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8.4.8 Partial Idempotents

Let $\operatorname{Idem}$ denote the category introduced in Construction 8.4.2.7. For each integer $n \geq 0$, we let $\operatorname{N}_{\leq n}( \operatorname{Idem})$ denote the $n$-skeleton of the simplicial set $\operatorname{N}_{\bullet }( \operatorname{Idem})$ (see Variant 1.2.1.5). If $\operatorname{\mathcal{C}}$ is an $\infty $-category, we will refer to a morphism $\operatorname{N}_{\leq n}(\operatorname{Idem}) \rightarrow \operatorname{\mathcal{C}}$ as a partial idempotent in $\operatorname{\mathcal{C}}$.

Example 8.4.8.1. The simplicial set $\operatorname{N}_{\leq 0}( \operatorname{Idem})$ is isomorphic to the standard simplex $\Delta ^0$. Consequently, if $\operatorname{\mathcal{C}}$ is an $\infty $-category, then a morphism $\operatorname{N}_{\leq 0}( \operatorname{Idem}) \rightarrow \operatorname{\mathcal{C}}$ can be identified with an object $X \in \operatorname{\mathcal{C}}$.

Example 8.4.8.2. The simplicial set $\operatorname{N}_{\leq 1}(\operatorname{Idem})$ can be identified with the simplicial circle $\Delta ^1 / \operatorname{\partial \Delta }^1$, obtained from the standard simplex $\Delta ^1$ by identifying its endpoints. Consequently, if $\operatorname{\mathcal{C}}$ is an $\infty $-category, then a morphism $\operatorname{N}_{\leq 1}( \operatorname{Idem}) \rightarrow \operatorname{\mathcal{C}}$ can be identified with a pair $(X,e)$, where $X$ is an object of $\operatorname{\mathcal{C}}$ and $e$ is an endomorphism of $X$.

Example 8.4.8.3. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then a morphism $\operatorname{N}_{\leq 2}( \operatorname{Idem}) \rightarrow \operatorname{\mathcal{C}}$ can be identified with a triple $(X,e, \sigma )$, where $X$ is an object of $\operatorname{\mathcal{C}}$, $e: X \rightarrow X$ is an endomorphism of $X$, and $\sigma $ is a $2$-simplex of $\operatorname{\mathcal{C}}$ with boundary indicated in the diagram

\[ \xymatrix@R =50pt@C=50pt{ & X \ar [dr]^{e} & \\ X \ar [ur]^{e} \ar [rr]^{e} & & X, } \]

so that $\sigma $ witnesses the identity $[e] = [e] \circ [e]$ in the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$.

Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $e: X \rightarrow X$ be an endomorphism in $\operatorname{\mathcal{C}}$, which we identify with a morphism $F_{\leq 1}: \operatorname{N}_{\leq 1}( \operatorname{Idem}) \rightarrow \operatorname{\mathcal{C}}$. The endomorphism $e$ is homotopy idempotent (in the sense of Definition 8.4.7.1) if and only if $F_{\leq 1}$ admits an extension $F_{\leq 2}: \operatorname{N}_{\leq 2}(\operatorname{Idem}) \rightarrow \operatorname{\mathcal{C}}$. Proposition 8.4.7.15 shows that this condition does not guarantee the existence of an idempotent $F: \operatorname{N}_{\bullet }( \operatorname{Idem}) \rightarrow \operatorname{\mathcal{C}}$ extending $F_{\leq 1}$. Our goal in this section is to show that a slightly stronger condition does suffice: namely, it is enough to assume that $F_{\leq 1}$ can be extended to a diagram $F_{\leq 3}: \operatorname{N}_{\leq 3}(\operatorname{Idem}) \rightarrow \operatorname{\mathcal{C}}$. This is a consequence of the following:

Theorem 8.4.8.4. Let $n \geq 3$ be an integer. The inclusion map $\operatorname{N}_{\leq n}( \operatorname{Idem}) \hookrightarrow \operatorname{N}_{\bullet }( \operatorname{Idem})$ admits a factorization

\[ \operatorname{N}_{\leq n}(\operatorname{Idem}) \xrightarrow { \iota } \operatorname{\mathcal{E}}\xrightarrow { U } \operatorname{N}_{\bullet }( \operatorname{Idem}), \]

where $\operatorname{\mathcal{E}}$ is an $\infty $-category, $\iota $ is an inner anodyne morphism which is bijective on simplices of dimension $< n$, and the functor $U$ admits a right inverse $V: \operatorname{N}_{\bullet }( \operatorname{Idem}) \rightarrow \operatorname{\mathcal{E}}$.

Remark 8.4.8.5. In the situation of Theorem 8.4.8.5, we can regard $\iota $ and $V|_{ \operatorname{N}_{\leq n}(\operatorname{Idem}) }$ as morphisms from $\operatorname{N}_{\leq n}(\operatorname{Idem})$ to $\operatorname{\mathcal{E}}$. By construction, these morphisms coincide after composing with the functor $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{N}_{\bullet }( \operatorname{Idem})$. Since $U$ is bijective on simplices of dimensions $< n$, it follows that $\iota $ and $V|_{ \operatorname{N}_{\leq n}( \operatorname{Idem}) }$ coincide when on the $(n-1)$-skeleton of $\operatorname{N}_{\bullet }( \operatorname{Idem})$. Beware that $\iota $ and $V$ do not coincide on the nondegenerate $n$-simplex of $\operatorname{N}_{\bullet }( \operatorname{Idem})$. In fact, we claim that $\iota $ and $V|_{ \operatorname{N}_{\leq n}( \operatorname{Idem}) }$ are not even isomorphic when viewed as an object of the $\infty $-category $\operatorname{Fun}( \operatorname{N}_{\leq n}( \operatorname{Idem}), \operatorname{\mathcal{E}})$. Assume, for a contradiction, that there exists an isomorphism $\alpha $ of $\iota = \operatorname{id}_{\operatorname{\mathcal{E}}} \circ \iota $ with $V|_{ \operatorname{N}_{\leq n}( \operatorname{Idem}) } = (V \circ U) \circ \iota $. Since $\iota $ is inner anodyne, we could then lift $\alpha $ to an isomorphism $\widetilde{\alpha }: \operatorname{id}_{\operatorname{\mathcal{E}}} \rightarrow V \circ U$ in the $\infty $-category $\operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Idem}), \operatorname{\mathcal{C}})$. It would follow that $U$ is an equivalence of $\infty $-categories (with homotopy inverse given by $V$). Then $(U \circ \iota ): \operatorname{N}_{\leq n}( \operatorname{Idem}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{Idem})$ would be a categorical equivalence of simplicial sets, which contradicts Remark 8.4.4.18.

We will give the proof of Theorem 8.4.8.4 at the end of this section. First, let us record some consequences.

Corollary 8.4.8.6. Let $n \geq 3$ be an integer, and let $\operatorname{\mathcal{C}}$ be an $\infty $-category equipped with a partial idempotent $F_{< n}: \operatorname{N}_{\leq n-1}( \operatorname{Idem}) \rightarrow \operatorname{\mathcal{C}}$. The following conditions are equivalent:

$(1)$

The morphism $F_{< n}$ extends to an idempotent $F: \operatorname{N}_{\bullet }( \operatorname{Idem}) \rightarrow \operatorname{\mathcal{C}}$.

$(2)$

The morphism $F_{< n}$ extends to a partial idempotent $F_{\leq n}: \operatorname{N}_{\leq n}( \operatorname{Idem}) \rightarrow \operatorname{\mathcal{C}}$.

Proof. The implication $(1) \Rightarrow (2)$ is immediate. To prove the converse, suppose that $F_{\leq n}: \operatorname{N}_{\leq n}(\operatorname{Idem}) \rightarrow \operatorname{\mathcal{C}}$ is an extension of $F_{< n}$. Let $\iota : \operatorname{N}_{\leq n}( \operatorname{Idem}) \hookrightarrow \operatorname{\mathcal{E}}$, $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{N}_{\bullet }( \operatorname{Idem})$, and $V: \operatorname{N}_{\bullet }( \operatorname{Idem}) \rightarrow \operatorname{\mathcal{E}}$ be as in Theorem 8.4.8.4. Since $\iota $ is inner anodyne, we can choose a functor $\overline{F}: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ satisfying $\overline{F} \circ \iota = F_{\leq n}$. Then $F = \overline{F} \circ V$ is a functor from $\operatorname{N}_{\bullet }( \operatorname{Idem})$ to $\operatorname{\mathcal{C}}$, and Remark 8.4.8.5 shows that $F$ coincides with $F_{< n}$ on the $(n-1)$-skeleton of $\operatorname{N}_{\bullet }( \operatorname{Idem})$. $\square$

Warning 8.4.8.7. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $F_{\leq n}: \operatorname{N}_{\leq n}( \operatorname{Idem}) \rightarrow \operatorname{\mathcal{C}}$ be a partial idempotent in $\operatorname{\mathcal{C}}$. Corollary 8.4.8.6 asserts that, if $n \geq 3$, then we can choose an idempotent $F: \operatorname{N}_{\bullet }( \operatorname{Idem}) \rightarrow \operatorname{\mathcal{C}}$ such that $F$ and $F_{\leq n}$ coincide on the $(n-1)$-skeleton $\operatorname{N}_{\bullet }( \operatorname{Idem})$. Beware that we generally cannot arrange that $F|_{ \operatorname{N}_{\leq n}( \operatorname{Idem}) }$ coincides with $F_{\leq n}$. For example, this always fails in the (universal) case $F_{\leq n}$ is the inner anodyne morphism $\iota : \operatorname{N}_{\leq n}( \operatorname{Idem}) \hookrightarrow \operatorname{\mathcal{E}}$ of Theorem 8.4.8.4 (see Remark 8.4.8.5).

Corollary 8.4.8.8. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $e: X \rightarrow X$ be an endomorphism in $\operatorname{\mathcal{C}}$. Then $e$ is idempotent if and only if it can be extended to a diagram $\operatorname{N}_{\leq 3}( \operatorname{Idem}) \rightarrow \operatorname{\mathcal{C}}$.

Corollary 8.4.8.9. Let $\{ \operatorname{\mathcal{C}}_{i} \} _{i \in \operatorname{\mathcal{I}}}$ be a diagram of simplicial sets indexed by a filtered category $\operatorname{\mathcal{I}}$. Suppose that each $\operatorname{\mathcal{C}}_{i}$ is an $\infty $-category. Then the tautological map

\[ \theta : \varinjlim _{i \in \operatorname{\mathcal{I}}} \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Idem}), \operatorname{\mathcal{C}}_ i ) \rightarrow \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Idem}), \varinjlim _{i \in \operatorname{\mathcal{I}}} \operatorname{\mathcal{C}}_ i ) \]

is an equivalence of $\infty $-categories.

Proof. Choose any integer $n \geq 3$, and let $\iota : \operatorname{N}_{\leq n}( \operatorname{Idem}) \hookrightarrow \operatorname{\mathcal{E}}$, $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{N}_{\bullet }( \operatorname{Idem})$, and $V: \operatorname{N}_{\bullet }( \operatorname{Idem}) \rightarrow \operatorname{\mathcal{E}}$ be as in Theorem 8.4.8.4. We then have a commutative diagram of $\infty $-categories

\[ \xymatrix@R =50pt@C=50pt{ \varinjlim \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Idem}), \operatorname{\mathcal{C}}_ i ) \ar [r]^-{\circ V} \ar [d]^{\theta } & \varinjlim \operatorname{Fun}( \operatorname{\mathcal{E}}, \operatorname{\mathcal{C}}_ i ) \ar [d]^{\theta '} \ar [r]^-{ \circ U} & \varinjlim \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Idem}) , \operatorname{\mathcal{C}}_{i} ) \ar [d]^{\theta } \\ \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Idem}), \varinjlim \operatorname{\mathcal{C}}_{i}) \ar [r]^-{ \circ V} & \operatorname{Fun}( \operatorname{\mathcal{E}}, \varinjlim \operatorname{\mathcal{C}}_{i} ) \ar [r]^-{ \circ U} & \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Idem}), \varinjlim \operatorname{\mathcal{C}}_{i} ), } \]

where the horizontal compositions are identity morphisms. Consequently, to show that $\theta $ is an equivalence of $\infty $-categories, it will suffice to show that $\theta '$ is an equivalence of $\infty $-categories (Proposition 8.4.1.6). The functor $\theta '$ fits into a commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ \varinjlim \operatorname{Fun}( \operatorname{\mathcal{E}}, \operatorname{\mathcal{C}}_ i ) \ar [r]^-{ \circ \iota } \ar [d]^{\theta '} & \varinjlim \operatorname{Fun}( \operatorname{N}_{\leq n}(\operatorname{Idem}), \operatorname{\mathcal{C}}_ i ) \ar [d]^{\theta ''} \\ \operatorname{Fun}( \operatorname{\mathcal{E}}, \varinjlim \operatorname{\mathcal{C}}_ i ) \ar [r]^-{ \circ \iota } & \operatorname{Fun}( \operatorname{N}_{\leq n}( \operatorname{Idem}), \varinjlim (\operatorname{\mathcal{C}}_ i) ). } \]

Since $\iota $ is inner anodyne, the horizontal maps are trivial Kan fibrations (Proposition 1.4.7.6). We conclude by observing that $\theta ''$ is an isomorphism of simplicial sets, since the simplicial set $\operatorname{N}_{\leq n}( \operatorname{Idem})$ is finite (Corollary 3.5.1.10). $\square$

Corollary 8.4.8.10. Let $\{ \operatorname{\mathcal{C}}_{i} \} _{i \in \operatorname{\mathcal{I}}}$ be a diagram of simplicial sets indexed by a filtered category $\operatorname{\mathcal{I}}$. Suppose that each $\operatorname{\mathcal{C}}_{i}$ is an idempotent complete $\infty $-category. Then the colimit $\operatorname{\mathcal{C}}= \varinjlim _{i} \operatorname{\mathcal{C}}_{i}$ is idempotent complete.

Proof. For each object $i \in \operatorname{\mathcal{I}}$, our assumption that $\operatorname{\mathcal{C}}_{i}$ is idempotent complete guarantees that the restriction functor $R_{i}: \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Ret}), \operatorname{\mathcal{C}}_ i ) \rightarrow \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Idem}), \operatorname{\mathcal{C}}_ i )$ is an equivalence of $\infty $-categories (Remark 8.4.4.8). Passing to filtered colimits, we deduce that the induced map $\varinjlim \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Ret}), \operatorname{\mathcal{C}}_ i ) \rightarrow \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Idem}), \operatorname{\mathcal{C}}_ i )$ is also an equivalence of $\infty $-categories (Corollary 4.5.7.2). This map fits into a commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ \varinjlim \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Ret}), \operatorname{\mathcal{C}}_ i ) \ar [r] \ar [d] & \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Ret}), \operatorname{\mathcal{C}}) \ar [d]^{R} \\ \varinjlim \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Idem}), \operatorname{\mathcal{C}}_ i ) \ar [r] & \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Idem}), \operatorname{\mathcal{C}}), } \]

where the horizontal maps are equivalences of $\infty $-categories (Corollaries 8.4.8.9 and 8.4.1.27). It follows that the restriction functor $R: \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Ret}), \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Idem}), \operatorname{\mathcal{C}})$ is also an equivalence of $\infty $-categories, so that $\operatorname{\mathcal{C}}$ is idempotent complete (Remark 8.4.4.8). $\square$

Corollary 8.4.8.11. Let $\operatorname{\mathcal{C}}$ be a small filtered $\infty $-category and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$ be a functor. Suppose that, for every object $C \in \operatorname{\mathcal{C}}$, the $\infty $-category $\mathscr {F}(C)$ is idempotent complete. Then the colimit $\varinjlim ( \mathscr {F} )$ (formed in the $\infty $-category $\operatorname{\mathcal{QC}}$) is also idempotent complete.

Proof. Using Theorem 7.2.7.2, we can choose a directed partially ordered set $(A, \leq )$ and a right cofinal functor $\operatorname{N}_{\bullet }(A) \rightarrow \operatorname{\mathcal{C}}$. Using Corollary 7.2.2.3 we can replace $\operatorname{\mathcal{C}}$ by $\operatorname{N}_{\bullet }(A)$ and thereby reduce to the case where $\operatorname{\mathcal{C}}$ is (the nerve of) a directed partially ordered set. Replacing $\mathscr {F}$ by an isomorphic functor if necessary, we can assume that it obtained from an $A$-indexed diagram in the ordinary category $\operatorname{QCat}$ (Corollary 5.7.5.16). In this case, the colimit $\varinjlim ( \mathscr {F} )$ in the $\infty $-category $\operatorname{\mathcal{QC}}$ can be identified with its colimit in the ordinary category $\operatorname{QCat}\subset \operatorname{Set_{\Delta }}$ (Corollary 7.5.9.3), so the desired result follows from Corollary 8.4.8.10. $\square$

We now turn to the proof of Theorem 8.4.8.4. The existence of an inner anodyne morphism $\iota : \operatorname{N}_{\leq n}( \operatorname{Idem}) \rightarrow \operatorname{\mathcal{E}}$ which is bijective on simplices of dimension $< n$ is essentially formal, by virtue of the following variant of Corollary 4.1.3.3:

Lemma 8.4.8.12. Let $n \geq 0$ be an integer and let $K$ be a simplicial set which satisfies the following condition for $0 \leq m \leq n$:

$(\ast _ m)$

For every integer $0 < i < m$, every morphism of simplicial sets $\sigma : \Lambda ^{m}_{i} \rightarrow K$ can be extended to an $m$-simplex of $K$.

Then there exists an inner anodyne morphism $\iota : K \hookrightarrow \operatorname{\mathcal{E}}$, where $\operatorname{\mathcal{E}}$ is an $\infty $-category and $\iota $ is bijective on simplices of dimension $< n$.

Proof. We construct $\operatorname{\mathcal{E}}$ as the colimit of a sequence of inner anodyne maps

\[ K = K(0) \hookrightarrow K(1) \hookrightarrow K(2) \hookrightarrow K(3) \hookrightarrow \cdots \]

Assume that $K(t)$ has been constructed for some $t \geq 0$, and let $S$ be the collection of all maps $\sigma : \Lambda ^{m}_{i} \rightarrow K(t)$ where $0 < i < m$ and $m > n$. For every $\sigma \in S$, let us write $C_{\sigma }$ for the simplicial set $\Lambda ^{m}_{i}$ which is the source of $\sigma $, and $D_{\sigma }$ for the simplex $\Delta ^{m}$. We then construct a pushout diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \coprod _{\sigma \in S} C_{\sigma } \ar [r] \ar [d] & K(t) \ar [d] \\ \coprod _{\sigma \in S} D_{\sigma } \ar [r] & K(t+1). } \]

By construction, the morphism $K(t) \rightarrow K(t+1)$ is inner anodyne and bijective on simplices of dimension $< n$. To complete the proof, it will suffice to show that the colimit $\operatorname{\mathcal{E}}= \varinjlim _{t} K(t)$ is an $\infty $-category. Fix a pair of integers $0 < i < m$ and a morphism $\sigma : \Lambda ^{m}_{i} \rightarrow \operatorname{\mathcal{E}}$; we wish to show that $\sigma $ can be extended to an $m$-simplex of $\operatorname{\mathcal{E}}$. Since $\Lambda ^{m}_{i}$ is a finite simplicial set, $\sigma $ factors (uniquely) through $K(t)$ for some integer $t \gg 0$. By construction, if $m > n$ then $\sigma $ can be extended to an $m$-simplex of $K(t+1)$. We may therefore assume that $m \leq n$. In this case, we can take $k = 0$, in which case the existence of the desired extension follows from assumption $(\ast _ m)$. $\square$

Example 8.4.8.13. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. For every integer $n \geq 0$, the skeleton $\operatorname{sk}_{n}(\operatorname{\mathcal{C}})$ satisfies condition $(\ast _ m)$ of Lemma 8.4.8.12 for $0 \leq m \leq n$. We can therefore choose an inner anodyne morphism $\iota : \operatorname{sk}_{n}(\operatorname{\mathcal{C}}) \hookrightarrow \operatorname{\mathcal{E}}$, where $\operatorname{\mathcal{E}}$ is an $\infty $-category and $f$ is bijective on simplices of dimension $< n$.

In what follows, we will write $X$ for the unique object of the category $\operatorname{Idem}$, and $e: X \rightarrow X$ for the unique non-identity morphism. The main content of Theorem 8.4.8.4 is contained in the following result:

Proposition 8.4.8.14. Let $\operatorname{\mathcal{E}}$ be an $\infty $-category and let $\iota : \operatorname{N}_{\leq 3}( \operatorname{Idem}) \rightarrow \operatorname{\mathcal{E}}$ be a morphism of simplicial sets which is bijective on simplices of dimension $\leq 2$. Then $\iota ( \widetilde{e} )$ is an idempotent morphism of $\operatorname{\mathcal{E}}$.

Proof. Let us regard the linearly ordered set $\operatorname{\mathbf{Z}}= \{ \cdots < -2 < -1 < 0 < 1 < 2 < \cdots \} $ as a category. Let $X \in \operatorname{Fun}( \operatorname{\mathbf{Z}}, \operatorname{Idem})$ denote the constant functor taking the value $X$, and let $Y \in \operatorname{Fun}( \operatorname{\mathbf{Z}}, \operatorname{Idem})$ denote the functor which carries each non-identity morphism of $\operatorname{\mathbf{Z}}$ to the morphism $\widetilde{e}$ of $\operatorname{Idem}$. We then have natural transformations

\[ e_{X}: X \rightarrow X \quad \quad i: Y \rightarrow X \quad \quad r: X \rightarrow Y \quad \quad e_{ Y }: Y \rightarrow Y \]

which carry each element of $\operatorname{\mathbf{Z}}$ to the morphism $\widetilde{e}$. Note that the linearly ordered set $(\operatorname{\mathbf{Z}}, \leq )$ can be identified with the homotopy category of the simplicial set $\operatorname{Spine}[\operatorname{\mathbf{Z}}]$ of Notation 8.4.4.12. Let $G: \operatorname{Fun}( \operatorname{Spine}[\operatorname{\mathbf{Z}}], \operatorname{N}_{\leq 3}( \operatorname{Idem}) ) \rightarrow \operatorname{Fun}( \operatorname{Spine}[\operatorname{\mathbf{Z}}], \operatorname{\mathcal{E}})$ be the morphism of simplicial sets given by composition with $\iota $. Since the simplicial set $\operatorname{Spine}[\operatorname{\mathbf{Z}}]$ is $1$-dimensional, the inclusion map

\[ \operatorname{Fun}( \operatorname{Spine}[\operatorname{\mathbf{Z}}], \operatorname{N}_{\leq 3}( \operatorname{Idem}) ) \hookrightarrow \operatorname{Fun}( \operatorname{Spine}[\operatorname{\mathbf{Z}}], \operatorname{N}_{\bullet }( \operatorname{Idem}) ) \simeq \operatorname{N}_{\bullet }( \operatorname{Fun}( \operatorname{\mathbf{Z}}, \operatorname{Idem}) ) \]

is bijective on simplices of dimension $\leq 2$, and therefore induces an equivalence of homotopy categories. It follows that $G$ induces a functor of homotopy categories $\mathrm{h} \mathit{G}: \operatorname{Fun}(\operatorname{\mathbf{Z}}, \operatorname{Idem}) \rightarrow \mathrm{h} \mathit{ \operatorname{Fun}( \operatorname{Spine}[\operatorname{\mathbf{Z}}], \operatorname{\mathcal{E}}) }$.

In what follows, we will identify the morphisms $X$ and $Y$ with vertices of the simplicial set $\operatorname{Fun}( \operatorname{Spine}[\operatorname{\mathbf{Z}}], \operatorname{N}_{\leq 3}( \operatorname{Idem}) )$, and the morphisms $i$, $r$, $e_{X }$, and $e_{Y }$ with edges of the simplicial set $\operatorname{Fun}( \operatorname{Spine}[\operatorname{\mathbf{Z}}], \operatorname{N}_{\leq 3}( \operatorname{Idem}) )$. Let $\delta _{\operatorname{\mathcal{E}}}: \operatorname{\mathcal{E}}\rightarrow \operatorname{Fun}( \operatorname{Spine}[\operatorname{\mathbf{Z}}], \operatorname{\mathcal{E}})$ be the diagonal map, so that we have $G( X ) = \delta _{\operatorname{\mathcal{C}}}( \iota ( \widetilde{X}) )$ and $G( e_{ X} ) = \delta _{\operatorname{\mathcal{C}}}( \iota ( \widetilde{e} ) )$. We will deduce Proposition 8.4.8.14 from the following:

$(\ast )$

There exists an $\infty $-category $\operatorname{\mathcal{C}}$ and a functor $T: \operatorname{Fun}( \operatorname{Spine}[\operatorname{\mathbf{Z}}], \operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{C}}$ such that $(T \circ \delta _{\operatorname{\mathcal{E}}}): \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is fully faithful and $(T \circ G)( e_{Y} )$ is an isomorphism in $\operatorname{\mathcal{C}}$.

Let us first assume $(\ast )$, and show that it implies Proposition 8.4.8.14. We wish to show that $\iota ( \widetilde{e})$ is an idempotent endomorphism in the $\infty $-category $\operatorname{\mathcal{E}}$. Since $T \circ \delta _{\operatorname{\mathcal{E}}}$ is fully faithful, this is equivalent to the statement that $(T \circ \delta _{C})( \iota ( \widetilde{e} ) ) = (T \circ G)( e_{ X} ) )$ is an idempotent endomorphism in the $\infty $-category $\operatorname{\mathcal{C}}$. By virtue of Proposition 8.4.6.10, it will suffice to show that the homotopy class $[ T( G( e_{ X } ) ) ] = (\mathrm{h} \mathit{T} \circ \mathrm{h} \mathit{G})(e_{X })$ is a split idempotent in the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$. By construction, the morphism $e_{ X }$ factors as a composition $i \circ r$ in the category $\operatorname{Fun}( \operatorname{\mathbf{Z}}, \operatorname{Idem})$. It will therefore suffice to show that the functor $\mathrm{h} \mathit{T} \circ \mathrm{h} \mathit{G}$ carries the commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ & X \ar [dr]^{ r } & \\ Y \ar [ur]^{i} \ar [rr]^{ e_{Y } } & & Y } \]

to a retraction diagram in $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$: that is, that $(\mathrm{h} \mathit{T} \circ \mathrm{h} \mathit{G})(e_{ Y })$ is an identity morphism. This follows from Remark 8.4.2.2, since the morphism $(\mathrm{h} \mathit{T} \circ \mathrm{h} \mathit{G})( e_{ Y} )$ is both idempotent (since $e_{ Y }$ is an idempotent in the category $\operatorname{Fun}( \operatorname{\mathbf{Z}}, \operatorname{Idem})$) and an isomorphism (by virtue of assumption $(\ast )$).

We now prove $(\ast )$. Using Corollary 8.2.5.12, we can choose a fully faithful functor of $\infty $-categories $H: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$, where the $\infty $-category $\operatorname{\mathcal{C}}$ admits sequential colimits. Let $\operatorname{\mathcal{D}}$ denote the full subcategory of $\operatorname{Fun}( \operatorname{Spine}[\operatorname{\mathbf{Z}}]^{\triangleright }, \operatorname{\mathcal{C}})$ spanned by the colimit diagrams. Our assumption that $\operatorname{\mathcal{C}}$ admits sequential colimits guarantees that the restriction functor

\[ \operatorname{\mathcal{D}}\rightarrow \operatorname{Fun}( \operatorname{Spine}[\operatorname{\mathbf{Z}}], \operatorname{\mathcal{C}}) \quad \quad U \mapsto U|_{ \operatorname{Spine}[\operatorname{\mathbf{Z}}] } \]

is a trivial Kan fibration of $\infty $-categories (Corollary 7.3.6.13). Let $\delta _{\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\hookrightarrow \operatorname{Fun}( \operatorname{Spine}[\operatorname{\mathbf{Z}}], \operatorname{\mathcal{C}})$ and $\widetilde{\delta }_{\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\hookrightarrow \operatorname{Fun}( \operatorname{Spine}[\operatorname{\mathbf{Z}}]^{\triangleright }, \operatorname{\mathcal{C}})$ be the diagonal embeddings. Since the simplicial set $\operatorname{Spine}[\operatorname{\mathbf{Z}}]$ is weakly contractible, the morphism $\widetilde{\delta }_{\operatorname{\mathcal{C}}}$ factors through $\operatorname{\mathcal{D}}$ (Corollary 7.2.3.5). Let $s: \operatorname{Fun}( \operatorname{Spine}[\operatorname{\mathbf{Z}}], \operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{D}}$ be a solution to the lifting problem

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [d]^{\delta _{\operatorname{\mathcal{C}}}} \ar [r]^-{ \widetilde{\delta }_{\operatorname{\mathcal{C}}} } & \operatorname{\mathcal{D}}\ar [d]^{U \mapsto U|_{ \operatorname{Spine}[\operatorname{\mathbf{Z}}] }} \\ \operatorname{Fun}( \operatorname{Spine}[\operatorname{\mathbf{Z}}], \operatorname{\mathcal{C}}) \ar [r]^-{\operatorname{id}} \ar@ {-->}[ur]^{ s } & \operatorname{Fun}( \operatorname{Spine}[\operatorname{\mathbf{Z}}], \operatorname{\mathcal{C}}). } \]

Let $\operatorname{ev}: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be the functor given by evaluation at the cone point of $\operatorname{Spine}[\operatorname{\mathbf{Z}}]^{\triangleright }$. We let $T: \operatorname{Fun}( \operatorname{Spine}[\operatorname{\mathbf{Z}}], \operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{C}}$ denote the functor given by the composition

\[ \operatorname{Fun}( \operatorname{Spine}[\operatorname{\mathbf{Z}}], \operatorname{\mathcal{E}}) \xrightarrow {H \circ } \operatorname{Fun}( \operatorname{Spine}[\operatorname{\mathbf{Z}}], \operatorname{\mathcal{C}}) \xrightarrow {s} \operatorname{Fun}( \operatorname{Spine}[\operatorname{\mathbf{Z}}]^{\triangleright }, \operatorname{\mathcal{C}}) \xrightarrow { \operatorname{ev}} \operatorname{\mathcal{C}}. \]

Stated more informally, the functor $T$ carries a diagram

\[ \cdots \rightarrow C_{-2} \rightarrow C_{-1} \rightarrow C_0 \rightarrow C_1 \rightarrow C_2 \rightarrow \cdots \]

in the $\infty $-category $\operatorname{\mathcal{E}}$ to a colimit of the diagram

\[ \cdots \rightarrow H( C_{-2} ) \rightarrow H( C_{-1} ) \rightarrow H( C_0 ) \rightarrow H( C_1 ) \rightarrow H(C_2 ) \rightarrow \cdots \]

in the $\infty $-category $\operatorname{\mathcal{C}}$. By construction, $T \circ \delta _{\operatorname{\mathcal{E}}}$ coincides with the fully faithful functor $H$.

We now complete the proof by showing that the functor $T$ carries $G( e_{ Y} )$ to an isomorphism in $\operatorname{\mathcal{C}}$. Let us regard $G( \widetilde{Y} )$ as a diagram $\operatorname{Spine}[\operatorname{\mathbf{Z}}] \rightarrow \operatorname{\mathcal{E}}$. Since the inclusion $\operatorname{Spine}[\operatorname{\mathbf{Z}}] \hookrightarrow \operatorname{N}_{\bullet }( \operatorname{\mathbf{Z}})$ is inner anodyne (Remark 8.4.4.14), we can extend $G( Y )$ to a functor $C: \operatorname{N}_{\bullet }( \operatorname{\mathbf{Z}}) \rightarrow \operatorname{\mathcal{E}}$. Let $a: \{ 0 < 1 \} \times \operatorname{\mathbf{Z}}\rightarrow \operatorname{\mathbf{Z}}$ be the morphism of partially ordered sets given by $a(i, n) = i+n$. Passing to nerves, we obtain a morphism of simplicial sets $A: \Delta ^1 \times \operatorname{N}_{\bullet }(\operatorname{\mathbf{Z}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathbf{Z}})$. The composition $M = C \circ A$ then corresponds to a diagram in $\operatorname{\mathcal{E}}$ which we display informally as

\[ \xymatrix@R =50pt@C=50pt{ \cdots \ar [r] & \iota (\widetilde{X}) \ar [r]^-{ \iota (\widetilde{e}) } \ar [d]^{ \iota (\widetilde{e}) } & \iota (\widetilde{X}) \ar [r]^-{ \iota ( \widetilde{e}) } \ar [d]^{ \iota (\widetilde{e}) } & \iota (\widetilde{X}) \ar [r]^-{ \iota (\widetilde{e}) } \ar [d]^{\iota (\widetilde{e}) } & \cdots \\ \cdots \ar [r] & \iota (\widetilde{X}) \ar [r]^-{ \iota (\widetilde{e}) } & \iota (\widetilde{X}) \ar [r]^-{ \iota (\widetilde{e}) } & \iota (\widetilde{X}) \ar [r]^-{ \iota (\widetilde{e}) } & \cdots } \]

Let $f$ denote the restriction $M|_{ \Delta ^1 \times \operatorname{Spine}[\operatorname{\mathbf{Z}}] }$, which we regard as a morphism in the $\infty $-category $\operatorname{Fun}( \operatorname{Spine}[\operatorname{\mathbf{Z}}], \operatorname{\mathcal{E}})$. Since $\iota $ is bijective on simplices of dimension $\leq 2$ and the simplicial set $\operatorname{Spine}[\operatorname{\mathbf{Z}}]$ has dimension $\leq 1$, the morphism $G$ is bijective on simplices of dimension $\leq 1$. We can therefore write $f = G( f_0 )$ for a unique edge $f_0$ of $\operatorname{Fun}( \operatorname{Spine}[\operatorname{\mathbf{Z}}], \operatorname{N}_{\leq 3}( \operatorname{Idem}) )$. It follows by inspection that $f_0$ must coincide with $e_{ \widetilde{Y} }$. We are therefore reduced to showing that $T(f)$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{C}}$.

Since the $\infty $-category $\operatorname{\mathcal{C}}$ admits sequential colimits, we can extend $(H \circ C): \operatorname{N}_{\bullet }(\operatorname{\mathbf{Z}}) \rightarrow \operatorname{\mathcal{C}}$ to a colimit diagram $\overline{C}: \operatorname{N}_{\bullet }(\operatorname{\mathbf{Z}})^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$. Note that $A$ extends uniquely to a morphism of simplicial sets $\overline{A}: \Delta ^1 \times \operatorname{N}_{\bullet }(\operatorname{\mathbf{Z}})^{\triangleright } \rightarrow \operatorname{N}_{\bullet }( \operatorname{\mathbf{Z}})^{\triangleright }$ (given on vertices by $\overline{A}(i,v) = v$, where $v$ is the cone point of $\operatorname{N}_{\bullet }(\operatorname{\mathbf{Z}})^{\triangleright }$). Let us identify the restriction $(\overline{C} \circ \overline{A})|_{ \Delta ^1 \times \operatorname{Spine}[\operatorname{\mathbf{Z}}]^{\triangleright } }$ with a morphism $\overline{f}: D \rightarrow D'$ in the $\infty $-category $\operatorname{Fun}( \operatorname{Spine}[\operatorname{\mathbf{Z}}]^{\triangleright }, \operatorname{\mathcal{C}})$. Since the inclusion $\operatorname{Spine}[\operatorname{\mathbf{Z}}] \hookrightarrow \operatorname{N}_{\bullet }(\operatorname{\mathbf{Z}})$ is right cofinal (Proposition 7.2.1.3), both $D$ and $D'$ are colimit diagrams in $\operatorname{\mathcal{D}}$ (Corollary 7.2.2.3). Consequently, we can view $\overline{f}$ as a morphism in the $\infty $-category $\operatorname{\mathcal{D}}$. By construction, the restriction functor $\operatorname{\mathcal{D}}\rightarrow \operatorname{Fun}( \operatorname{Spine}[\operatorname{\mathbf{Z}}], \operatorname{\mathcal{C}})$ carries $\overline{f}$ to $H(f)$. It follows that $T(f) = (\operatorname{ev}\circ s \circ H)(f)$ is isomorphic to $\operatorname{ev}( \overline{f} )$ as an object of the $\infty $-category $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$. We are therefore reduced to showing that $\operatorname{ev}( \overline{f} )$ is an isomorphism in $\operatorname{\mathcal{C}}$. This is clear: the morphism $\operatorname{ev}( \overline{f} )$ is an identity morphism in $\operatorname{\mathcal{C}}$, since the functor $\overline{A}$ carries $\Delta ^1 \times \{ v\} $ to a degenerate edge of $\operatorname{N}_{\bullet }( \operatorname{\mathbf{Z}})^{\triangleright }$. $\square$

Proof of Theorem 8.4.8.4. Fix an integer $n \geq 0$. Using Example 8.4.8.13, we can choose an $\infty $-category $\operatorname{\mathcal{E}}$ and an inner anodyne morphism $\iota : \operatorname{N}_{\leq n}(\operatorname{Idem}) \hookrightarrow \operatorname{\mathcal{E}}$ which is bijective on simplices of dimension $< n$. Since $\iota $ is inner anodyne, there is a unique functor $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{N}_{\bullet }( \operatorname{Idem})$ for which $U \circ \iota $ is coincides with the inclusion map $\operatorname{N}_{\leq n}( \operatorname{Idem}) \hookrightarrow \operatorname{N}_{\bullet }( \operatorname{Idem})$. If $n \geq 3$, then Proposition 8.4.8.14 guarantees that $\iota (\widetilde{e})$ is an idempotent endomorphism in $\operatorname{\mathcal{E}}$: that is, there exists a functor $V: \operatorname{N}_{\bullet }( \operatorname{Idem}) \rightarrow \operatorname{\mathcal{E}}$ satisfying $V(\widetilde{e}) = \iota (\widetilde{e})$. To complete the proof, it will suffice to show that the composition $\operatorname{N}_{\bullet }( \operatorname{Idem}) \xrightarrow {V} \operatorname{\mathcal{E}}\xrightarrow {U} \operatorname{N}_{\bullet }( \operatorname{Idem})$ is the identity functor. This follows from the universal property of Remark 8.4.2.8 (together with Proposition 1.2.2.1), it since the functor $U \circ V$ carries the morphism $e$ to itself. $\square$