Kerodon

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 8.5.8.4. 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 8.5.8.5 shows that $F$ coincides with $F_{< n}$ on the $(n-1)$-skeleton of $\operatorname{N}_{\bullet }( \operatorname{Idem})$. $\square$

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 8.5.8.4. We then have a commutative diagram of $\infty $-categories

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 8.5.1.7). The functor $\theta '$ fits into a commutative diagram

Since $\iota $ is inner anodyne, the horizontal maps are trivial Kan fibrations (Proposition 1.5.7.6). 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 3.6.1.10). $\square$

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 8.5.4.8). 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

where the horizontal maps are equivalences of $\infty $-categories (Corollaries 8.5.8.9 and 8.5.1.32). 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 8.5.4.8). $\square$

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 5.6.5.16). 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 8.5.8.10. $\square$

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

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

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 $(\ast _ m)$. $\square$

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

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 8.5.4.12. 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

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 8.5.8.14 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 8.5.8.14. 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 8.5.6.10, 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

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 8.5.2.2, 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 8.3.3.17, 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

is a trivial Kan fibration of $\infty $-categories (Corollary 7.3.6.15). 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

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

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

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

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 8.5.4.14), 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

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 8.5.8.4. Fix an integer $n \geq 0$. Using Example 8.5.8.13, 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 8.5.8.14 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 8.5.2.8 (together with Proposition 1.3.3.1), it since the functor $U \circ V$ carries the morphism $e$ to itself. $\square$