Kerodon

Proof. Combine Theorem 7.2.5.5 with Remark 7.2.5.4. $\square$

Proof. Combine Theorem 7.2.5.5 with Example 7.2.5.3. $\square$

Proof. Combine Theorem 7.2.5.5 with Corollary 4.6.9.20. $\square$

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

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

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

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

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 1.1.4.12, we can choose a pushout diagram

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

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

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

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

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 $\operatorname{\mathcal{C}}_{f/} = \operatorname{\mathcal{C}}_{ f' / } \times _{ \operatorname{\mathcal{C}}_{ f'_0 / } } \operatorname{\mathcal{C}}_{ f_0 / }$. $\square$

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

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

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

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

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

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

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

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 5.2.8.7 and 4.6.9.16, we deduce that the diagram of Kan complexes

commutes up to homotopy, where $\iota _{X,Y}^{\mathrm{R}}$ and $\iota _{X,Z}^{\mathrm{R}}$ are the right-pinch inclusion morphisms of Construction 4.6.5.7. Since $\iota ^{\mathrm{R}}_{X,Z}$ is a homotopy equivalence (Proposition 4.6.5.10), it follows that the composite map $\operatorname{\partial \Delta }^{n-1} \xrightarrow {\widetilde{g}} \{ X\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/f} \rightarrow \{ X\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/Z}$ is nullhomotopic, and can therefore be extended to an $(n-1)$-simplex $g': \Delta ^{n-1} \rightarrow \{ X\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/Z}$ (Variant 3.2.4.13). 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 $f$. $\square$