Kerodon

Proof. Without loss of generality, we may assume that the morphism $e$ is obtained from a morphism $(X,x) \rightarrow (X,x)$ in the ordinary category of pointed Kan complexes (see Proposition 5.5.3.8). Then the diagram $T_{e}: \operatorname{Spine}[\operatorname{\mathbf{Z}}] \rightarrow \operatorname{\mathcal{S}}_{\ast }$ of Notation 8.5.6.11 lifts to a functor of ordinary categories $\mathscr {F}: (\operatorname{\mathbf{Z}}, \leq ) \rightarrow \operatorname{Kan}_{\ast }$, which we display as

Let us abuse notation by identifying $\mathscr {F}$ with its image in the category of Kan complexes $\operatorname{Kan}$. Applying Variant 7.5.3.6, we can choose a levelwise homotopy equivalence $\alpha : \mathscr {F} \rightarrow \mathscr {G}$, where $\mathscr {G}: (\operatorname{\mathbf{Z}}, \leq ) \rightarrow \operatorname{Kan}$ is an isofibrant diagram of Kan complexes. Note that we can also regard $\mathscr {G}$ as a diagram of pointed Kan complexes, by equipping each $\mathscr {G}(n)$ with the base point $y_{n} = \alpha (n)(x)$. Let us extend $\mathscr {G}$ to a functor $\mathscr {G}^{\pm }: ( \operatorname{\mathbf{Z}}\cup \{ -\infty , \infty \} , \leq ) \rightarrow \operatorname{Kan}_{\ast }$ by setting $\mathscr {G}^{\pm }(-\infty ) = \varprojlim ( \mathscr {G} )$ and $\mathscr {G}^{\pm }(\infty ) = \varinjlim ( \mathscr {G} )$, where the limit and colimit are formed in the category of (pointed) simplicial sets; we denote the base points of $\mathscr {G}^{\pm }(-\infty )$ and $\mathscr {G}^{\pm }(\infty )$ by $y_{-\infty }$ and $y_{\infty }$, respectively. Passing to nerves, the functor $\mathscr {G}^{\pm }$ determines a diagram $S: \{ - \infty \} \star \operatorname{Spine}[\operatorname{\mathbf{Z}}] \star \{ \infty \} \rightarrow \operatorname{\mathcal{S}}_{\ast }$.

Let $U: \operatorname{\mathcal{S}}_{\ast } \rightarrow \operatorname{\mathcal{S}}$ be the forgetful functor (given on objects by $U(X,x) = X$). Since the diagram $\mathscr {G}$ is isofibrant and the inclusion $\operatorname{Spine}[\operatorname{\mathbf{Z}}] \hookrightarrow \operatorname{N}_{\bullet }(\operatorname{\mathbf{Z}})$ is left cofinal (Remark 8.5.4.14), the restriction $(U \circ S)|_{ \{ - \infty \} \star \operatorname{Spine}[\operatorname{\mathbf{Z}}] }$ is a limit diagram in the $\infty $-category $\operatorname{\mathcal{S}}$ (Corollary 7.5.4.7). Applying Corollary 7.1.3.20, we see that $S|_{ \{ - \infty \} \star \operatorname{Spine}[\operatorname{\mathbf{Z}}] }$ is a limit diagram in the $\infty $-category $\operatorname{\mathcal{S}}_{\ast }$. Since the $\infty $-category $\operatorname{N}_{\bullet }(\operatorname{\mathbf{Z}})$ is filtered and the inclusion $\operatorname{Spine}[\operatorname{\mathbf{Z}}] \hookrightarrow \operatorname{N}_{\bullet }(\operatorname{\mathbf{Z}})$ is right cofinal, the restriction $(U \circ S)|_{ \operatorname{Spine}[\operatorname{\mathbf{Z}}] \star \{ \infty \} }$ is a colimit diagram in the $\infty $-category $\operatorname{\mathcal{S}}$ (Corollary 7.5.9.3). Since the spine $\operatorname{Spine}[\operatorname{\mathbf{Z}}]$ is weakly contractible (Remark 8.5.4.15), it follows that $S|_{ \operatorname{Spine}[\operatorname{\mathbf{Z}}] \star \{ \infty \} }$ is a colimit diagram in the $\infty $-category $\operatorname{\mathcal{S}}_{\ast }$. Moreover, the natural transformation $\alpha $ induces an isomorphism $T_{e} \rightarrow S|_{ \operatorname{Spine}[\operatorname{\mathbf{Z}}] }$ in the $\infty $-category $\operatorname{Fun}( \operatorname{Spine}[\operatorname{\mathbf{Z}}], \operatorname{\mathcal{S}}_{\ast } )$. It follows that the morphism $e$ is idempotent if and only if the composition

is an isomorphism in $\operatorname{\mathcal{S}}_{\ast }$: that is, if and only if the map of Kan complexes

is a homotopy equivalence of (pointed) Kan complexes (Corollary 8.5.7.6).

Since each $\mathscr {G}(n)$ is a connected Kan complex, it follows that the colimit $\varinjlim ( \mathscr {G} )$ is also connected. By virtue of Theorem 3.2.7.1, it will suffice to show that, for every integer $d \geq 0$, $\theta $ induces a bijection $\pi _{d}( \varprojlim (\mathscr {G}), y_{-\infty } ) \rightarrow \pi _{d}( \varinjlim ( \mathscr {G}), y_{\infty } )$ (note that, in the case $d = 0$, this guarantees that the Kan complex $\varprojlim (\mathscr {G} )$ is also connected, so that a similar conclusion holds for any choice of base point). Let $\overleftarrow {G}$ denote the diagram of sets

Note that $\alpha $ determines an isomorphism of $\overleftarrow {G}$ with the diagram

where each of the transition maps is induced by $e$. Since $e$ is homotopic to $e \circ e$ (in the homotopy category of pointed Kan complexes), it follows that $f_ d = f_ d \circ f_ d$, so that the tautological map $v: \varprojlim ( \overleftarrow {G} ) \rightarrow \varinjlim ( \overleftarrow {G} )$ is a bijection. Unwinding the definition, we see that $\pi _{d}(\theta )$ factors as a composition

where the map $w$ is also bijective (Remark 3.2.2.16). It will therefore suffice to show that the map $u$ is bijective. By virtue of the Milnor exact sequence (Proposition ), this is equivalent to the assertion that the set

has a single element. This is a special case of Proposition , since the inverse system of groups

is Mittag-Leffler (since $f_{d+1}$ is idempotent, its image coincides with the image of $f_{d+1}^{n}$ for every integer $n > 0$). $\square$