Definition 8.5.4.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. We say that $\operatorname{\mathcal{C}}$ is idempotent complete if every idempotent $\operatorname{N}_{\bullet }( \operatorname{Idem}) \rightarrow \operatorname{\mathcal{C}}$ splits (see Definition 8.5.3.5).
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8.5.4 Idempotent Completeness
We now study $\infty $-categories in which every idempotent splits.
Example 8.5.4.2. Let $\operatorname{\mathcal{C}}$ be a category. If $\operatorname{\mathcal{C}}$ admits equalizers (or coequalizers), then the $\infty $-category $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is idempotent complete (this is a restatement of Corollary 8.5.2.6). In particular, if $\operatorname{\mathcal{C}}$ admits finite limits or finite colimits, then $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is idempotent complete.
Warning 8.5.4.3. An $\infty $-category which admits finite limits (or colimits) need not be idempotent complete. See Example 
.
Example 8.5.4.4. Let $X$ be a Kan complex. Since the simplicial set $\operatorname{N}_{\bullet }(\operatorname{Idem})$ is weakly contractible (Remark 8.5.3.4), every morphism of simplicial sets $\operatorname{N}_{\bullet }( \operatorname{Idem}) \rightarrow X$ is homotopic to a constant map. It follows that $X$ is idempotent complete when viewed as an $\infty $-category.
Example 8.5.4.5. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is finite-dimensional when viewed as a simplicial set. Then $\operatorname{\mathcal{C}}$ is idempotent complete. See Example 8.5.3.14.
Remark 8.5.4.6. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then $\operatorname{\mathcal{C}}$ is idempotent complete if and only if the opposite $\infty $-category $\operatorname{\mathcal{C}}^{\operatorname{op}}$ is idempotent complete.
Remark 8.5.4.7. Let $\{ \operatorname{\mathcal{C}}_ i \} _{i \in I}$ be a collection of $\infty $-categories with product $\prod _{i \in I} \operatorname{\mathcal{C}}_{i}$. Then an idempotent in $\operatorname{\mathcal{C}}$ is split if and only if its image in each factor $\operatorname{\mathcal{C}}_{i}$ is split. In particular, if each of the $\infty $-categories $\operatorname{\mathcal{C}}_{i}$ is idempotent complete, then $\operatorname{\mathcal{C}}$ is idempotent complete.
Proposition 8.5.4.8. Let $\operatorname{\mathcal{C}}$ be an idempotent complete $\infty $-category and let $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ be a full subcategory. Suppose that, for every object $X \in \operatorname{\mathcal{C}}_0$ and every object $Y \in \operatorname{\mathcal{C}}$ which is a retract of $X$, there exists an object $Y' \in \operatorname{\mathcal{C}}_0$ which is isomorphic to $Y$. Then $\operatorname{\mathcal{C}}_0$ is idempotent complete.
Proof. Let $\operatorname{Ret}$ denote the category of Construction 8.5.0.2. Suppose we are given an idempotent $F: \operatorname{N}_{\bullet }( \operatorname{Idem}) \rightarrow \operatorname{\mathcal{C}}_0$, carrying the object $\widetilde{X} \in \operatorname{Idem}$ to an object $X = F( \widetilde{X} ) \in \operatorname{\mathcal{C}}_0$. We wish to show that $F$ is a split idempotent in $\operatorname{\mathcal{C}}_0$. Since $\operatorname{\mathcal{C}}$ is idempotent complete, we can extend $F$ to a functor $\overline{F}: \operatorname{N}_{\bullet }( \operatorname{Ret}) \rightarrow \operatorname{\mathcal{C}}$, carrying the object $\widetilde{Y} \in \operatorname{Ret}$ to an object $Y = \overline{F}( \widetilde{Y} )$ which is a retract of $X$. By assumption, we can choose an isomorphism $\alpha _0: Y \rightarrow Y'$, where $Y'$ belongs to $\operatorname{\mathcal{C}}_0$. Using Corollary 4.4.5.3, we can lift $\alpha _0$ to an isomorphism of functors $\alpha : \overline{F} \rightarrow \overline{F}'$ in $\operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Ret}), \operatorname{\mathcal{C}})$, whose image in $\operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Idem}), \operatorname{\mathcal{C}})$ is the identity transformation from $F$ to itself. Then $\overline{F}': \operatorname{N}_{\bullet }( \operatorname{Ret}) \rightarrow \operatorname{\mathcal{C}}_0$ is a splitting of the idempotent $F$. $\square$
Proposition 8.5.4.9. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. The following conditions are equivalent:
The $\infty $-category $\operatorname{\mathcal{C}}$ is idempotent complete.
The $\infty $-category $\operatorname{\mathcal{C}}$ admits limits indexed by $\operatorname{N}_{\bullet }( \operatorname{Idem})$.
The $\infty $-category $\operatorname{\mathcal{C}}$ admits colimits indexed by $\operatorname{N}_{\bullet }( \operatorname{Idem})$.
Proof. This is an immediate consequence of Corollary 8.5.3.11. $\square$
Remark 8.5.4.10. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then $\operatorname{\mathcal{C}}$ is idempotent complete if and only if the restriction functor is an equivalence of $\infty $-categories. This is an immediate consequence of Corollary 8.5.3.10.
\[ \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Ret}), \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Idem}), \operatorname{\mathcal{C}}) \]
Corollary 8.5.4.11. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty $-categories which are equivalent. Then $\operatorname{\mathcal{C}}$ is idempotent complete if and only if $\operatorname{\mathcal{D}}$ is idempotent complete.
Corollary 8.5.4.12. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $K$ be a simplicial set. If $\operatorname{\mathcal{C}}$ is idempotent complete, then $\operatorname{Fun}(K, \operatorname{\mathcal{C}})$ is idempotent complete.
To apply the criterion of Proposition 8.5.4.9, it is often useful to replace $\operatorname{N}_{\bullet }( \operatorname{Idem})$ by a simpler simplicial set.
Notation 8.5.4.13. Let $\operatorname{Spine}[\operatorname{\mathbf{Z}}]$ denote the $1$-dimensional simplicial set associated to the directed graph We let $Q: \operatorname{Spine}[\operatorname{\mathbf{Z}}] \rightarrow \operatorname{N}_{\bullet }( \operatorname{Idem})$ be the morphism of simplicial sets corresponding to the diagram in the category $\operatorname{Idem}$.
\[ \cdots \rightarrow -2 \rightarrow -1 \rightarrow 0 \rightarrow 1 \rightarrow 2 \rightarrow \cdots \]
\[ \cdots \rightarrow X \xrightarrow {e} X \xrightarrow {e} X \xrightarrow {e} X \xrightarrow {e} X \rightarrow \cdots \]
Remark 8.5.4.14. Since the simplicial set $\operatorname{Spine}[\operatorname{\mathbf{Z}}]$ is $1$-dimensional, the morphism $Q$ of Notation 8.5.4.13 factors (uniquely) through the $1$-skeleton of $\operatorname{N}_{\bullet }( \operatorname{Idem})$, which we can identify with the simplicial circle $\Delta ^1 / \operatorname{\partial \Delta }^1$. Under this identification, $Q$ corresponds to a morphism of simplicial sets $q: \operatorname{Spine}[\operatorname{\mathbf{Z}}] \rightarrow \Delta ^1 / \operatorname{\partial \Delta }^1$. This is a covering map (see Definition 3.1.4.1), which exhibits the simplicial circle $\Delta ^1 / \operatorname{\partial \Delta }^1$ as the quotient of $\operatorname{Spine}[\operatorname{\mathbf{Z}}]$ by a free action of the group $(\operatorname{\mathbf{Z}},+)$ by translations. The induced map of geometric realizations $| \operatorname{Spine}[\operatorname{\mathbf{Z}}] | \rightarrow | \Delta ^1 / \operatorname{\partial \Delta }^1 |$ can be identified with the standard covering map $\mathbf{R} \rightarrow S^1$ in the category of topological spaces.
Remark 8.5.4.15. In the situation of Notation 8.5.4.13, we can regard $\operatorname{Spine}[\operatorname{\mathbf{Z}}]$ as a simplicial subset of the nerve $\operatorname{N}_{\bullet }(\operatorname{\mathbf{Z}})$, where we regard the set of integers $\operatorname{\mathbf{Z}}= \{ \cdots < -2 < -1 < 0 < 1 < 2 < \cdots \} $ as equipped with its usual linear ordering. Moreover, the inclusion $\operatorname{Spine}[\operatorname{\mathbf{Z}}] \hookrightarrow \operatorname{N}_{\bullet }( \operatorname{\mathbf{Z}})$ is inner anodyne (this is a special case of Proposition 1.5.7.3).
Remark 8.5.4.16. The simplicial set $\operatorname{Spine}[\operatorname{\mathbf{Z}}]$ is weakly contractible. In fact, the inclusion $\{ 0\} \hookrightarrow \operatorname{Spine}[\operatorname{\mathbf{Z}}]$ is anodyne, since it can be realized as an iterated pushout of the inclusion maps $\{ 0\} \hookrightarrow \Delta ^1$ and $\{ 1\} \hookrightarrow \Delta ^1$. Alternatively, it can be deduced from Example 3.6.4.4, since the geometric realization $| \operatorname{Spine}[\operatorname{\mathbf{Z}}] |$ is homeomorphic to the set of real numbers $\mathbf{R}$ (endowed with its usual topology).
Proposition 8.5.4.17. The morphism $Q: \operatorname{Spine}[\operatorname{\mathbf{Z}}] \rightarrow \operatorname{N}_{\bullet }( \operatorname{Idem})$ of Notation 8.5.4.13 is both left and right cofinal.
Proof. We will show that $Q$ is left cofinal; a similar argument will show that it is right cofinal. By virtue of Theorem 7.2.3.1, it will suffice to show that the simplicial set $K = \operatorname{Spine}[\operatorname{\mathbf{Z}}] \times _{ \operatorname{N}_{\bullet }( \operatorname{Idem}) } \operatorname{N}_{\bullet }( \operatorname{Idem})_{/ \widetilde{X}}$ is weakly contractible. Let us identify the vertices of $K$ with pairs $(n,f)$, where $n$ is an integer and $f: \widetilde{X} \rightarrow \widetilde{X}$ is a morphism in the category $\operatorname{Idem}$. Unwinding the definitions, we see that $K$ is the $1$-dimensional simplicial set associated to the direct graph given in the diagram
\[ \xymatrix@R =50pt@C=50pt{ \cdots \ar [dr] \ar [r] & (-1, \widetilde{e}) \ar [dr] \ar [r] & (0, \widetilde{e}) \ar [dr] \ar [r] & (1, \widetilde{e}) \ar [dr] \ar [r] & \cdots \\ \cdots & (-1,\operatorname{id}_{\widetilde{X}} ) & (0, \operatorname{id}_{\widetilde{X}} ) & (1, \operatorname{id}_{\widetilde{X}} ) & \cdots . } \]
The inclusion of the upper part of the diagram determines a monomorphism of simplicial sets $\operatorname{Spine}[\operatorname{\mathbf{Z}}] \hookrightarrow K$ which is left anodyne (since it is a pushout of a coproduct of countably many copies of the inclusion map $\{ 0\} \hookrightarrow \Delta ^1$), and therefore a weak homotopy equivalence (Proposition 3.1.6.14). The desired result now follows from the weak contractibility of the simplicial set $\operatorname{Spine}[\operatorname{\mathbf{Z}}]$ (Remark 8.5.4.16). $\square$
Corollary 8.5.4.18. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits sequential limits (or colimits). Then $\operatorname{\mathcal{C}}$ is idempotent complete.
Proof. It follows from Proposition 8.5.4.17 (and Corollary 7.2.2.12) that the $\infty $-category $\operatorname{\mathcal{C}}$ admits limits (or colimits) indexed by the $\infty $-category $\operatorname{N}_{\bullet }( \operatorname{Idem})$, and is therefore idempotent complete by virtue of Proposition 8.5.4.9. $\square$
Remark 8.5.4.19. Broadly speaking, Proposition 8.5.4.17 will be useful to us because it shows that the $\infty $-category $\operatorname{N}_{\bullet }( \operatorname{Idem})$ admits a (left and right) cofinal diagram $Q: K \rightarrow \operatorname{N}_{\bullet }( \operatorname{Idem})$, where the simplicial set $K$ is finite-dimensional. Beware that it is not possible to arrange that the simplicial set $K$ is finite, since an $\infty $-category which admits finite colimits need not be idempotent complete (Warning 8.5.4.3). In particular, there does not exist a categorical equivalence $K \rightarrow \operatorname{N}_{\bullet }( \operatorname{Idem})$, where $K$ is a finite simplicial set.
Example 8.5.4.20. Let $\operatorname{\mathcal{QC}}$ denote the $\infty $-category of (small) $\infty $-categories. Then $\operatorname{\mathcal{QC}}$ is idempotent complete. More generally, for every uncountable cardinal $\kappa $, the $\infty $-category $\operatorname{\mathcal{QC}}_{< \kappa }$ of $\kappa $-small $\infty $-categories is idempotent complete. To prove this, we can use Propositions 8.5.4.8 and 8.5.1.15 to reduce to the case where $\kappa $ has uncountable cofinality. In this case, the $\infty $-category $\operatorname{\mathcal{QC}}_{< \kappa }$ admits sequential colimits (Example 7.6.6.8), so the desired result follows from Corollary 8.5.4.18.
Example 8.5.4.21. Let $\operatorname{\mathcal{S}}$ denote the $\infty $-category of spaces. Then $\operatorname{\mathcal{S}}$ is idempotent complete. More generally, for every uncountable cardinal $\kappa $, the $\infty $-category $\operatorname{\mathcal{S}}_{< \kappa }$ of $\kappa $-small spaces is idempotent complete. This follows from Example 8.5.4.20 and Proposition 8.5.4.8, since the full subcategory $\operatorname{\mathcal{S}}_{< \kappa } \subseteq \operatorname{\mathcal{QC}}_{< \kappa }$ is closed under the formation of retracts (Remark 8.5.1.16).
Warning 8.5.4.22. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. If $\operatorname{\mathcal{C}}$ is idempotent complete, then its homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ need not be idempotent complete. For example, the $\infty $-category of spaces $\operatorname{\mathcal{S}}$ is idempotent complete (Example 8.5.4.21), but its homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{S}}} = \mathrm{h} \mathit{\operatorname{Kan}}$ is not (see Proposition 8.5.8.15).
We now record some stability properties of idempotent completeness.
Proposition 8.5.4.23. Let $F_0: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{C}}$ and $F_1: \operatorname{\mathcal{C}}_1 \rightarrow \operatorname{\mathcal{C}}$ be functors of $\infty $-categories, and let $\operatorname{\mathcal{C}}_{01}$ denote the oriented fiber product $\operatorname{\mathcal{C}}_0 \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$ (see Definition 4.6.4.1). Then an idempotent $E: \operatorname{N}_{\bullet }(\operatorname{Idem}) \rightarrow \operatorname{\mathcal{C}}_{01}$ splits if and only if its images in $\operatorname{\mathcal{C}}_0$ and $\operatorname{\mathcal{C}}_1$ split. In particular, if $\operatorname{\mathcal{C}}_0$ and $\operatorname{\mathcal{C}}_1$ are idempotent complete, then $\operatorname{\mathcal{C}}_{01}$ is idempotent complete.
Proof. Let us identify $E$ with a triple $(E_0, E_1, u)$, where $E_0: \operatorname{N}_{\bullet }(\operatorname{Idem}) \rightarrow \operatorname{\mathcal{C}}_0$ and $E_1: \operatorname{N}_{\bullet }(\operatorname{Idem}) \rightarrow \operatorname{\mathcal{C}}_1$ are idempotents in the $\infty $-categories $\operatorname{\mathcal{C}}_0$ and $\operatorname{\mathcal{C}}_1$, and $u: (F_0 \circ E_0) \rightarrow (F_1 \circ E_1)$ is a natural transformation. Suppose that the idempotents $E_0$ and $E_1$ split; we wish to show that $E$ splits (the reverse implication is trivial). Choose functors $\overline{E}_0: \operatorname{N}_{\bullet }(\operatorname{Ret}) \rightarrow \operatorname{\mathcal{C}}_0$ and $\overline{E}_1: \operatorname{N}_{\bullet }( \operatorname{Ret}) \rightarrow \operatorname{\mathcal{C}}_1$ extending $E_0$ and $E_1$, respectively. It follows from Corollary 8.5.3.10 that $u$ admits an (essentially unique) extension to a natural transformation $\overline{u}: (F_0 \circ \overline{E}_0) \rightarrow (F_1 \circ \overline{E}_1)$. The triple $(\overline{E}_0, \overline{E}_1, \overline{u} )$ can then be regarded as a diagram $\operatorname{N}_{\bullet }(\operatorname{Ret}) \rightarrow \operatorname{\mathcal{C}}_{01}$ extending $E$. $\square$
Corollary 8.5.4.24. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a left fibration of $\infty $-categories. If $\operatorname{\mathcal{C}}$ is idempotent complete, then $\operatorname{\mathcal{E}}$ is idempotent complete.
Proof. Choose an uncountable regular cardinal $\kappa $ such that $U$ is essentially $\kappa $-small, and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}_{< \kappa }$ be a covariant transport representation for $U$. Then the $\infty $-category $\operatorname{\mathcal{E}}$ is equivalent to the oriented fiber $\{ \Delta ^0 \} \operatorname{\vec{\times }}_{\operatorname{\mathcal{S}}} \operatorname{\mathcal{C}}$, which is idempotent complete by virtue of Proposition 8.5.4.23. $\square$
Corollary 8.5.4.25. Let $F_0: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{C}}$ and $F_1: \operatorname{\mathcal{C}}_1 \rightarrow \operatorname{\mathcal{C}}$ be functors of $\infty $-categories. If $\operatorname{\mathcal{C}}_0$ and $\operatorname{\mathcal{C}}_1$ are idempotent complete, then the homotopy fiber product $\operatorname{\mathcal{C}}_0 \times _{\operatorname{\mathcal{C}}}^{\mathrm{h}} \operatorname{\mathcal{C}}_1$ is also idempotent complete.
Proof. It follows from Proposition 8.5.1.7 that $\operatorname{\mathcal{C}}_0 \times _{\operatorname{\mathcal{C}}}^{\mathrm{h}} \operatorname{\mathcal{C}}_1$ is closed under retracts in the oriented fiber product $\operatorname{\mathcal{C}}_0 \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$. The desired result now follows from Propositions 8.5.4.23 and 8.5.4.8. $\square$
Corollary 8.5.4.26. Suppose we are given a categorical pullback diagram of $\infty $-categories If $\operatorname{\mathcal{C}}_0$ and $\operatorname{\mathcal{C}}_1$ are idempotent complete, then $\operatorname{\mathcal{C}}_{01}$ is idempotent complete.
8.66
\begin{equation} \begin{gathered}\label{equation:pullback-idempotent-complete} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{01} \ar [r]^{G_0} \ar [d]^{G_1} & \operatorname{\mathcal{C}}_0 \ar [d]^{F_0} \\ \operatorname{\mathcal{C}}_1 \ar [r]^{ F_1 } & \operatorname{\mathcal{C}}. } \end{gathered} \end{equation}
Proof. Combine Corollaries 8.5.4.25 and Corollaries 8.5.4.11. $\square$
Corollary 8.5.4.27. Let $\operatorname{\mathcal{QC}}'$ be the full subcategory of $\operatorname{\mathcal{QC}}$ spanned by the (small) idempotent complete $\infty $-categories. Then $\operatorname{\mathcal{QC}}'$ is closed under small limits in $\operatorname{\mathcal{QC}}$.
Proof. By virtue of Proposition 7.6.6.9 (with Remark 7.6.6.10 and Exercise 7.6.6.11), it will suffice to show that $\operatorname{\mathcal{QC}}'$ is closed under pullbacks and small products. The case of pullbacks follows from Corollary 8.5.4.25 (see Corollary 7.6.3.9), and the case of products follows from Remark 8.5.4.7 (see Example 7.6.1.17). $\square$
Warning 8.5.4.28. The collection of idempotent complete $\infty $-categories is not closed under the formation of colimits in $\operatorname{\mathcal{QC}}$. (In fact, every $\infty $-category $\operatorname{\mathcal{C}}$ can be realized as a colimit of standard simplices $\Delta ^ n$, which are idempotent complete $\infty $-categories: see Proposition .) However, we will see in ยง8.5.9 that the collection of idempotent complete $\infty $-categories is closed under filtered colimits: see Corollary 8.5.9.10.