5.1.1 Cartesian Edges of Simplicial Sets
Our first goal is to adapt Definition 5.0.0.1 to the setting of $\infty $-categories.
Definition 5.1.1.1. Let $q: X \rightarrow S$ be a morphism of simplicial sets, and let $e$ be an edge of $X$. We say that $e$ is $q$-cartesian if every lifting problem
\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{n}_{n} \ar [r]^-{\sigma _0} \ar [d] & X \ar [d]^-{q} \\ \Delta ^ n \ar [r]^-{ \overline{\sigma } } \ar@ {-->}[ur] & S } \]
admits a solution, provided that $n \geq 2$ and the composite map
\[ \Delta ^1 \simeq \operatorname{N}_{\bullet }( \{ n-1 < n \} ) \hookrightarrow \Lambda ^ n_{n} \xrightarrow {\sigma _0} X \]
corresponds to the edge $e$.
We say that $e$ is $q$-cocartesian if every lifting problem
\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{n}_{0} \ar [r]^-{\sigma _0} \ar [d] & X \ar [d]^-{q} \\ \Delta ^ n \ar [r]^-{ \overline{\sigma } } \ar@ {-->}[ur] & S } \]
admits a solution, provided that $n \geq 2$ and the composite map
\[ \Delta ^1 \simeq \operatorname{N}_{\bullet }( \{ 0 < 1 \} ) \hookrightarrow \Lambda ^ n_{0} \xrightarrow {\sigma _0} X \]
corresponds to the edge $e$.
Example 5.1.1.3. Let $q: X \rightarrow S$ be a right fibration of simplicial sets. Then every edge of $X$ is $q$-cartesian. Similarly, if $q: X \rightarrow S$ is a left fibration of simplicial sets, then every edge of $X$ is $q$-cocartesian.
Example 5.1.1.4. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $q: \operatorname{\mathcal{C}}\rightarrow \Delta ^0$ be the projection map, and let $e: X \rightarrow Y$ be a morphism in $\operatorname{\mathcal{C}}$. The following conditions are equivalent:
The morphism $e$ is an isomorphism.
The morphism $e$ is $q$-cartesian.
The morphism $e$ is $q$-cocartesian.
This is a restatement of Theorem 4.4.2.6.
Example 5.1.1.5. Let $q: X \rightarrow S$ be a morphism of simplicial sets which restricts to an isomorphism from $X$ to a full simplicial subset of $S$ (see Definition 4.1.2.15). Then every edge of $X$ is both $q$-cartesian and $q$-cocartesian.
Proposition 5.1.1.8. Let $q: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of $\infty $-categories and let $e: X \rightarrow Y$ be a morphism in $\operatorname{\mathcal{C}}$. The following conditions are equivalent:
- $(1)$
The morphism $e$ is an isomorphism in $\operatorname{\mathcal{C}}$.
- $(2)$
The morphism $e$ is $q$-cartesian and $q(e)$ is an isomorphism in $\operatorname{\mathcal{D}}$.
- $(3)$
The morphism $e$ is $q$-cocartesian and $q(e)$ is an isomorphism in $\operatorname{\mathcal{D}}$.
Proof.
We will prove the equivalence $(1) \Leftrightarrow (2)$; the proof of the equivalence $(1) \Leftrightarrow (3)$ is similar. The implication $(1) \Rightarrow (2)$ follows from Proposition 4.4.2.13 and Remark 1.5.1.6. To prove the converse, let $p: \operatorname{\mathcal{D}}\rightarrow \Delta ^0$ denote the projection map. If $q(e)$ is an isomorphism in $\operatorname{\mathcal{C}}$, then it is $p$-cartesian (Example 5.1.1.4). If, in addition, the morphism $e$ is $q$-cartesian, then it is also $(p \circ q)$-cartesian (Remark 5.1.1.6) and is therefore an isomorphism in the $\infty $-category $\operatorname{\mathcal{C}}$ (Example 5.1.1.4).
$\square$
Corollary 5.1.1.9. Let $q: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of $\infty $-categories. For every object $X \in \operatorname{\mathcal{C}}$, the identity morphism $\operatorname{id}_{X}: X \rightarrow X$ is $q$-cartesian and $q$-cocartesian.
Corollary 5.1.1.10. Let $q: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of $\infty $-categories, where $\operatorname{\mathcal{D}}$ is a Kan complex, and let $e: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$. The following conditions are equivalent:
- $(1)$
The morphism $e$ is an isomorphism in $\operatorname{\mathcal{C}}$.
- $(2)$
The morphism $e$ is $q$-cartesian.
- $(3)$
The morphism $e$ is $q$-cocartesian.
Proof.
Combine Propositions 5.1.1.8 and 1.4.6.10.
$\square$
Proposition 5.1.1.13. Let $q: X \rightarrow S$ be a morphism simplicial sets and let $e: x \rightarrow y$ be an edge of $X$. Then:
The edge $e$ is $q$-cartesian if and only if the natural map
\[ X_{/e} \rightarrow X_{/y} \times _{ S_{/q(y)} } S_{ /q(e) } \]
is a trivial Kan fibration of simplicial sets.
The edge $e$ is $q$-cocartesian if and only if the natural map
\[ X_{e/} \rightarrow X_{x/} \times _{ S_{q(x)/} } S_{ q(e) / } \]
is a trivial Kan fibration of simplicial sets.
Proof.
We will prove the first assertion; the proof of the second is similar. By definition, the natural map $X_{/e} \rightarrow X_{/y} \times _{ S_{/q(y)} } S_{ / q(e) }$ is a trivial Kan fibration if and only if, for every integer $n \geq 0$, every lifting problem
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\partial \Delta }^{n} \ar [r] \ar [d] & X_{/e} \ar [d] \\ \Delta ^{n} \ar@ {-->}[ur] \ar [r] & X_{/y} \times _{ S_{/q(y)} } S_{ /q(e) } } \]
admits a solution. By virtue of Lemma 4.3.6.15, this is equivalent to the datum of a lifting problem
\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{n+2}_{n+2} \ar [r]^-{\sigma _0} \ar [d] & X \ar [d] \\ \Delta ^{n+2} \ar@ {-->}[ur] \ar [r] & S, } \]
where $\sigma _0$ carries the final edge $\operatorname{N}_{\bullet }( \{ n+1 < n+2 \} ) \subseteq \Lambda ^{n+2}_{n+2}$ to $e$.
$\square$
Corollary 5.1.1.14. Let $q: X \rightarrow S$ and $f: K \rightarrow X$ be morphisms of simplicial sets, and let $q': X_{f/} \rightarrow S_{ (q \circ f)/ }$ be the morphism induced by $q$. Let $\overline{e}: \overline{x} \rightarrow \overline{y}$ be an edge of the simplicial set $X_{f/}$, and let $e: x \rightarrow y$ be its image in $X$. If $e$ is $q$-cartesian, then $\overline{e}$ is $q'$-cartesian.
Proof.
Since $e$ is $q$-cartesian, the restriction map
\[ \theta : X_{/e} \rightarrow X_{/y} \times _{ S_{/q(y)} } S_{/q(e)} \]
is a trivial Kan fibration (Proposition 5.1.1.13). We wish to show that the restriction map
\[ \overline{\theta }: (X_{f/} )_{ / \overline{e} } \rightarrow (X_{f/} )_{ / \overline{y} } \times _{ (S_{(q \circ f)/} )_{ / q'( \overline{y} )}} (S_{ (q \circ f)/} )_{ / q'( \overline{e} ) }. \]
is also a trivial Kan fibration. We can identify $\overline{e}$ with a morphism of simplicial sets $\overline{f}: K \rightarrow X_{/e}$, and $\overline{\theta }$ with the induced map
\[ ( X_{/e} )_{\overline{f} / } \rightarrow (X_{/y} \times _{ S_{/q(y)} } S_{/q(e)})_{ (\theta \circ \overline{f}) / }. \]
The desired result now follows from Corollary 4.3.7.17.
$\square$
Corollary 5.1.1.15. Let $q: X \rightarrow S$ be a morphism of simplicial sets and let $e: x \rightarrow y$ be an edge of $X$. The following conditions are equivalent:
- $(1)$
The edge $e$ is $q$-cartesian.
- $(2)$
Let $f: B \rightarrow X$ be a morphism of simplicial sets, let $A$ be a simplicial subset of $B$, and let $Y$ denote the fiber product $X_{f|_{A} / } \times _{ S_{ (q \circ f|_{A})/} } S_{ (q \circ f)/ }$, so that the restriction map $X_{f/} \rightarrow X$ factors as a composition $X_{f/} \xrightarrow {\theta } Y \xrightarrow {\rho } X$. Then every lifting problem
\[ \xymatrix@R =50pt@C=50pt{ \{ 1\} \ar [r] \ar [d] & X_{f/} \ar [d]^-{\theta } \\ \Delta ^1 \ar [r]^-{e'} \ar@ {-->}[ur] & Y } \]
admits a solution, provided that $\rho (e') = e$.
Proof.
For a fixed simplicial set $B$ with a simplicial subset $A \subseteq B$, condition $(2)$ is equivalent to the requirement that every lifting problem
\[ \xymatrix@R =50pt@C=50pt{ A \ar [r] \ar [d] & X_{/e} \ar [d] \\ B \ar [r] \ar@ {-->}[ur] & X_{/y} \times _{ S_{/q(y)} } S_{ /q(e) } } \]
admits a solution. This condition is satisfied for every inclusion of simplicial sets $A \subseteq B$ if and only if the map $X_{/e} \rightarrow X_{/y} \times _{ S_{/q(y)} } S_{ /q(e) }$ is a trivial Kan fibration: that is, if and only if $e$ is $q$-cartesian (Proposition 5.1.1.13).
$\square$