.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. An idempotent in $\operatorname{\mathcal{C}}$ is a functor of $\infty $-categories $\operatorname{N}_{\bullet }( \operatorname{Idem}) \rightarrow \operatorname{\mathcal{C}}$. Here $\operatorname{Idem}$ denotes the category introduced in Construction 8.5.2.7.
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8.5.3 Idempotents in $\infty $-Categories
We now consider an $\infty $-categorical counterpart of Definition 8.5.2.1.
Remark 8.5.3.2. Let $\operatorname{\mathcal{C}}$ be a category. It follows from Remark 8.5.2.8 (and Proposition 1.3.3.1) that evaluation on the morphism $\widetilde{e} \in \operatorname{Hom}_{\operatorname{Idem}}( \widetilde{X}, \widetilde{X})$ supplies a bijection from the set of idempotents in the $\infty $-category $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ (in the sense of Definition 8.5.3.1) to the set of idempotent endomorphisms $(X,e)$ in the category $\operatorname{\mathcal{C}}$ (in the sense of Definition 8.5.2.1). We can therefore view Definition 8.5.3.1 as a generalization of Definition 8.5.2.1.
Remark 8.5.3.3 (The Structure of $\operatorname{N}_{\bullet }( \operatorname{Idem})$). For every integer $n \geq 0$, the simplicial set $\operatorname{N}_{\bullet }( \operatorname{Idem})$ contains a unique nondegenerate $n$-simplex $\sigma _{n}$, given by the diagram Moreover, the face morphisms of $\operatorname{N}_{\bullet }( \operatorname{Idem})$ satisfy $d^{n}_{i}( \sigma _ n) = \sigma _{n-1}$ for $0 \leq i \leq n$. Applying Corollary 3.3.1.8, we obtain an isomorphism of $\operatorname{N}_{\bullet }( \operatorname{Idem})$ with the simplicial set $(\Delta ^0)^{+}$ introduced in Construction 3.3.1.6. Here we abuse notation by identifying $\Delta ^0$ with its underlying semisimplicial set.
\[ \widetilde{X} \xrightarrow { \widetilde{e}} \widetilde{X} \xrightarrow {\widetilde{e}} \widetilde{X} \xrightarrow {\widetilde{e}} \cdots \rightarrow \widetilde{X} \xrightarrow { \widetilde{e}} \widetilde{X}. \]
Remark 8.5.3.4. The simplicial set $\operatorname{N}_{\bullet }( \operatorname{Idem})$ is weakly contractible. This is a special case of Lemma 3.4.5.9, applied to the (discrete) category $[0]$.
Definition 8.5.3.5 (Split Idempotents). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $F: \operatorname{N}_{\bullet }( \operatorname{Idem}) \rightarrow \operatorname{\mathcal{C}}$ be an idempotent in the $\infty $-category $\operatorname{\mathcal{C}}$. A splitting of $F$ is a functor $\overline{F}: \operatorname{N}_{\bullet }( \operatorname{Ret}) \rightarrow \operatorname{\mathcal{C}}$ satisfying $\overline{F}|_{ \operatorname{N}_{\bullet }( \operatorname{Idem}) } = F$. We say that $F$ is split if there exists a splitting of $F$.
Example 8.5.3.6. Let $\operatorname{\mathcal{C}}$ be a category and let $e: X \rightarrow X$ be an idempotent endomorphism in $\operatorname{\mathcal{C}}$. Then $e$ is split (in the sense of Example 8.5.2.3) if and only if the induced map $\operatorname{N}_{\bullet }( \operatorname{Idem}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is a split idempotent in the $\infty $-category $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ (in the sense of Definition 8.5.3.5).
Remark 8.5.3.7. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $F,F': \operatorname{N}_{\bullet }( \operatorname{Idem}) \rightarrow \operatorname{\mathcal{C}}$ be idempotents which are isomorphic (when regarded as objects of the $\infty $-category $\operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Idem}), \operatorname{\mathcal{C}})$). Then $F$ is split if and only if $F'$ is split. See Corollary 4.4.5.3.
Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $F: \operatorname{N}_{\bullet }( \operatorname{Idem}) \rightarrow \operatorname{\mathcal{C}}$ be an idempotent in $\operatorname{\mathcal{C}}$. If $F$ is split, then the splitting is essentially unique.
Proposition 8.5.3.8. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\overline{F}: \operatorname{N}_{\bullet }( \operatorname{Ret}) \rightarrow \operatorname{\mathcal{C}}$ be a functor. Then $\overline{F}$ is both left and right Kan extended from the full subcategory $\operatorname{N}_{\bullet }( \operatorname{Idem}) \subset \operatorname{N}_{\bullet }( \operatorname{Ret})$.
Proof. This is a special case of Proposition 8.5.1.8. $\square$
Remark 8.5.3.9. The category $\operatorname{Ret}$ of Construction 8.5.0.2 contains an initial object $\widetilde{Y}$. It follows that the inclusion map $\operatorname{Idem}\hookrightarrow \operatorname{Ret}$ has a unique extension $T: \operatorname{Idem}^{\triangleleft } \rightarrow \operatorname{Ret}$ carrying the cone point of $\operatorname{Idem}^{\triangleleft }$ to the object $\widetilde{Y}$. Unwinding the definitions, we see that a functor of $\infty $-categories $\overline{F}: \operatorname{N}_{\bullet }( \operatorname{Ret}) \rightarrow \operatorname{\mathcal{C}}$ is right Kan extended from $\operatorname{N}_{\bullet }( \operatorname{Idem})$ if and only if the composition is a limit diagram. Proposition 8.5.3.8 asserts that this condition is automatically satisfied. In particular, the object $\overline{F}( \widetilde{Y} )$ is a limit of the underlying diagram $F = \overline{F}|_{ \operatorname{N}_{\bullet }( \operatorname{Idem})}$. Similarly, $\overline{F}( \widetilde{Y} )$ is a colimit of the diagram $F$.
\[ \operatorname{N}_{\bullet }( \operatorname{Idem})^{\triangleleft } \xrightarrow { \operatorname{N}_{\bullet }(T) } \operatorname{N}_{\bullet }( \operatorname{Ret}) \xrightarrow { F } \operatorname{\mathcal{C}} \]
Corollary 8.5.3.10 (Uniqueness of Splittings). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then the restriction functor is fully faithful, and its essential image is the full subcategory consists of the split idempotents in $\operatorname{\mathcal{C}}$.
\[ \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Ret}), \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Idem}), \operatorname{\mathcal{C}}) \]
Proof. Combine Proposition 8.5.3.8 with Corollary 7.3.6.15. $\square$
Corollary 8.5.3.11. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $F: \operatorname{N}_{\bullet }(\operatorname{Idem}) \rightarrow \operatorname{\mathcal{C}}$ be an idempotent in $\operatorname{\mathcal{C}}$. The following conditions are equivalent:
The idempotent $F$ is split: that is, it can be extended to a functor $\operatorname{N}_{\bullet }( \operatorname{Ret}) \rightarrow \operatorname{\mathcal{C}}$.
The diagram $F$ admits a limit in $\operatorname{\mathcal{C}}$.
The diagram $F$ admits a colimit in $\operatorname{\mathcal{C}}$.
Proof. The implications $(1) \Rightarrow (2)$ and $(1) \Rightarrow (3)$ follow from Remark 8.5.3.9, and the reverse implications follow from Corollary 7.3.5.8. $\square$
Corollary 8.5.3.12. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $F: \operatorname{N}_{\bullet }( \operatorname{Idem}) \rightarrow \operatorname{\mathcal{C}}$ be an idempotent in $\operatorname{\mathcal{C}}$. If $F$ admits a limit (or colimit) in $\operatorname{\mathcal{C}}$, then it is preserved by any functor of $\infty $-categories $G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$.
Proof. Suppose that $F$ admits a limit in $\operatorname{\mathcal{C}}$. Then $F$ splits (Corollary 8.5.3.11): that is, it extends to a diagram $\overline{F}: \operatorname{N}_{\bullet }( \operatorname{Ret}) \rightarrow \operatorname{\mathcal{C}}$. Let $T: \operatorname{Idem}^{\triangleleft } \rightarrow \operatorname{Ret}$ be as in Remark 8.5.3.9, so that $(\overline{F} \circ \operatorname{N}_{\bullet }(T)): \operatorname{N}_{\bullet }( \operatorname{Idem})^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ is a limit diagram in the $\infty $-category $\operatorname{\mathcal{C}}$. We wish to show that the functor $(G \circ \overline{F} \circ \operatorname{N}_{\bullet }(T)): \operatorname{N}_{\bullet }( \operatorname{Idem})^{\triangleleft } \rightarrow \operatorname{\mathcal{D}}$ is a limit diagram in the $\infty $-category $\operatorname{\mathcal{D}}$. By virtue of Proposition 8.5.3.8, this is automatic (Remark 8.5.3.9). $\square$
Example 8.5.3.13 (Idempotent Automorphisms). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing an idempotent $F: \operatorname{N}_{\bullet }(\operatorname{Idem}) \rightarrow \operatorname{\mathcal{C}}$ and let $e: X \rightarrow X$ be the morphism in $\operatorname{\mathcal{C}}$ given by the image of the nondegenerate edge of $\operatorname{N}_{\bullet }(\operatorname{Idem})$. The following conditions are equivalent:
The morphism $e$ is an isomorphism in $\operatorname{\mathcal{C}}$.
The functor $F$ factors through the core $\operatorname{\mathcal{C}}^{\simeq }$.
There is an isomorphism from $F$ to the constant functor $\underline{X}: \operatorname{N}_{\bullet }(\operatorname{Idem}) \rightarrow \operatorname{\mathcal{C}}$.
In particular, if $e$ is an isomorphism, then $F$ is a split idempotent. The equivalence $(1) \Leftrightarrow (2)$ is immediate from the definition of the core $\operatorname{\mathcal{C}}^{\simeq }$, and the equivalence $(2) \Leftrightarrow (3)$ follows from the weak contractibility of $\operatorname{N}_{\bullet }(\operatorname{Idem})$ (Remark 8.5.3.4).
Example 8.5.3.14. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $F: \operatorname{N}_{\bullet }(\operatorname{Idem}) \rightarrow \operatorname{\mathcal{C}}$ be an idempotent in $\operatorname{\mathcal{C}}$, and let $e: X \rightarrow X$ be the morphism in $\operatorname{\mathcal{C}}$ given by the image of the nondegenerate edge of $\operatorname{N}_{\bullet }(\operatorname{Idem})$. Suppose that $\operatorname{\mathcal{C}}$ is finite-dimensional when viewed as a simplicial set. Then, for $n > \dim (\operatorname{\mathcal{C}})$, the functor $F$ carries the unique nondegenerate $n$-simplex $\sigma _{n}$ of $\operatorname{N}_{\bullet }(\operatorname{Idem})$ to a degenerate $n$-simplex of $\operatorname{\mathcal{C}}$. Since the composition $(F \circ \sigma _{n}): \Delta ^ n \rightarrow \operatorname{\mathcal{C}}$ carries every nondegenerate edge of $\Delta ^ n$ to the morphism $e$, we conclude that $e = \operatorname{id}_{X}$. Invoking Example 8.5.3.13, we conclude that $F$ is isomorphic to the constant functor $\underline{X}$; in particular, it is a split idempotent.