Proposition 8.5.6.19 (041U)

Proposition 8.5.6.19. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then the restriction functor

\[ \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Ret}), \operatorname{\mathcal{C}}) \rightarrow \operatorname{End}_{\operatorname{\mathcal{C}}}^{w} \]

admits a left homotopy inverse.

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. $\square$