Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

7.2.6 Left Fibrations over Filtered $\infty $-Categories

Our goal in this section is to prove the following:

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 Example 5.2.2.6).

$(4)$

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 5.2.2.1. 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). $\square$

Corollary 7.2.6.2. Suppose we are given a pullback diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}' \ar [r] \ar [d]^{U'} & \operatorname{\mathcal{E}}\ar [d]^{U} \\ \operatorname{\mathcal{C}}' \ar [r]^-{V} & \operatorname{\mathcal{C}}, } \]

where $U$ and $V$ are left fibrations. If $\operatorname{\mathcal{C}}$, $\operatorname{\mathcal{C}}'$, and $\operatorname{\mathcal{E}}$ are filtered $\infty $-categories, then $\operatorname{\mathcal{E}}'$ is also a filtered $\infty $-category.

Proof. Since $U': \operatorname{\mathcal{E}}' \rightarrow \operatorname{\mathcal{C}}'$ is a pullback of $U$, it is a left fibration. It will therefore suffice to show that $U'$ satisfies condition $(4)$ of Theorem 7.2.6.1. Suppose we are given an object $X' \in \operatorname{\mathcal{C}}'$ and a morphism of simplicial sets $e: \operatorname{\partial \Delta }^{n} \rightarrow \operatorname{\mathcal{E}}'_{X'} = \{ X' \} \times _{\operatorname{\mathcal{C}}'} \operatorname{\mathcal{E}}'$. Set $X = V(X')$, so that we can identify $e$ with a morphism from $\operatorname{\partial \Delta }^{n}$ to the fiber $\operatorname{\mathcal{E}}_{X} = \{ X\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$. Since $\operatorname{\mathcal{E}}$ and $\operatorname{\mathcal{C}}$ are filtered, Theorem 7.2.6.1 guarantees that we can choose a morphism $f: X \rightarrow Y$ in $\operatorname{\mathcal{C}}$ for which the composite map $\operatorname{\partial \Delta }^ n \xrightarrow {e} \operatorname{\mathcal{E}}_{X} \xrightarrow { f_{!} } \operatorname{\mathcal{E}}_{Y}$, where $f_{!}$ is given by covariant transport along $f$. Since $V$ is a left fibration, we can write $f = V(f')$ for some morphism $f': X' \rightarrow Y'$ in the $\infty $-category $\operatorname{\mathcal{C}}'$. Under the canonical isomorphisms $\operatorname{\mathcal{E}}'_{X'} \simeq \operatorname{\mathcal{E}}_{X}$ and $\operatorname{\mathcal{E}}'_{Y'} \simeq \operatorname{\mathcal{E}}_{Y}$, the morphism $f_{!}: \operatorname{\mathcal{E}}_{X} \rightarrow \operatorname{\mathcal{E}}_{Y}$ corresponds to a functor $f'_{!}: \operatorname{\mathcal{E}}'_{X'} \rightarrow \operatorname{\mathcal{E}}'_{Y'}$ given by covariant transport along $f'$ (Remark 5.2.2.4 ), so that the composition $(f'_{!} \circ e): \operatorname{\partial \Delta }^{n} \rightarrow \operatorname{\mathcal{E}}'_{Y'}$ is also nullhomotopic. $\square$

Using Corollary 7.2.6.2, we obtain another characterization of the class of filtered $\infty $-categories:

Corollary 7.2.6.3. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then $\operatorname{\mathcal{C}}$ is filtered if and only if it satisfies the following pair of conditions:

$(a)$

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

$(b)$

Let $U: \widetilde{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{C}}$, $V_0: \widetilde{\operatorname{\mathcal{C}}}_{0} \rightarrow \widetilde{\operatorname{\mathcal{C}}}$, and $V_{1}: \widetilde{\operatorname{\mathcal{C}}}_{1} \rightarrow \widetilde{\operatorname{\mathcal{C}}}$ be left fibrations of $\infty $-categories. If $\widetilde{\operatorname{\mathcal{C}}}$, $\widetilde{\operatorname{\mathcal{C}}}_0$, and $\widetilde{\operatorname{\mathcal{C}}}_{1}$ are weakly contractible, then the fiber product $\widetilde{\operatorname{\mathcal{C}}}_0 \times _{ \widetilde{\operatorname{\mathcal{C}}} } \widetilde{\operatorname{\mathcal{C}}}_{1}$ is also weakly contractible.

Proof. Suppose first that $\operatorname{\mathcal{C}}$ is filtered. Assertion $(a)$ follows from Proposition 7.2.4.9. To prove $(b)$, suppose we are given left fibrations $U: \widetilde{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{C}}$, $V_0: \widetilde{\operatorname{\mathcal{C}}}_{0} \rightarrow \widetilde{\operatorname{\mathcal{C}}}$, and $V_{1}: \widetilde{\operatorname{\mathcal{C}}}_{1} \rightarrow \widetilde{\operatorname{\mathcal{C}}}$, where $\widetilde{\operatorname{\mathcal{C}}}$, $\widetilde{\operatorname{\mathcal{C}}}_{0}$, and $\widetilde{\operatorname{\mathcal{C}}}_{1}$ are weakly contractible. Applying Theorem 7.2.6.1, we deduce that the $\infty $-categories $\widetilde{\operatorname{\mathcal{C}}}$, $\widetilde{\operatorname{\mathcal{C}}}_{0}$, and $\widetilde{\operatorname{\mathcal{C}}}_{1}$ are filtered. Applying Corollary 7.2.6.2 to the diagram of left fibrations

\[ \xymatrix@R =50pt@C=50pt{ \widetilde{\operatorname{\mathcal{C}}}_{0} \times _{ \widetilde{\operatorname{\mathcal{C}}} } \widetilde{\operatorname{\mathcal{C}}}_{1} \ar [r] \ar [d] & \widetilde{\operatorname{\mathcal{C}}}_{0} \ar [d]^{V_0} \\ \widetilde{\operatorname{\mathcal{C}}}_{1} \ar [r]^-{V_1} & \widetilde{\operatorname{\mathcal{C}}}, } \]

we conclude that the fiber product $\widetilde{\operatorname{\mathcal{C}}}_{0} \times _{ \widetilde{\operatorname{\mathcal{C}}}} \widetilde{\operatorname{\mathcal{C}}}_{1}$ is also filtered; in particular, it is weakly contractible (Proposition 7.2.4.9).

We now prove the converse. Assume that $\operatorname{\mathcal{C}}$ satisfies conditions $(a)$ and $(b)$; we wish to show that $\operatorname{\mathcal{C}}$ is filtered. We will prove this using the criterion of Lemma 7.2.5.13. Fix an integer $n \geq 0$ and a diagram $e: \operatorname{\partial \Delta }^ n \rightarrow \operatorname{\mathcal{C}}$; we wish to show that the coslice $\infty $-category $\operatorname{\mathcal{C}}_{e/}$ is nonempty. In fact, we will prove the following stronger assertion: for every simplicial subset $K \subseteq \operatorname{\partial \Delta }^{n}$, the coslice $\infty $-category $\operatorname{\mathcal{C}}_{ e_ K / }$ is weakly contractible, where $e_{K}$ denotes the restriction $e|_{K}$. Our proof proceeds by induction on the number of nondegenerate simplices of $K$. If $K = \emptyset $, then the desired result follows from assumption $(a)$. If $K$ is not isomorphic to a standard simplex, then we can use Proposition 1.1.3.13 to write $K$ as a union $K(0) \cup K(1)$, where $K(0), K(1) \subsetneq K$ are proper simplicial subsets. Setting $K(01) = K(0) \cap K(1)$, we have a pullback diagram of left fibrations

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{ e_ K / } \ar [r] \ar [d] & \operatorname{\mathcal{C}}_{ e_{K(0)} / } \ar [d] \\ \operatorname{\mathcal{C}}_{ e_{ K(1) } / } \ar [r] & \operatorname{\mathcal{C}}_{ e_{ K(01) } / }, } \]

where the $\infty $-categories $\operatorname{\mathcal{C}}_{ e_{K(0)} / }$, $\operatorname{\mathcal{C}}_{ e_{K(1)} / }$, and $\operatorname{\mathcal{C}}_{ e_{K(01)} / }$ are weakly contractible by virtue of our inductive hypothesis. Applying $(b)$, we deduce that $\operatorname{\mathcal{C}}_{ e_ K / }$ is weakly contractible. We may therefore assume without loss of generality that $K \simeq \Delta ^{m}$ is a standard simplex. In particular, $K$ contains a final vertex $v$ for which the inclusion $\{ v\} \hookrightarrow K$ is right anodyne (Example 4.3.7.11), so that the restriction map $\operatorname{\mathcal{C}}_{ e_ K / } \rightarrow \operatorname{\mathcal{C}}_{ e(v) / }$ is a trivial Kan fibration (Corollary 4.3.6.11). It will therefore suffice to show that the $\infty $-category $\operatorname{\mathcal{C}}_{ e(v) / }$ is weakly contractible. This follows from Corollary 7.2.3.5, since the $\infty $-category $\operatorname{\mathcal{C}}_{e(v) / }$ has an initial object (Proposition 7.1.2.8). $\square$