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8.4.4 Idempotent Completeness

We now study $\infty $-categories in which every idempotent splits.

Definition 8.4.4.1. 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 8.4.3.5).

Example 8.4.4.2. 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 8.4.2.6). In particular, if $\operatorname{\mathcal{C}}$ admits finite limits or finite colimits, then $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is idempotent complete.

Warning 8.4.4.3. An $\infty $-category which admits finite limits (or colimits) need not be idempotent complete. See Example .

Example 8.4.4.4. Let $X$ be a Kan complex. Since the simplicial set $\operatorname{N}_{\bullet }(\operatorname{Idem})$ is weakly contractible (Remark 8.4.3.4), 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.

Remark 8.4.4.5. 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.

Proposition 8.4.4.6. 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 8.4.0.1. 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 8.4.4.7. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. The following conditions are equivalent:

$(1)$

The $\infty $-category $\operatorname{\mathcal{C}}$ is idempotent complete.

$(2)$

The $\infty $-category $\operatorname{\mathcal{C}}$ admits limits indexed by $\operatorname{N}_{\bullet }( \operatorname{Idem})$.

$(3)$

The $\infty $-category $\operatorname{\mathcal{C}}$ admits colimits indexed by $\operatorname{N}_{\bullet }( \operatorname{Idem})$.

Proof. This is an immediate consequence of Corollary 8.4.3.11. $\square$

Remark 8.4.4.8. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then $\operatorname{\mathcal{C}}$ is idempotent complete if and only if 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 an equivalence of $\infty $-categories. This is an immediate consequence of Corollary 8.4.3.10.

Corollary 8.4.4.9. 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 8.4.4.10. 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.

Corollary 8.4.4.11. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f: K \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. If $\operatorname{\mathcal{C}}$ is idempotent complete, then the slice and coslice $\infty $-categories $\operatorname{\mathcal{C}}_{/f}$ and $\operatorname{\mathcal{C}}_{f/}$ are idempotent complete.

To apply the criterion of Proposition 8.4.4.7, it is often useful to replace $\operatorname{N}_{\bullet }( \operatorname{Idem})$ by a simpler simplicial set.

Notation 8.4.4.12. Let $\operatorname{Spine}[\operatorname{\mathbf{Z}}]$ denote the $1$-dimensional simplicial set associated to the directed graph

\[ \cdots \rightarrow -2 \rightarrow -1 \rightarrow 0 \rightarrow 1 \rightarrow 2 \rightarrow \cdots \]

We let $Q: \operatorname{Spine}[\operatorname{\mathbf{Z}}] \rightarrow \operatorname{N}_{\bullet }( \operatorname{Idem})$ be the morphism of simplicial sets corresponding to the diagram

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

in the category $\operatorname{Idem}$.

Remark 8.4.4.13. Since the simplicial set $\operatorname{Spine}[\operatorname{\mathbf{Z}}]$ is $1$-dimensional, the morphism $Q$ of Notation 8.4.4.12 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 8.4.4.14. In the situation of Notation 8.4.4.12, 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 1.4.7.3).

Remark 8.4.4.15. The simplicial set $\operatorname{Spine}[\operatorname{\mathbf{Z}}]$ is weakly contractible. This follows from Remark 8.4.4.14, since the $\infty $-category $\operatorname{N}_{\bullet }( \operatorname{\mathbf{Z}})$ is filtered and therefore weakly contractible (Proposition 7.2.4.9). Alternatively, it can be deduced from Example 3.5.4.4, 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 8.4.4.16. The morphism $Q: \operatorname{Spine}[\operatorname{\mathbf{Z}}] \rightarrow \operatorname{N}_{\bullet }( \operatorname{Idem})$ of Notation 8.4.4.12 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 8.4.4.15). $\square$

Corollary 8.4.4.17. 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 8.4.4.16 (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 8.4.4.7. $\square$

Remark 8.4.4.18. Broadly speaking, Proposition 8.4.4.16 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 8.4.4.3). In particular, there does not exist a categorical equivalence $K \rightarrow \operatorname{N}_{\bullet }( \operatorname{Idem})$, where $K$ is a finite simplicial set.

Example 8.4.4.19. 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 8.4.4.6 and 8.4.1.12 to reduce to the case where $\kappa $ has uncountable cofinality. In this case, the $\infty $-category $\operatorname{\mathcal{QC}}^{< \kappa }$ admits sequential colimits (Example 7.6.7.7), so the desired result follows from Corollary 8.4.4.17.

Example 8.4.4.20. 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 8.4.4.19 and Proposition 8.4.4.6, since the full subcategory $\operatorname{\mathcal{S}}^{< \kappa } \subseteq \operatorname{\mathcal{QC}}^{< \kappa }$ is closed under the formation of retracts (Remark 8.4.1.13).

Warning 8.4.4.21. 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 8.4.4.20), but its homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{S}}} = \mathrm{h} \mathit{\operatorname{Kan}}$ is not (see Proposition 8.4.7.15).