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Theorem 7.2.6.1. Let $U: \widetilde{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{C}}$ be a left fibration of $\infty $-categories, where the $\infty $-category $\operatorname{\mathcal{C}}$ is filtered. For each object $X \in \operatorname{\mathcal{C}}$, let $\widetilde{\operatorname{\mathcal{C}}}_{X}$ denote the fiber $\{ X\} \times _{\operatorname{\mathcal{C}}} \widetilde{\operatorname{\mathcal{C}}}$. The following conditions are equivalent:

$(1)$

The $\infty $-category $\widetilde{\operatorname{\mathcal{C}}}$ is filtered.

$(2)$

The $\infty $-category $\widetilde{\operatorname{\mathcal{C}}}$ is weakly contractible.

$(3)$

For every object $X \in \operatorname{\mathcal{C}}$ and every diagram $e: K \rightarrow \widetilde{\operatorname{\mathcal{C}}}_{X}$ where $K$ is a finite simplicial set, there exists a morphism $f: X \rightarrow Y$ in $\operatorname{\mathcal{C}}$ for which the composite map $K \xrightarrow {e} \widetilde{\operatorname{\mathcal{C}}}_{X} \xrightarrow { f_{!} } \widetilde{\operatorname{\mathcal{C}}}_{Y}$ is nullhomotopic; here $f_{!}: \widetilde{\operatorname{\mathcal{C}}}_{X} \rightarrow \widetilde{\operatorname{\mathcal{C}}}_{Y}$ is given by covariant transport along $f$ (see Notation 5.2.2.9).

$(4)$

For every object $X \in \operatorname{\mathcal{C}}$, every integer $n \geq 0$, and every diagram $e: \operatorname{\partial \Delta }^{n} \rightarrow \widetilde{\operatorname{\mathcal{C}}}_{X}$, there exists a morphism $f: X \rightarrow Y$ in $\operatorname{\mathcal{C}}$ for which the composite map $\operatorname{\partial \Delta }^{n} \xrightarrow {e} \widetilde{\operatorname{\mathcal{C}}}_{X} \xrightarrow { f_{!} } \widetilde{\operatorname{\mathcal{C}}}_{Y}$ is nullhomotopic.

Proof. The implication $(1) \Rightarrow (2)$ follows from Proposition 7.2.4.9 and the implication $(3) \Rightarrow (4)$ is immediate. We next show that $(4)$ implies $(1)$. Assume that condition $(4)$ is satisfied; we wish to prove that $\widetilde{\operatorname{\mathcal{C}}}$ is filtered. By virtue of Lemma 7.2.5.13 (and Remark 7.2.5.14), it will suffice to show that for every integer $n \geq 0$ and every diagram $e: \operatorname{\partial \Delta }^{n} \rightarrow \widetilde{\operatorname{\mathcal{C}}}$, there exists a natural transformation from $e$ to a constant diagram. Set $\overline{e} = U \circ e$, which we regard as an object of the $\infty $-category $\operatorname{Fun}( \operatorname{\partial \Delta }^ n, \operatorname{\mathcal{C}})$. Since $\operatorname{\mathcal{C}}$ is filtered, there exists an object $X \in \operatorname{\mathcal{C}}$ and a morphism $\overline{\alpha }: \overline{e} \rightarrow \underline{X}$ in the $\infty $-category $\operatorname{Fun}( \operatorname{\partial \Delta }^ n, \operatorname{\mathcal{C}})$, where $\underline{X}: \operatorname{\partial \Delta }^ n \rightarrow \operatorname{\mathcal{C}}$ denotes the constant morphism taking the value $X$. Since $U$ is a left fibration, we can lift $\overline{\alpha }$ to a morphism $\alpha : e \rightarrow e'$ in $\operatorname{Fun}( \operatorname{\partial \Delta }^ n, \widetilde{\operatorname{\mathcal{C}}} )$, where $e'$ is a morphism from $\operatorname{\partial \Delta }^ n$ to the Kan complex $\widetilde{\operatorname{\mathcal{C}}}_{X}$ (see Remark 4.2.6.3). Invoking assumption $(4)$, we can choose a morphism $f: X \rightarrow Y$ in $\operatorname{\mathcal{C}}$ and a covariant transport functor $f_{!}: \widetilde{\operatorname{\mathcal{C}}}_{X} \rightarrow \widetilde{\operatorname{\mathcal{C}}}_{Y}$ for which the composite map $f_{!} \circ u'$ is nullhomotopic. It follows that there exists a natural transformation $\beta : e' \rightarrow e''$ in $\operatorname{Fun}( \operatorname{\partial \Delta }^ n, \widetilde{\operatorname{\mathcal{C}}} )$, where $e'': \operatorname{\partial \Delta }^ n \rightarrow \widetilde{\operatorname{\mathcal{C}}}_{Y}$ is a constant map. Any choice of composition of $\alpha $ and $\beta $ then determines a natural transformation from $e$ to the constant diagram $e''$.

We now complete the proof by showing that $(2)$ implies $(3)$. Assume that the $\infty $-category $\widetilde{\operatorname{\mathcal{C}}}$ is weakly contractible, and suppose that we are given an object $X \in \operatorname{\mathcal{C}}$ and a diagram $e: K \rightarrow \widetilde{\operatorname{\mathcal{C}}}_{X}$, where the simplicial set $K$ is finite. We wish to show that there exists a morphism $f: X \rightarrow Y$ in $\operatorname{\mathcal{C}}$ for which the composite map $K \xrightarrow {e} \widetilde{\operatorname{\mathcal{C}}}_{X} \xrightarrow {f_{!} } \widetilde{\operatorname{\mathcal{C}}}_{Y}$ is nullhomotopic. Choose an embedding $K \hookrightarrow L$, where $L$ is another finite simplicial set which is weakly contractible (for example, we can take $L = K^{\triangleright }$). Let $\operatorname{Ex}^{\infty }( \widetilde{\operatorname{\mathcal{C}}} )$ be the simplicial set given by Construction 3.3.6.1, so that $\operatorname{Ex}^{\infty }( \widetilde{\operatorname{\mathcal{C}}} )$ is a Kan complex (Proposition 3.3.6.9). Let $\rho ^{\infty }: \widetilde{\operatorname{\mathcal{C}}} \rightarrow \operatorname{Ex}^{\infty }( \widetilde{\operatorname{\mathcal{C}}} )$ be the weak homotopy equivalence of Proposition 3.3.6.7. Since $\widetilde{\operatorname{\mathcal{C}}}$ is weakly contractible, the Kan complex $\operatorname{Ex}^{\infty }( \widetilde{\operatorname{\mathcal{C}}} )$ is contractible. It follows that the composite map $K \xrightarrow {e} \widetilde{\operatorname{\mathcal{C}}}_{X} \xrightarrow { \rho ^{\infty } } \operatorname{Ex}^{\infty }( \widetilde{\operatorname{\mathcal{C}}} )$ can be extended to a map $e^{+}: L \rightarrow \operatorname{Ex}^{\infty }( \widetilde{\operatorname{\mathcal{C}}} )$. Since the simplicial set $L$ is finite, the morphism $\overline{e}$ factors through $\operatorname{Ex}^{m}( \widetilde{\operatorname{\mathcal{C}}} )$ for some $m \gg 0$ (see Proposition 3.5.1.9). By virtue of Proposition 3.3.4.8, we can replace $K$ and $L$ by the iterated subdivisions subdivision $\operatorname{Sd}^{m}(K)$ and $\operatorname{Sd}^{m}(L)$ (and $e$ by the composite map $\operatorname{Sd}^{m}(K) \twoheadrightarrow K \xrightarrow {e} \widetilde{\operatorname{\mathcal{C}}}_{X}$) and thereby reduce to the case $m=0$, so that $e$ admits an extension $e^{+}: L \rightarrow \widetilde{\operatorname{\mathcal{C}}}$.

Set $\overline{e}^{+} = U \circ e^{+}$, which we regard as an object of the $\infty $-category $\operatorname{Fun}(L, \operatorname{\mathcal{C}})$. Since $\operatorname{\mathcal{C}}$ is filtered, there exists an object $Y \in \operatorname{\mathcal{C}}$ and a natural transformation $\overline{\alpha }: \overline{e}^{+} \rightarrow \underline{Y}$, where $\underline{Y} \in \operatorname{Fun}(L, \operatorname{\mathcal{C}})$ denotes the constant diagram taking the value $Y$ (Remark 7.2.4.7). Let $\overline{\alpha }_0$ denote the image of $\overline{\alpha }$ in $\operatorname{Fun}(K, \operatorname{\mathcal{C}})$. Then $\overline{\alpha }_{0}$ can be identified with a morphism from $K$ to the morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$. Since $\operatorname{\mathcal{C}}$ is filtered, Theorem 7.2.5.5 guarantees the existence of a morphism $g: Y \rightarrow Z$ of $\operatorname{\mathcal{C}}$ for which the composite map

\[ K \xrightarrow { \overline{\alpha }_0 } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,U) \xrightarrow { g \circ } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z) \]

is nullhomotopic. Let $\underline{Z}: L \rightarrow \operatorname{\mathcal{C}}$ denote the constant diagram taking the value $Z$, so that $g$ determines a morphism $\underline{g}: \underline{Y} \rightarrow \underline{Z}$ in the $\infty $-category $\operatorname{Fun}(L,\operatorname{\mathcal{C}})$. Replacing $Y$ by $Z$ and $\overline{\alpha }$ by its composition with $\underline{g}$, we can reduce to the case where the morphism $\overline{\alpha }_0: K \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ is nullhomotopic. Note that the restriction map $\operatorname{Fun}(L, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}(K, \operatorname{\mathcal{C}})$ is an isofibration of $\infty $-categories (Corollary 4.4.5.3), and therefore induces a Kan fibration of morphism spaces $\operatorname{Hom}_{ \operatorname{Fun}(L,\operatorname{\mathcal{C}})}( \overline{e}^{+}, \underline{Y} ) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}(K,\operatorname{\mathcal{C}})}( \overline{e}^{+}|_ K, \underline{Y}_ K )$ (Exercise 4.6.1.21). We may therefore modify $\overline{\alpha }$ by a homotopy and thereby reduce to the case where $\overline{\alpha }_0: K \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ is the constant map taking some value $f \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$. Since $U$ is a left fibration, we can lift $\overline{\alpha }$ to a natural transformation $\alpha : e^{+} \rightarrow e'^{+}$, for some diagram $e'^{+}: L \rightarrow \widetilde{\operatorname{\mathcal{C}}}_{Y} \subseteq \widetilde{\operatorname{\mathcal{C}}}$. Set $e' = e'^{+}|_{K}$, so that $\alpha $ restricts to a natural transformation $\alpha _0: e \rightarrow e'$ which witnesses $e'$ as given by covariant transport along $f$, in the sense of Definition 5.2.2.4. To complete the proof, it will suffice to show that the morphism $e': K \rightarrow \widetilde{\operatorname{\mathcal{C}}}_{Y}$ is nullhomotopic. This is clear: already the morphism $e'^{+}: L \rightarrow \widetilde{\operatorname{\mathcal{C}}}_{Y}$ is nullhomotopic, since $L$ is weakly contractible and $\widetilde{\operatorname{\mathcal{C}}}_ Y$ is a Kan complex (see Variant 3.2.6.10). $\square$