Notation 8.5.6.1. Let $\Delta ^1 / \operatorname{\partial \Delta }^1$ denote the simplicial circle (Example 1.5.7.10). For every $\infty $-category $\operatorname{\mathcal{C}}$, we let $\operatorname{End}_{\operatorname{\mathcal{C}}}$ denote the $\infty $-category of diagrams $\operatorname{Fun}( \Delta ^1 / \operatorname{\partial \Delta }^1, \operatorname{\mathcal{C}})$. Note that objects of $\operatorname{End}_{\operatorname{\mathcal{C}}}$ can be identified with pairs $(X,e)$, where $X$ is an object of $\operatorname{\mathcal{C}}$ and $e: X \rightarrow X$ is an endomorphism of $X$. We will refer to $\operatorname{End}_{\operatorname{\mathcal{C}}}$ as the $\infty $-category of endomorphisms in $\operatorname{\mathcal{C}}$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
8.5.6 Idempotent Endomorphisms
Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $F: \operatorname{N}_{\bullet }( \operatorname{Idem}) \rightarrow \operatorname{\mathcal{C}}$ be an idempotent in $\operatorname{\mathcal{C}}$. Then $F$ carries the unique object of $\operatorname{Idem}$ to an object $X \in \operatorname{\mathcal{C}}$, and the unique non-identity morphism of $\operatorname{Idem}$ to an endomorphism $e: X \rightarrow X$ in $\operatorname{\mathcal{C}}$. If $\operatorname{\mathcal{C}}$ is (the nerve of) an ordinary category, then the functor $F$ is uniquely determined by the pair $(X,e)$ (Remark 8.5.2.8). In more general situations, this is false: the simplicial set $\operatorname{N}_{\bullet }( \operatorname{Idem})$ contains a nondegenerate simplex of each dimension (Remark 8.5.3.3), so the specification of the functor $F$ requires an infinite amount of data. Our goal in this section is to show that, nevertheless, the idempotent $F: \operatorname{N}_{\bullet }( \operatorname{Idem}) \rightarrow \operatorname{\mathcal{C}}$ can be recovered up to isomorphism from the underlying endomorphism $(X,e)$. We begin by introducing some terminology.
Remark 8.5.6.2. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Evaluation on the unique vertex of $\Delta ^1 / \operatorname{\partial \Delta }^1$ induces an isofibration of $\infty $-categories $\operatorname{End}_{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{C}}$. Moreover, for each object $X \in \operatorname{\mathcal{C}}$, the fiber $\{ X\} \times _{\operatorname{\mathcal{C}}} \operatorname{End}_{\operatorname{\mathcal{C}}}$ can be identified with the endomorphism space $\operatorname{End}_{\operatorname{\mathcal{C}}}(X) = \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,X)$ of Variant 4.6.1.3.
Definition 8.5.6.3 (Idempotent Endomorphisms). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $e: X \rightarrow X$ be an endomorphism in $\operatorname{\mathcal{C}}$. We will say that $e$ is idempotent if there exists a functor $F: \operatorname{N}_{\bullet }( \operatorname{Idem}) \rightarrow \operatorname{\mathcal{C}}$ satisfying $F( \widetilde{e} ) = e$; here $\widetilde{e}$ denotes the (unique) non-identity morphism in the category $\operatorname{Idem}$. We let $\operatorname{End}_{\operatorname{\mathcal{C}}}^{\mathrm{idm}}$ denote the full subcategory of $\operatorname{End}_{\operatorname{\mathcal{C}}}$ spanned by the idempotent endomorphisms.
We can now formulate our main result.
Proposition 8.5.6.4. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then the restriction functor has a left homotopy inverse.
\[ \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Idem}), \operatorname{\mathcal{C}}) \rightarrow \operatorname{End}_{\operatorname{\mathcal{C}}}^{\mathrm{idm}} \]
Stated more informally, Proposition 8.5.6.4 asserts that if $e: X \rightarrow X$ is an endomorphism in the $\infty $-category $\operatorname{\mathcal{C}}$ which can be extended to an idempotent $F: \operatorname{N}_{\bullet }( \operatorname{Idem}) \rightarrow \operatorname{\mathcal{C}}$, then $F$ is uniquely determined up to isomorphism and can be chosen to depend functorially on the pair $(X,e)$.
Corollary 8.5.6.5. For every $\infty $-category $\operatorname{\mathcal{C}}$, evaluation on the non-identity morphism of $\operatorname{Idem}$ induces a bijection
\[ \theta : \pi _0( \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Idem}), \operatorname{\mathcal{C}})^{\simeq } ) \rightarrow \pi _0( (\operatorname{End}_{\operatorname{\mathcal{C}}}^{\mathrm{idm} })^{\simeq } ). \]
Proof. The surjectivity of $\theta $ follows from the definition of an idempotent endomorphism, and the injectivity from Proposition 8.5.6.4. $\square$
Warning 8.5.6.6. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $R: \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Idem}), \operatorname{\mathcal{C}}) \rightarrow \operatorname{End}_{\operatorname{\mathcal{C}}}^{\mathrm{idm}}$ be the restriction functor. Proposition 8.5.6.4 asserts that there exists a functor $S: \operatorname{End}_{\operatorname{\mathcal{C}}}^{\mathrm{idm}} \rightarrow \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Idem}), \operatorname{\mathcal{C}})$ for which the composition is isomorphic to the identity functor. Let $e: X \rightarrow X$ be an idempotent endomorphism in $\operatorname{\mathcal{C}}$, so that $e$ can be extended to a morphism $F: \operatorname{N}_{\bullet }( \operatorname{Idem}) \rightarrow \operatorname{\mathcal{C}}$. Then $S(e) = (S \circ R)(F)$ is isomorphic to $F$, so there is an isomorphism of $(R \circ S)(e)$ with $e$ in the category $\operatorname{End}_{\operatorname{\mathcal{C}}}^{\mathrm{idm}}$. Beware that this isomorphism usually cannot be chosen to depend functorially on $e$. In general, the functor $R$ is not an equivalence of $\infty $-categories, so the composition is not isomorphic to the identity functor on $\operatorname{End}_{\operatorname{\mathcal{C}}}^{\mathrm{idm}}$.
\[ \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Idem}), \operatorname{\mathcal{C}}) \xrightarrow {R} \operatorname{End}_{\operatorname{\mathcal{C}}}^{\mathrm{idm}} \xrightarrow {S} \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Idem}),\operatorname{\mathcal{C}}). \]
\[ \operatorname{End}_{\operatorname{\mathcal{C}}}^{\mathrm{idm}} \xrightarrow {S} \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Idem}),\operatorname{\mathcal{C}}) \xrightarrow {R} \operatorname{End}_{\operatorname{\mathcal{C}}}^{\mathrm{idm}} \]
Example 8.5.6.7. For any $\infty $-category $\operatorname{\mathcal{C}}$, we have a commutative diagram where the vertical maps are the diagonal embeddings. If $\operatorname{\mathcal{C}}$ is a Kan complex, then the left vertical map is a homotopy equivalence of Kan complexes (since the simplicial set $\operatorname{N}_{\bullet }( \operatorname{Idem})$ is weakly contractible; see Remark 8.5.3.4). In this case, Proposition 8.5.6.4 reduces to the assertion that the diagonal map has a left homotopy inverse. This is clear: the map $\delta $ has a left inverse in the category of simplicial sets, given by evaluation at the vertex of $\Delta ^1 / \operatorname{\partial \Delta }^1$. Beware that $\delta $ is usually not a homotopy equivalence, since the simplicial set $\Delta ^1 / \operatorname{\partial \Delta }^1$ is not contractible.
\[ \xymatrix@R =50pt@C=50pt{ & \operatorname{\mathcal{C}}\ar [dl] \ar [dr]^{\delta } & \\ \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Idem}), \operatorname{\mathcal{C}}) \ar [rr] & & \operatorname{End}^{\mathrm{idm}}_{\operatorname{\mathcal{C}}}, } \]
\[ \delta : \operatorname{\mathcal{C}}\rightarrow \operatorname{End}^{\mathrm{idm}}_{\operatorname{\mathcal{C}}} \subseteq \operatorname{Fun}( \Delta ^1 / \operatorname{\partial \Delta }^1, \operatorname{\mathcal{C}}) \quad \quad X \mapsto (X, \operatorname{id}_{X} ) \]
We will give the proof of Proposition 8.5.6.4 at the end of this section. First, let us introduce an important class of idempotent endomorphisms.
Definition 8.5.6.8. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. We say that an endomorphism $e: X \rightarrow X$ in $\operatorname{\mathcal{C}}$ is split idempotent if the homotopy class $[e]$ is a split idempotent in the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ (see Example 8.5.2.3).
Remark 8.5.6.9. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then an endomorphism $e: X \rightarrow X$ is split idempotent if and only there exists a retraction diagram in the $\infty $-category $\operatorname{\mathcal{C}}$, where $e$ factors as a composition $X \xrightarrow {r} Y \xrightarrow {i} X$.
\[ \xymatrix@R =50pt@C=50pt{ & X \ar [dr]^{r} & \\ Y \ar [ur]^{i} \ar [rr]^{ \operatorname{id}_{Y} } & & Y. } \]
Proposition 8.5.6.10 (Lifting Split Idempotents). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $e: X \rightarrow X$ be an endomorphism in $\operatorname{\mathcal{C}}$. Then $e$ is split idempotent endomorphism if and only if it extends to a split idempotent $\operatorname{N}_{\bullet }( \operatorname{Idem}) \rightarrow \operatorname{\mathcal{C}}$, in the sense of Definition 8.5.3.5. In particular, every split idempotent endomorphism is an idempotent endomorphism.
Proof. Assume that the endomorphism $e$ is split idempotent; we will show that $e$ can be extended to a split idempotent $F: \operatorname{N}_{\bullet }( \operatorname{Idem}) \rightarrow \operatorname{\mathcal{C}}$ (the reverse implication follows immediately from the definitions). Choose a retraction diagram
8.69
\begin{equation} \begin{gathered}\label{equation:lifting-split-idempotents} \xymatrix@R =50pt@C=50pt{ & X \ar [dr]^{r} & \\ Y \ar [ur]^{i} \ar [rr]^{ \operatorname{id}_{Y} } & & Y } \end{gathered} \end{equation}
in the $\infty $-category $\operatorname{\mathcal{C}}$, where $[e] = [i] \circ [r]$ in the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$. Using Corollary 8.5.1.28, we can extend the diagram (8.69) to a functor $\overline{F}: \operatorname{N}_{\bullet }( \operatorname{Ret}) \rightarrow \operatorname{\mathcal{C}}$. By construction, $\overline{F}$ carries the unique non-identity morphism of $\operatorname{Idem}$ to a morphism $e': X \rightarrow X$ of $\operatorname{\mathcal{C}}$ which is homotopic to $e$. Replacing $\overline{F}$ by an isomorphic functor if necessary, we may assume that $e' = e$ (see Corollary 4.4.5.3). Then $F = \overline{F}|_{ \operatorname{N}_{\bullet }( \operatorname{Idem})}$ is a split idempotent in $\operatorname{\mathcal{C}}$ extending $e$. $\square$
Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. When restricted to split idempotents, Proposition 8.5.6.4 asserts every retraction diagram
\[ \xymatrix@R =50pt@C=50pt{ & X \ar [dr]^{r} & \\ Y \ar [ur]^{i} \ar [rr]^{ \operatorname{id}_{Y} } & & Y } \]
can be recovered (up to canonical isomorphism) from a choice of composition $e = (i \circ r)$ in the $\infty $-category $\operatorname{\mathcal{C}}$. To prove this, we will exploit the observation that $Y$ can be realized as the limit (and colimit) of the diagram
\[ \cdots \rightarrow X \xrightarrow {e} X \xrightarrow {e} X \xrightarrow {e} X \xrightarrow {e} X \rightarrow \cdots , \]
indexed by the $1$-dimensional simplicial set $\operatorname{Spine}[\operatorname{\mathbf{Z}}]$ of Notation 8.5.4.12.
Notation 8.5.6.11. Let $q: \operatorname{Spine}[\operatorname{\mathbf{Z}}] \rightarrow \Delta ^1 / \operatorname{\partial \Delta }^1$ be the covering map of Remark 8.5.4.13. For every $\infty $-category $\operatorname{\mathcal{C}}$, precomposition with $q$ induces a functor More informally, the functor $T$ carries each endomorphism $e: X \rightarrow X$ in the $\infty $-category $\operatorname{\mathcal{C}}$ to the associated sequential diagram
\[ T: \operatorname{End}_{\operatorname{\mathcal{C}}} = \operatorname{Fun}( \Delta ^1 / \operatorname{\partial \Delta }^1, \operatorname{\mathcal{C}}) \hookrightarrow \operatorname{Fun}( \operatorname{Spine}[\operatorname{\mathbf{Z}}], \operatorname{\mathcal{C}}) \quad \quad (X,e) \mapsto T_{e}. \]
\[ \cdots \rightarrow X \xrightarrow {e} X \xrightarrow {e} X \xrightarrow {e} X \xrightarrow {e} X \rightarrow \cdots \]
Proposition 8.5.6.12. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $e: X \rightarrow X$ be an idempotent endomorphism in $\operatorname{\mathcal{C}}$. Then $e$ splits if and only if the diagram $T_{e}: \operatorname{Spine}[\operatorname{\mathbf{Z}}] \rightarrow \operatorname{\mathcal{C}}$ admits a limit.
Proof. Since $e$ is idempotent, it can be extended to a functor $F: \operatorname{N}_{\bullet }( \operatorname{Idem}) \rightarrow \operatorname{\mathcal{C}}$. Then $T_{e} = F \circ Q$, where $Q: \operatorname{Spine}[\operatorname{\mathbf{Z}}] \rightarrow \operatorname{N}_{\bullet }( \operatorname{Idem})$ is the left cofinal morphism of Proposition 8.5.4.16. Using Corollary 7.2.2.10, we see that $T_{e}$ has a limit in $\operatorname{\mathcal{C}}$ if and only if $F$ has a limit in $\operatorname{\mathcal{C}}$. The desired result now follows from the criterion of Corollary 8.5.3.11 $\square$
Remark 8.5.6.13. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $e: X \rightarrow X$ be an split idempotent endomorphism in $\operatorname{\mathcal{C}}$, so that the diagram admits both a limit and colimit in $\operatorname{\mathcal{C}}$. The limit and colimit of this diagram are automatically preserved by any functor of $\infty $-categories $\operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$. This follows by combining Corollary 8.5.3.12 with Proposition 8.5.4.16.
\[ \cdots \rightarrow X \xrightarrow {e} X \xrightarrow {e} X \xrightarrow {e} X \xrightarrow {e} X \rightarrow \cdots \]
Motivated by Proposition 8.5.6.12, we introduce a variant of Definition 8.5.6.8.
Definition 8.5.6.14. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $e: X \rightarrow X$ be an endomorphism in $\operatorname{\mathcal{C}}$. We will say that $e$ is weakly split if it satisfies the following conditions:
The diagram $T_{e}$ of Notation 8.5.6.11 can be extended to a limit diagram in $\operatorname{\mathcal{C}}$, which we depict as
The diagram $T_{e}$ of Notation 8.5.6.11 can be extended to a colimit diagram in $\operatorname{\mathcal{C}}$, which we depict as
The composition $Y \xrightarrow {i} X \xrightarrow {r} Z$ is an isomorphism in $\operatorname{\mathcal{C}}$.
\[ \xymatrix@R =50pt@C=30pt{ & & & Y \ar [dll] \ar [dl] \ar [d]^{ i} \ar [dr] \ar [drr] & & & \\ \cdots \ar [r] & X \ar [r]^-{e} & X \ar [r]^-{ e} & X \ar [r]^-{e} & X \ar [r]^-{e} & X \ar [r]^-{e} & \cdots } \]
\[ \xymatrix@R =50pt@C=30pt{ \cdots \ar [r] & X \ar [r]^-{e} \ar [drr] & X \ar [r]^-{ e} \ar [dr] & X \ar [r]^-{e} \ar [d]^{r} & X \ar [r]^-{e} \ar [dl] & X \ar [r]^-{e} \ar [dll] & \cdots \\ & & & Z. & & & } \]
Our next goal is to show that every split idempotent endomorphism is weakly split.
Notation 8.5.6.15. Let $\operatorname{Ret}$ denote the category of Construction 8.5.0.2. Then the object $\widetilde{Y} \in \operatorname{Ret}$ is both initial and final. It follows that the diagram $Q: \operatorname{Spine}[\operatorname{\mathbf{Z}}] \rightarrow \operatorname{N}_{\bullet }(\operatorname{Idem})$ of Proposition 8.5.4.16 admits unique extensions which carry the cone points to the object $\widetilde{Y}$.
\[ Q^{-}: \operatorname{Spine}[\operatorname{\mathbf{Z}}]^{\triangleleft } \rightarrow \operatorname{N}_{\bullet }(\operatorname{Ret}) \quad \quad Q^{+}: \operatorname{Spine}[\operatorname{\mathbf{Z}}]^{\triangleright } \rightarrow \operatorname{N}_{\bullet }(\operatorname{Ret}) \]
Lemma 8.5.6.16. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $F: \operatorname{N}_{\bullet }( \operatorname{Ret}) \rightarrow \operatorname{\mathcal{C}}$ be a functor. Then the composition $\operatorname{Spine}[\operatorname{\mathbf{Z}}]^{\triangleleft } \xrightarrow { Q^{-} } \operatorname{N}_{\bullet }( \operatorname{Ret}) \xrightarrow {F} \operatorname{\mathcal{C}}$ is a limit diagram in $\operatorname{\mathcal{C}}$, and the composition $\operatorname{Spine}[\operatorname{\mathbf{Z}}]^{\triangleright } \xrightarrow {Q^{+}} \operatorname{N}_{\bullet }( \operatorname{Ret}) \xrightarrow { F} \operatorname{\mathcal{C}}$ is a colimit diagram in $\operatorname{\mathcal{C}}$.
Proof. Combine Remark 8.5.3.9, Corollary 7.2.2.3, and Proposition 8.5.4.16. $\square$
Proposition 8.5.6.17. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $e: X \rightarrow X$ be a split idempotent endomorphism in $\operatorname{\mathcal{C}}$. Then $e$ is weakly split.
Proof. Let $\operatorname{Ret}$ denote the category introduced in Construction 8.5.0.2. Using Proposition 8.5.6.10, we can choose a functor $F: \operatorname{N}_{\bullet }(\operatorname{Ret}) \rightarrow \operatorname{\mathcal{C}}$ satisfying $F(\widetilde{X} ) = X$ and $F( \widetilde{e} )= e$.
Let $Q: \operatorname{Spine}[\operatorname{\mathbf{Z}}] \rightarrow \operatorname{N}_{\bullet }( \operatorname{Idem})$ denote the (left and right) cofinal morphism of Proposition 8.5.4.16, and let $Q^{-}: \operatorname{Spine}[\operatorname{\mathbf{Z}}]^{\triangleleft } \rightarrow \operatorname{N}_{\bullet }(\operatorname{Ret})$ and $Q^{+}: \operatorname{Spine}[\operatorname{\mathbf{Z}}]^{\triangleright } \rightarrow \operatorname{N}_{\bullet }( \operatorname{Ret})$ be the extensions of Notation 8.5.6.15. Lemma 8.5.6.16 guarantees that $F \circ Q^{-}$ is a limit diagram in $\operatorname{\mathcal{C}}$ extending $F \circ Q = T_{e}$, so that $e$ satisfies condition $(1)$ of Definition 8.5.6.14. Similarly, $F \circ Q^{+}$ is a colimit diagram extending $T_{e}$, so that $e$ satisfies condition $(2)$ of Definition 8.5.6.14. Condition $(3)$ follows from the observation that any composition of $F( \widetilde{i} )$ with $F( \widetilde{r} )$ is homotopic to the morphism $F( \widetilde{r} \circ \widetilde{i} ) = F( \operatorname{id}_{ \widetilde{Y} } ) = \operatorname{id}_{ F( \widetilde{Y} ) }$, and is therefore an isomorphism. $\square$
Warning 8.5.6.18. The converse of Proposition 8.5.6.17 is false. For example, every isomorphism $e: X \rightarrow X$ is weakly split, but is split idempotent only if $e$ is homotopic to the identity morphism $\operatorname{id}_{X}$ (see Example 8.5.2.2).
Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, and let $\operatorname{End}_{\operatorname{\mathcal{C}}}^{w}$ denote the full subcategory of $\operatorname{End}_{\operatorname{\mathcal{C}}}$ spanned by the weakly split endomorphisms in $\operatorname{\mathcal{C}}$. It follows from Proposition 8.5.6.17 that the restriction functor
\[ \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Ret}), \operatorname{\mathcal{C}}) \rightarrow \operatorname{End}_{\operatorname{\mathcal{C}}} \quad \quad F \mapsto F(e) \]
factors through $\operatorname{End}_{\operatorname{\mathcal{C}}}^{w}$. We will deduce Proposition 8.5.6.4 from the following:
Proposition 8.5.6.19. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then the restriction functor admits a left homotopy inverse.
\[ \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Ret}), \operatorname{\mathcal{C}}) \rightarrow \operatorname{End}_{\operatorname{\mathcal{C}}}^{w} \]
Proof. Let $\operatorname{\mathcal{D}}\subseteq \operatorname{Fun}( \operatorname{Spine}[\operatorname{\mathbf{Z}}], \operatorname{\mathcal{C}})$ denote the full subcategory spanned by those diagrams $S: \operatorname{Spine}[\operatorname{\mathbf{Z}}] \rightarrow \operatorname{\mathcal{C}}$ which admit both a limit and a colimit. Let $u$ and $v$ be auxiliary symbols, and let $\widetilde{\operatorname{\mathcal{D}}}$ denote the full subcategory of $\operatorname{Fun}( \{ u\} \star \operatorname{Spine}[\operatorname{\mathbf{Z}}] \star \{ v\} , \operatorname{\mathcal{C}})$ spanned by those diagrams $S^{\pm }: \{ u\} \star \operatorname{Spine}[\operatorname{\mathbf{Z}}] \star \{ v\} \rightarrow \operatorname{\mathcal{C}}$ which satisfy the following conditions:
- $(1)$
The restriction $S^{-} = S^{\pm }|_{ \{ u\} \star \operatorname{Spine}[\operatorname{\mathbf{Z}}] }$ is a limit diagram in $\operatorname{\mathcal{C}}$.
- $(2)$
The restriction $S^{+} = S^{\pm }|_{ \operatorname{Spine}[\operatorname{\mathbf{Z}}] \star \{ v\} }$ is a colimit diagram in $\operatorname{\mathcal{C}}$.
Note that the simplicial set $\operatorname{Spine}[\operatorname{\mathbf{Z}}]$ is weakly contractible (Remark 8.5.4.15), so that the inclusion map $\operatorname{Spine}[\operatorname{\mathbf{Z}}] \hookrightarrow \{ u\} \star \operatorname{Spine}[\operatorname{\mathbf{Z}}]$ is right anodyne (Proposition 4.3.7.9). Applying Corollary 7.2.2.3, we can replace $(2)$ by the condition that $S^{\pm }$ is a colimit diagram in $\operatorname{\mathcal{C}}$. Moreover, the functor $S^{-}$ admits a colimit if and only if $S = S^{\pm }|_{ \operatorname{Spine}[\operatorname{\mathbf{Z}}] }$ admits a colimit (Corollary 7.2.2.10). Invoking Corollary 7.3.6.15 twice, we deduce that the restriction functor
\[ R: \widetilde{\operatorname{\mathcal{D}}} \rightarrow \operatorname{\mathcal{D}}\quad \quad S^{\pm } \mapsto S^{\pm }|_{ \operatorname{Spine}[\operatorname{\mathbf{Z}}] } \]
is a trivial Kan fibration of $\infty $-categories.
Let $\widetilde{\operatorname{\mathcal{D}}}^{w}$ denote the replete full subcategory of $\widetilde{\operatorname{\mathcal{D}}}$ spanned by those functors $S^{\pm }$ for which the composition
\[ \Delta ^1 \simeq \{ u \} \star \{ v\} \hookrightarrow \{ u\} \star \operatorname{Spine}[\operatorname{\mathbf{Z}}] \star \{ v\} \xrightarrow { S^{\pm } } \operatorname{\mathcal{C}} \]
is an isomorphism in $\operatorname{\mathcal{C}}$. Let $\operatorname{\mathcal{D}}^{w} \subseteq \operatorname{\mathcal{D}}$ be the essential image of $\widetilde{\operatorname{\mathcal{D}}}^{w}$ under $R$, so that $R$ restricts to a trivial Kan fibration $R^ w: \widetilde{\operatorname{\mathcal{D}}}^{w} \rightarrow \operatorname{\mathcal{D}}^{w}$.
Let $T: \operatorname{End}_{\operatorname{\mathcal{C}}} \rightarrow \operatorname{Fun}( \operatorname{Spine}[\operatorname{\mathbf{Z}}], \operatorname{\mathcal{C}})$ be the functor given by precomposition with the covering map $\operatorname{Spine}[\operatorname{\mathbf{Z}}] \rightarrow \Delta ^1 / \operatorname{\partial \Delta }^1$ (see Notation 8.5.4.12). By definition, and endomorphism $e$ of $\operatorname{\mathcal{C}}$ is weakly split if and only if the associated diagram $T_{e}: \operatorname{Spine}[\operatorname{\mathbf{Z}}] \rightarrow \operatorname{\mathcal{C}}$ is an object of $\operatorname{\mathcal{D}}^{w}$. Consequently, the functor $T$ restricts to a functor $T^{w}: \operatorname{End}_{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{D}}_{w}$.
Since the object $\widetilde{Y} \in \operatorname{Ret}$ is both initial and final, the diagram $Q: \operatorname{Spine}[\operatorname{\mathbf{Z}}] \rightarrow \operatorname{N}_{\bullet }( \operatorname{Idem})$ admits an unique extension $Q^{\pm }: \{ u\} \star \operatorname{Spine}[\operatorname{\mathbf{Z}}] \star \{ v\} \rightarrow \operatorname{N}_{\bullet }( \operatorname{Ret})$ carrying both $u$ and $v$ to the object $\widetilde{Y}$. It follows from Lemma 8.5.6.16 that precomposition with $Q^{\pm }$ induces a functor
\[ \widetilde{T}: \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Ret}), \operatorname{\mathcal{C}}) \rightarrow \widetilde{\operatorname{\mathcal{D}}}^{w} \subseteq \operatorname{Fun}( \{ u\} \star \operatorname{Spine}[\operatorname{\mathbf{Z}}] \star \{ v\} , \operatorname{\mathcal{C}}). \]
By construction, we have a commutative diagram of $\infty $-categories
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Ret}), \operatorname{\mathcal{C}}) \ar [r]^-{ \widetilde{T} } \ar [d] & \widetilde{\operatorname{\mathcal{D}}}^{w} \ar [d]^{R^{w}}_{\sim } \\ \operatorname{End}_{\operatorname{\mathcal{C}}}^{w} \ar [r]^-{ T^{w} } & \operatorname{\mathcal{D}}^{w}, } \]
where the right vertical map is a trivial Kan fibration. Consequently, to show that the left vertical map has a left homotopy inverse, it will suffice to show that the functor $\widetilde{T}$ has a left homotopy inverse.
Note that precomposition with the map
\[ \Delta ^2 \simeq \{ u\} \star \{ 0\} \star \{ v\} \hookrightarrow \{ u\} \star \operatorname{Spine}[\operatorname{\mathbf{Z}}] \star \{ v\} \]
determines an evaluation functor $\operatorname{ev}: \widetilde{\operatorname{\mathcal{D}}} \rightarrow \operatorname{Fun}( \Delta ^2, \operatorname{\mathcal{C}})$. Let $\operatorname{Fun}^{w}( \Delta ^2, \operatorname{\mathcal{C}})$ denote the full subcategory of $\operatorname{Fun}( \Delta ^2, \operatorname{\mathcal{C}})$ spanned by those diagrams
\[ \xymatrix@R =50pt@C=50pt{ & X \ar [dr]^{r} & \\ Y \ar [ur]^{i} \ar [rr]^{ u} & & Y' } \]
where $u$ is an isomorphism, so that $\operatorname{ev}$ restricts to a functor $\operatorname{ev}^{w}: \widetilde{\operatorname{\mathcal{D}}}^{w} \rightarrow \operatorname{Fun}^{w}( \Delta ^2, \operatorname{\mathcal{C}})$. It will therefore suffice to show that the composite functor
\[ \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Ret}), \operatorname{\mathcal{C}}) \xrightarrow { \widetilde{T} } \widetilde{\operatorname{\mathcal{D}}}^{w} \xrightarrow { \operatorname{ev}^{w} } \operatorname{Fun}^{w}( \Delta ^2, \operatorname{\mathcal{C}}) \]
has a left homotopy inverse. We conclude by observing that this composite functor is an equivalence of $\infty $-categories, by virtue of Corollary 8.5.1.30. $\square$
Proof of Proposition 8.5.6.4. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. We wish to show that the restriction functor
\[ R: \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Idem}), \operatorname{\mathcal{C}}) \rightarrow \operatorname{End}_{\operatorname{\mathcal{C}}}^{\mathrm{idm}} \]
has a left homotopy inverse. Using Corollary 8.5.5.3, we can choose a fully faithful functor $H: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}'$, where $\operatorname{\mathcal{C}}'$ is idempotent complete. Replacing $\operatorname{\mathcal{C}}$ by the essential image of $H$, we may assume without loss of generality that $\operatorname{\mathcal{C}}$ is a full subcategory of $\operatorname{\mathcal{C}}'$ (and $H$ is the inclusion functor). Then $R$ is the restriction of a functor $R': \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Idem}), \operatorname{\mathcal{C}}' ) \rightarrow \operatorname{End}_{\operatorname{\mathcal{C}}'}^{\mathrm{idm}}$. Since $\operatorname{\mathcal{C}}'$ is idempotent complete, every idempotent endomorphism in $\operatorname{\mathcal{C}}'$ is split, and therefore weakly split. Applying Proposition 8.5.6.19, we deduce that the composition
\[ \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Ret}), \operatorname{\mathcal{C}}' ) \rightarrow \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Idem}), \operatorname{\mathcal{C}}' ) \xrightarrow {R'} \operatorname{End}_{\operatorname{\mathcal{C}}'}^{\mathrm{idm}} \subseteq \operatorname{End}_{\operatorname{\mathcal{C}}}^{w} \]
admits a left homotopy inverse. The restriction map $\operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Ret}), \operatorname{\mathcal{C}}' ) \rightarrow \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Idem}), \operatorname{\mathcal{C}}' )$ is an equivalence of $\infty $-categories (Remark 8.5.4.8), so $R'$ admits a left homotopy inverse $S': \operatorname{End}_{\operatorname{\mathcal{C}}'}^{\mathrm{idm}} \rightarrow \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Idem}), \operatorname{\mathcal{C}}' )$. Restricting to the full subcategory $\operatorname{End}_{\operatorname{\mathcal{C}}}^{\mathrm{idm}} \subseteq \operatorname{End}_{\operatorname{\mathcal{C}}'}^{\mathrm{idm}}$, we obtain a functor $S: \operatorname{End}_{\operatorname{\mathcal{C}}}^{\mathrm{idm}} \rightarrow \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Idem}), \operatorname{\mathcal{C}})$ which is left homotopy inverse to $R$. $\square$