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7.2.5 Local Characterization of Filtered $\infty $-Categories

Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ denote the homotopy category of $\operatorname{\mathcal{C}}$, which we view as enriched over the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$ of Kan complexes (see Construction 4.6.7.13). In this section, we show that the condition that $\operatorname{\mathcal{C}}$ is filtered can be formulated entirely in terms of $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$, together with its $\mathrm{h} \mathit{\operatorname{Kan}}$-enrichment (Theorem 7.2.5.5).

Definition 7.2.5.1. Let $\mathrm{h} \mathit{\operatorname{Kan}}$ denote the homotopy category of Kan complexes (Construction 7.2.1.20), and let $\operatorname{\mathcal{C}}$ be a category which is enriched over $\mathrm{h} \mathit{\operatorname{Kan}}$. We will say that $\operatorname{\mathcal{C}}$ is homotopy filtered if it is nonempty and satisfies the following condition for each $n \geq 1$:

$(\ast _ n)$

For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$ and for every morphism of simplicial sets $\sigma : \operatorname{\partial \Delta }^{n-1} \rightarrow \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)$, there exists morphism $v: Y \rightarrow Z$ for which the composite morphism

\[ \operatorname{\partial \Delta }^{n-1} \xrightarrow { \sigma } \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}( X, Y) \xrightarrow { v \circ } \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Z) \]

is nullhomotopic.

Warning 7.2.5.2. In the formulation of condition $(\ast _ n)$ of Definition 7.2.5.1, postcomposition with $v$ defines a map of Kan complexes $V: \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}( X, Y) \rightarrow \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Z)$ which is only well-defined up to homotopy. However, the condition that $V \circ \sigma $ is nullhomotopic depends only on the homotopy class of $V$.

Example 7.2.5.3. Let $\operatorname{\mathcal{C}}$ be an ordinary category, which we regard as an $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched category in which each of the Kan complexes $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)$ is equal to $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ (regarded as a constant simplicial set). In this case, condition $(\ast _ n)$ of Definition 7.2.5.1 is automatically satisfied for $n \geq 3$. Moreover, we can state conditions $(\ast _1)$ and $(\ast _2)$ more concretely as follows:

$(\ast _1)$

For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, there exists an object $Z \in \operatorname{\mathcal{C}}$ equipped with morphisms $u: X \rightarrow Z$ and $v: Y \rightarrow Z$.

$(\ast _2)$

For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$ and every pair of morphisms $f_0, f_1: X \rightarrow Y$, there exists a morphism $v: Y \rightarrow Z$ satisfying $v \circ f_0 = v \circ f_1$.

It follows that $\operatorname{\mathcal{C}}$ is homotopy filtered (in the sense of Definition 7.2.5.1) if and only if is filtered (in the sense of Definition 7.2.4.1).

Remark 7.2.5.4. Let $\operatorname{\mathcal{C}}$ be an $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched category. If $\operatorname{\mathcal{C}}$ is homotopy filtered (in the sense of Definition 7.2.5.1), then it is filtered when regarded as an ordinary category (in the sense of Definition 7.2.4.1). Beware that the converse is false in general (see Warning 7.2.5.7).

We can now state the main result of this section:

Theorem 7.2.5.5. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then $\operatorname{\mathcal{C}}$ is filtered (in the sense of Definition 7.2.4.3) if and only if the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ is homotopy filtered (in the sense of Definition 7.2.5.1), when regarded as an $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched category by means of Construction 4.6.7.13.

Before giving the proof of Theorem 7.2.5.5, let us note some of its consequences.

Corollary 7.2.5.6. Let $\operatorname{\mathcal{C}}$ be a filtered $\infty $-category (in the sense of Definition 7.2.4.3). Then $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ is a filtered category (in the sense of Definition 7.2.4.1).

Warning 7.2.5.7. The converse of Corollary 7.2.5.6 is false. For example, if $\operatorname{\mathcal{C}}$ is a simply connected Kan complex, then the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ is automatically filtered. However, $\operatorname{\mathcal{C}}$ is filtered if and only if it is contractible (Corollary 7.2.4.12).

Corollary 7.2.5.8. Let $\operatorname{\mathcal{C}}$ be a category. Then the category $\operatorname{\mathcal{C}}$ is filtered (in the sense of Definition 7.2.4.1) if and only if the $\infty $-category $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is filtered (in the sense of Definition 7.2.4.3).

Example 7.2.5.9. Let $(A, \leq )$ be a partially ordered set. Combining Exercise 7.2.4.2 with Corollary 7.2.5.8, we see that the $\infty $-category $\operatorname{N}_{\bullet }(A)$ is filtered if and only if the partially ordered set $(A, \leq )$ is directed.

Corollary 7.2.5.10. Let $\operatorname{\mathcal{C}}$ be a locally Kan simplicial category. Then the $\infty $-category $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ is filtered if and only if the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ is homotopy filtered, when regarded as an $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched category.

Exercise 7.2.5.11. Let $\operatorname{\mathcal{C}}$ be a $(2,1)$-category (Definition 2.2.8.5). Show that the Duskin nerve $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ is a filtered $\infty $-category if and only if $\operatorname{\mathcal{C}}$ satisfies the following conditions:

  • The $2$-category $\operatorname{\mathcal{C}}$ is nonempty.

  • For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, there exists an object $Z \in \operatorname{\mathcal{C}}$ and a pair of $1$-morphisms $f: X \rightarrow Z$ and $g: Y \rightarrow Z$.

  • For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$ and every pair of $1$-morphisms $f,g: X \rightarrow Y$, there exists a $1$-morphism $h: Y \rightarrow Z$ such that the $1$-morphisms $h \circ f$ and $h \circ g$ are isomorphic (when viewed as objects of the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Z)$).

  • For every $1$-morphism $f: X \rightarrow Y$ in $\operatorname{\mathcal{C}}$ and every $2$-morphism $\gamma : f \Rightarrow f$, there exists a $1$-morphism $g: Y \rightarrow Z$ for which the horizontal composition $\operatorname{id}_{g} \circ \gamma $ is equal to the identity $2$-morphism $\operatorname{id}_{g \circ f}$.

We now turn to the proof of Theorem 7.2.5.5. The easy part is to show that if $\operatorname{\mathcal{C}}$ is a filtered $\infty $-category, then the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ is homotopy filtered. Condition $(\ast _ n)$ of Definition 7.2.5.1 is a special case of the following assertion:

Lemma 7.2.5.12. Let $\operatorname{\mathcal{C}}$ be a filtered $\infty $-category containing objects $X$ and $Y$, and let $K$ be a finite simplicial set equipped with a morphism $f: K \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$. Then there exists a morphism $v: Y \rightarrow Z$ of $\operatorname{\mathcal{C}}$ for which the composition $K \xrightarrow {f} \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \xrightarrow { v \circ } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)$ is nullhomotopic.

Proof. Let $\Sigma (K)$ denote the iterated coproduct

\[ \{ x\} \coprod _{ ( \{ 0\} \times K ) } (\Delta ^1 \times K ) \coprod _{ ( \{ 1\} \times K ) } \{ y\} , \]

so that we can identify $f$ with a morphism of simplicial sets $F: \Sigma (K) \rightarrow \operatorname{\mathcal{C}}$ satisfying $F(x) = X$ and $F(y) = Y$. Our assumption that $\operatorname{\mathcal{C}}$ is filtered guarantees that we can extend $F$ to a morphism of simplicial set $\overline{F}: \Sigma (K) \star \{ z\} \rightarrow \operatorname{\mathcal{C}}$. Set $Z = \overline{F}(z)$. Then $\overline{F}$ carries $\{ x\} \star \{ z\} $ and $\{ y\} \star \{ z\} $ to morphisms $u: X \rightarrow Z$ and $v: Y \rightarrow Z$ in $\operatorname{\mathcal{C}}$. Moreover, the natural map $\Delta ^1 \times K \rightarrow \Sigma (K)$ admits a unique extension $q: \Delta ^2 \times K \rightarrow \Sigma (K) \star \{ z\} $ carrying $\{ 2\} \times K$ to the vertex $z$, and the composition

\[ \Delta ^2 \times K \xrightarrow {q} \Sigma (K) \star \{ z\} \xrightarrow { \overline{F} } \operatorname{\mathcal{C}} \]

determines a morphism of simplicial sets $g: K \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y,Z)$. Unwinding the definitions, we see that the diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ K \ar [dr]^{g} \ar [r]^-{(v,f)} \ar [dd] & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \\ & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y,Z) \ar [u] \ar [d] \\ \Delta ^{0} \ar [r]^-{u} & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z ) } \]

is strictly commutative, from which we immediately deduce (from the definition of the composition law on $\operatorname{\mathcal{C}}$) that the composition $K \xrightarrow {f} \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow {v \circ } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)$ is homotopic to the constant map taking the value $u$. $\square$

The difficult half of Theorem 7.2.5.5 will require some further preliminaries. We first note that, to verify that an $\infty $-category $\operatorname{\mathcal{C}}$ is filtered, it suffices to verify the extension condition of Definition 7.2.4.3 in the special case where $K = \operatorname{\partial \Delta }^ n$ is the boundary of a simplex.

Lemma 7.2.5.13. An $\infty $-category $\operatorname{\mathcal{C}}$ is filtered if and only if it satisfies the following condition for every integer $n \geq 0$:

$(\ast '_ n)$

Every morphism of simplicial sets $\operatorname{\partial \Delta }^ n \rightarrow \operatorname{\mathcal{C}}$ can be extended to a morphism $( \operatorname{\partial \Delta }^ n )^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$.

Proof. The necessity of condition $(\ast '_ n)$ is clear. For the converse, suppose that $\operatorname{\mathcal{C}}$ satisfies $(\ast '_ n)$ for each $n \geq 0$. We wish to prove that $\operatorname{\mathcal{C}}$ is filtered. Let $f: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram where $K$ is a finite simplicial set; we wish to show that the $\infty $-category $\operatorname{\mathcal{C}}_{f/}$ is nonempty. If $K = \emptyset $, then this follows immediately from assumption $(\ast '_{0} )$. Otherwise, the simplicial set $K$ has dimension $m$ for some integer $m \geq 0$. We proceed by induction on $m$ and on the number of nondegenerate $m$-simplices of $K$. Choose a nondegenerate $m$-simplex $\sigma : \Delta ^ m \rightarrow K$. Using Proposition 1.1.3.13, we can choose a pushout diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\partial \Delta }^{m} \ar [r] \ar [d] & \Delta ^{m} \ar [d]^{\sigma } \\ K' \ar [r] & K } \]

where $K' \subseteq K$ is a simplicial subset having a smaller number of nondegenerate $m$-simplices. Set $f' = f|_{K'}$, $f_{0} = f \circ \sigma $, and $f'_0 = f \circ \sigma |_{ \operatorname{\partial \Delta }^ m}$, so that we have a pullback diagram of $\infty $-categories

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{f/} \ar [r] \ar [d] & \operatorname{\mathcal{C}}_{ f' / } \ar [d]^{\Phi } \\ \operatorname{\mathcal{C}}_{ f_0 / } \ar [r]^-{\Psi } & \operatorname{\mathcal{C}}_{ f'_0 / }. } \]

Applying our inductive hypothesis, we deduce that the $\infty $-category $\operatorname{\mathcal{C}}_{f'/}$ is nonempty. Choose an object $X$ of $\operatorname{\mathcal{C}}_{f'/}$, so that $\Phi (X) \in \operatorname{\mathcal{C}}_{ f_{\operatorname{\partial \Delta }} / }$ can be identified with a morphism of simplicial sets $g: (\operatorname{\partial \Delta }^ m)^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$. Amalgamating $f \circ \sigma $ with $g$, we obtain a morphism of simplicial sets

\[ \overline{g}: \operatorname{\partial \Delta }^{m+1} \simeq ( \operatorname{\partial \Delta }^{m} )^{\triangleright } \coprod _{ \operatorname{\partial \Delta }^{m} } \Delta ^ m \rightarrow \operatorname{\mathcal{C}}. \]

Invoking $(\ast _{m+1})$, we conclude that $\overline{g}$ can be extended to a morphism of simplicial sets $( \operatorname{\partial \Delta }^{m+1 } )^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$. Unwinding the definitions, we see that this extension supplies an object $Y \in \operatorname{\mathcal{C}}_{ f_0 / }$ together with a morphism $u: \Phi (X) \rightarrow \Psi (Y)$ in the $\infty $-category $\operatorname{\mathcal{C}}_{ f'_0 / }$.

Note that the projection maps $\operatorname{\mathcal{C}}_{f' / } \rightarrow \operatorname{\mathcal{C}}\leftarrow \operatorname{\mathcal{C}}_{ f'_0 / }$ are left fibrations (Proposition 4.3.6.1). Let $\overline{X}$ denote the image of $X$ in the $\infty $-category $\operatorname{\mathcal{C}}$, so that Corollary 4.3.7.13 guarantees that the vertical maps in the diagram

\[ \xymatrix@R =50pt@C=50pt{ (\operatorname{\mathcal{C}}_{f'/})_{X/} \ar [rr]^{ \Phi _{X/} } \ar [dr] & & (\operatorname{\mathcal{C}}_{ f'_0 / } )_{U(X) / } \ar [dl] \\ & \operatorname{\mathcal{C}}_{ \overline{X} / } & } \]

are trivial Kan fibrations. In particular, they are equivalences of $\infty $-categories, so that the functor $\Phi _{X/}$ is also an equivalence of $\infty $-categories. It follows that we can choose a morphism $w: X \rightarrow Z$ in the $\infty $-category $\operatorname{\mathcal{C}}_{f'/}$ and a $2$-simplex

\[ \xymatrix@R =50pt@C=50pt{ & \Psi (Y) \ar [dr]^{v} & \\ \Phi (X) \ar [ur]^{u} \ar [rr]^{ \Phi (w) } & & \Phi (Z) } \]

in the $\infty $-category $\operatorname{\mathcal{C}}_{ f'_0 / }$, where $v$ is an isomorphism. Since $\Psi $ is a left fibration (Corollary 4.3.6.11), we can lift $v$ to a morphism $\widetilde{v}: Y \rightarrow \widetilde{Z}$ of the $\infty $-category $\operatorname{\mathcal{C}}_{ f_0 / }$. The pair $(Z, \widetilde{Z})$ can then be regarded as an object of the $\infty $-category $\operatorname{\mathcal{C}}_{f/} = \operatorname{\mathcal{C}}_{ f' / } \times _{ \operatorname{\mathcal{C}}_{ f'_0 / } } \operatorname{\mathcal{C}}_{ f_0 / }$. $\square$

Remark 7.2.5.14. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $n \geq 0$ be a nonnegative integer. Condition $(\ast '_{n})$ of Lemma 7.2.5.13 is equivalent to the assertion that, for every morphism of simplicial sets $f: \operatorname{\partial \Delta }^{n} \rightarrow \operatorname{\mathcal{C}}$, the coslice $\infty $-category $\operatorname{\mathcal{C}}_{f/}$ is nonempty. By virtue of Theorem 4.6.4.16, this is equivalent to the requirement that the oriented fiber product $\{ f\} \operatorname{\vec{\times }}_{ \operatorname{Fun}( \operatorname{\partial \Delta }^ n, \operatorname{\mathcal{C}}) } \operatorname{\mathcal{C}}$ is nonempty. We can therefore reformulate $(\ast '_{n})$ as follows:

$(\ast '_ n)$

For every diagram $f: \operatorname{\partial \Delta }^{n} \rightarrow \operatorname{\mathcal{C}}$, there exists an object $C \in \operatorname{\mathcal{C}}$ and a natural transformation $f \rightarrow \underline{C}$, where $\underline{C}: \operatorname{\partial \Delta }^{n} \rightarrow \operatorname{\mathcal{C}}$ is the constant morphism taking the value $C$.

For each integer $n \geq 1$, let us identify the standard simplex $\Delta ^{n-1}$ with its image in $\operatorname{\partial \Delta }^{n} \subset \Delta ^ n$ (given by the face opposite the $n$th vertex).

Lemma 7.2.5.15. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $n \geq 1$ be an integer. Then condition $(\ast '_ n)$ of Lemma 7.2.5.13 is equivalent to the following:

$(\ast ''_ n)$

Let $f: \operatorname{\partial \Delta }^ n \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets for which the restriction $f|_{ \Delta ^{n-1} }$ is constant. Then $f$ can be extended to a morphism $\overline{f}: ( \operatorname{\partial \Delta }^{n} )^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$.

Proof. The implication $(\ast '_ n) \Rightarrow (\ast ''_{n} )$ is immediate. We will prove the converse. Assume that $(\ast ''_ n)$ is satisfied, and let $g: \operatorname{\partial \Delta }^ n \rightarrow \operatorname{\mathcal{C}}$ be an arbitrary morphism of simplicial sets; we wish to show that $g$ can be extended to a morphism $\overline{g}: ( \operatorname{\partial \Delta }^{n} ) \rightarrow \operatorname{\mathcal{C}}$. If $n=1$, this follows immediately form $(\ast ''_ n)$; we will therefore assume that $n \geq 2$. Note that we can write $\operatorname{\partial \Delta }^{n}$ as the union of $\Delta ^{n-1}$ and the horn $\Lambda ^{n}_{n}$, whose intersection is the simplicial subset $\operatorname{\partial \Delta }^{n-1} \subset \Delta ^{n-1}$. Set

\[ g_{-} = g|_{ \Delta ^{n-1} } \quad \quad g_{\pm } = g|_{ \operatorname{\partial \Delta }^{n-1} } \quad \quad g_{+} = g|_{ \Lambda ^{n}_{n} }. \]

Let $X = g(0)$ and $Y = g(n)$ and let $\pi : \operatorname{\mathcal{C}}_{/Y} \rightarrow \operatorname{\mathcal{C}}$ denote the projection map, so that we can identify $g_{+}$ with a morphism $\widetilde{g}_{\pm }: \operatorname{\partial \Delta }^{n-1} \rightarrow \operatorname{\mathcal{C}}_{/Y}$ satisfying $\pi \circ \widetilde{g}_{\pm } = g_{\pm }$.

Let $f_{-}: \Delta ^{n-1} \rightarrow \operatorname{\mathcal{C}}$ be the constant morphism taking the value $X$, and let $h_{-}: f_{-} \rightarrow g_{-}$ be the natural transformation given by the composite map

\[ \Delta ^{1} \times \Delta ^{n-1} \xrightarrow {(i,j) \mapsto ij} \Delta ^{n-1} \xrightarrow { g_{-} } \operatorname{\mathcal{C}}. \]

Set $f_{\pm } = f_{-} |_{ \operatorname{\partial \Delta }^{n-1} }$ and $h_{\pm } = h_{-} |_{ \Delta ^1 \times \operatorname{\partial \Delta }^{n-1} }$, so that $h_{\pm }$ can be regarded as a natural transformation from $f_{\pm }$ to $g_{\pm }$. Since $\pi $ is a right fibration, we can lift $h_{\pm }$ to a natural transformation $\widetilde{h}_{\pm }: \widetilde{f}_{\pm } \rightarrow \widetilde{g}_{\pm }$ in the $\infty $-category $\operatorname{Fun}( \operatorname{\partial \Delta }^{n-1}, \operatorname{\mathcal{C}}_{/Y} )$. Let us identify $\widetilde{f}_{\pm }$ with a morphism of simplicial sets $f_{+}: \Lambda ^{n}_{n} \rightarrow \operatorname{\mathcal{C}}$ satisfying $f_{+}(n) = Y$. Then $\widetilde{h}_{\pm }$ determines a natural transformation $h_{+}: f_{+} \rightarrow g_{+}$, given by the composition

\[ \Delta ^1 \times \Lambda ^{n}_{n} \simeq \Delta ^1 \times (\operatorname{\partial \Delta }^{n-1} )^{\triangleright } \rightarrow ( \Delta ^1 \times \operatorname{\partial \Delta }^{n-1} )^{\triangleright } \xrightarrow { \widetilde{h}_{\pm } } (\operatorname{\mathcal{C}}_{/Y})^{\triangleright } \rightarrow \operatorname{\mathcal{C}}. \]

Note that $f_{-}$ and $f_{+}$ can be amalgamated to a morphism $f: \operatorname{\partial \Delta }^{n} \rightarrow \operatorname{\mathcal{C}}$, and that $h_{-}$ and $h_{+}$ can be amalgamated to a natural transformation $h: f \rightarrow g$ in $\operatorname{Fun}( \operatorname{\partial \Delta }^{n}, \operatorname{\mathcal{C}})$.

Invoking hypothesis $(\ast ''_{n})$, we see that $f$ can be extended to a morphism $\overline{f}: ( \operatorname{\partial \Delta }^{n} )^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$. Let $Z \in \operatorname{\mathcal{C}}$ denote the image under $\overline{f}$ of the cone point and let $\varphi : \operatorname{\mathcal{C}}_{/Z} \rightarrow \operatorname{\mathcal{C}}$ denote the projection map, so that $\overline{f}$ can be identified with a morphism of simplicial sets $f': \operatorname{\partial \Delta }^{n} \rightarrow \operatorname{\mathcal{C}}_{/Z}$ satisfying $\varphi \circ f' = f$. Let us identify the vertex $f'(n) \in \operatorname{\mathcal{C}}_{/Z}$ with a morphism $v: Y \rightarrow Z$ in the $\infty $-category $\operatorname{\mathcal{C}}$, so that we have a commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{/v} \ar [r]^-{\varphi '} \ar [d]^{\pi '} & \operatorname{\mathcal{C}}_{/Y} \ar [d]^{\pi } \\ \operatorname{\mathcal{C}}_{/Z} \ar [r]^-{ \varphi } & \operatorname{\mathcal{C}}. } \]

Set $f'_{+} = f'|_{ \Lambda ^{n}_{n} }$ and $f'_{\pm } = f'|_{ \operatorname{\partial \Delta }^{n-1} }$, so that we can identify $f'_{+}$ with a morphism $\widetilde{f}'_{\pm }: \operatorname{\partial \Delta }^{n-1} \rightarrow \operatorname{\mathcal{C}}_{/v}$ satisfying $\pi ' \circ \widetilde{f}'_{\pm } = f'_{+}$. Since the inclusion $\{ 0\} \hookrightarrow \Delta ^{1}$ is left anodyne, the morphism $\varphi ': \operatorname{\mathcal{C}}_{/v} \rightarrow \operatorname{\mathcal{C}}_{/Y}$ is a trivial Kan fibration (Corollary 4.3.6.13). We can therefore lift $\widetilde{h}_{\pm }$ to a natural transformation $\widetilde{h}'_{\pm }: \widetilde{f}'_{\pm } \rightarrow \widetilde{g}'_{\pm }$ for some morphism $\widetilde{g}'_{\pm }: \operatorname{\partial \Delta }^{n-1} \rightarrow \operatorname{\mathcal{C}}_{/v}$. Let us identify $\widetilde{g}'_{\pm }$ with a morphism $g'_{+}: \Lambda ^{n}_{n} \rightarrow \operatorname{\mathcal{C}}_{/Z}$ satisfying $\varphi \circ g'_{+} = g_{+}$. Then $\widetilde{h}'_{\pm }$ determines a natural transformation $h'_{+}: f'_{+} \rightarrow g'_{+}$, given by the composition

\[ \Delta ^1 \times \Lambda ^{n}_{n} \simeq \Delta ^1 \times (\operatorname{\partial \Delta }^{n-1} )^{\triangleright } \rightarrow ( \Delta ^1 \times \operatorname{\partial \Delta }^{n-1} )^{\triangleright } \xrightarrow { \widetilde{h}'_{\pm } } (\operatorname{\mathcal{C}}_{/v})^{\triangleright } \rightarrow \operatorname{\mathcal{C}}_{/Z}. \]

Let $e$ denote the restriction $h'_{+} |_{ \Delta ^1 \times \{ 0\} }$, which we regard as an edge of the simplicial set $\operatorname{\mathcal{C}}_{/Z}$. By construction, $\varphi (e)$ is the degenerate edge $\operatorname{id}_{X}$ of $\operatorname{\mathcal{C}}$. Since $\varphi $ is a right fibration (Proposition 4.3.6.1), it follows that $e$ is an isomorphism in $\operatorname{\mathcal{C}}_{/Z}$ (Proposition 4.4.2.11). Applying Proposition 4.4.5.8, we deduce that the lifting problem

\[ \xymatrix@C =50pt@R=50pt{ (\Delta ^1 \times \Lambda ^{n}_{n} ) \coprod _{ ( \{ 0\} \times \Lambda ^{n}_{n} )} ( \{ 0\} \times \operatorname{\partial \Delta }^{n} ) \ar [r]^-{(h'_{+}, f')} \ar [d] & \operatorname{\mathcal{C}}_{/Z} \ar [d]^{\varphi } \\ \Delta ^1 \times \operatorname{\partial \Delta }^{n} \ar [r]^-{h} \ar@ {-->}[ur]^{h'} & \operatorname{\mathcal{C}}} \]

admits a solution. The morphism $h'$ is then a natural transformation from $f'$ to a morphism $g': \operatorname{\partial \Delta }^{n} \rightarrow \operatorname{\mathcal{C}}_{/Z}$, which we can identify with a map $\overline{g}: ( \operatorname{\partial \Delta }^{n} )^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ satisfying $\overline{g}|_{ \operatorname{\partial \Delta }^{n} } = g$. $\square$

Proof of Theorem 7.2.5.5. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and suppose that the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ is homotopy filtered; we wish to show that $\operatorname{\mathcal{C}}$ is filtered (the reverse implication follows from Lemma 7.2.5.12). By virtue of Lemma 7.2.5.13, it will suffice to show that for every integer $n \geq 0$, every morphism of simplicial sets $f: \operatorname{\partial \Delta }^ n \rightarrow \operatorname{\mathcal{C}}$ can be extended to a morphism $\overline{f}: (\operatorname{\partial \Delta }^ n)^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$. For $n = 0$, this follows from our assumption that $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ is nonempty. We will therefore assume that $n > 0$. By virtue of Lemma 7.2.5.15, we may assume without loss of generality that the restriction $f_{-} = f|_{ \Delta ^{n-1} }$ is the constant map taking the value $X$ for some object $X \in \operatorname{\mathcal{C}}$. Set $Y = f(n)$ and let $\operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{R}}(X,Y) = \{ X\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/Y}$ denote the right-pinched morphism space of Construction 4.6.5.1, so that we can identify $f|_{ \Lambda ^{n}_{n} }$ with a morphism of simplicial sets $g: \operatorname{\partial \Delta }^{n-1} \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{R}}(X,Y)$. Invoking assumption $(\ast _ n)$ of Definition 7.2.5.1, we deduce that there exists a morphism $v: Y \rightarrow Z$ in $\operatorname{\mathcal{C}}$ for which the composite map

\[ \operatorname{\partial \Delta }^{n-1} \xrightarrow {g} \operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{R}}(X,Y) \hookrightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \xrightarrow {[v] \circ } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z) \]

is nullhomotopic. Since the projection map $\operatorname{\mathcal{C}}_{/f} \rightarrow \operatorname{\mathcal{C}}_{/Y}$ is a trivial Kan fibration (Corollary 4.3.6.13), we can lift $g$ to a morphism $\widetilde{g}: \operatorname{\partial \Delta }^{n-1} \rightarrow \{ X\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/f}$. Combining Propositions 5.2.7.7 and 4.6.7.16, we deduce that the diagram of Kan complexes

\[ \xymatrix@R =50pt@C=50pt{ \{ X\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/Y} \ar [d]^{\iota _{X,Y}^{\mathrm{R}}} & \{ X\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/f} \ar [r] \ar [l] & \{ X\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/Z} \ar [d]^{\iota ^{\mathrm{R}}_{X,Z}} \\ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \ar [rr]^{[v] \circ } & & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z) } \]

commutes up to homotopy, where $\iota _{X,Y}^{\mathrm{R}}$ and $\iota _{X,Z}^{\mathrm{R}}$ are the right-pinch inclusion morphisms of Construction 4.6.5.6. Since $\iota ^{\mathrm{R}}_{X,Z}$ is a homotopy equivalence (Proposition 4.6.5.9), it follows that the composite map $\operatorname{\partial \Delta }^{n-1} \xrightarrow {\widetilde{g}} \operatorname{\mathcal{C}}_{/f} \rightarrow \operatorname{\mathcal{C}}_{/Z}$ is nullhomotopic, and can therefore be extended to an $(n-1)$-simplex $g': \Delta ^{n-1} \rightarrow \operatorname{\mathcal{C}}_{/Z}$ (Proposition 3.2.6.11). Unwinding the definitions, we can identify $\widetilde{g}$ and $g'$ with morphisms $(\Lambda ^{n}_{n})^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ and $( \Delta ^{n-1 })^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$, which can be amalgamated to a single morphism $\overline{f}: ( \operatorname{\partial \Delta }^{n} )^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ extending $f$. $\square$

Exercise 7.2.5.16. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $n \geq 1$ be an integer. Show that the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ satisfies condition $(\ast _ n)$ of Definition 7.2.5.1 if and only if $\operatorname{\mathcal{C}}$ satisfies condition $(\ast '_{n})$ of Lemma 7.2.5.13.