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8.4.3 Idempotents in $\infty $-Categories

We now consider an $\infty $-categorical counterpart of Definition

Definition 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

Remark Let $\operatorname{\mathcal{C}}$ be a category. It follows from Remark (and Proposition 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 to the set of idempotent endomorphisms $(X,e)$ in the category $\operatorname{\mathcal{C}}$ (in the sense of Definition We can therefore view Definition as a generalization of Definition

Remark (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

\[ \widetilde{X} \xrightarrow { \widetilde{e}} \widetilde{X} \xrightarrow {\widetilde{e}} \widetilde{X} \xrightarrow {\widetilde{e}} \cdots \rightarrow \widetilde{X} \xrightarrow { \widetilde{e}} \widetilde{X}. \]

Moreover, the face maps of $\operatorname{N}_{\bullet }( \operatorname{Idem})$ satisfy $d_{i}( \sigma _ n) = \sigma _{n-1}$ for $0 \leq i \leq n$. Applying Corollary, we obtain an isomorphism of $\operatorname{N}_{\bullet }( \operatorname{Idem})$ with the simplicial set $(\Delta ^0)^{+}$ introduced in Construction Here we abuse notation by identifying $\Delta ^0$ with its underlying semisimplicial set.

Remark The simplicial set $\operatorname{N}_{\bullet }( \operatorname{Idem})$ is weakly contractible. This is a special case of Lemma, applied to the (discrete) category $[0]$.

Definition (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 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 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

Remark 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

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 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 $\square$

Remark The category $\operatorname{Ret}$ of Construction 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

\[ \operatorname{N}_{\bullet }( \operatorname{Idem})^{\triangleleft } \xrightarrow { \operatorname{N}_{\bullet }(T) } \operatorname{N}_{\bullet }( \operatorname{Ret}) \xrightarrow { F } \operatorname{\mathcal{C}} \]

is a limit diagram. Proposition 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 $F$.

Corollary (Uniqueness of Splittings). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then the restriction functor

\[ \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Ret}), \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Idem}), \operatorname{\mathcal{C}}) \]

is fully faithful, and its essential image is the full subcategory consists of the split idempotents in $\operatorname{\mathcal{C}}$.

Corollary 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, and the reverse implications follow from Corollary $\square$

Corollary 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 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, 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, this is automatic (Remark $\square$