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5.2.3 Locally Cartesian Morphisms

It will often be convenient to consider a variant of Definition 5.2.1.1.

Definition 5.2.3.1. Let $q: X \rightarrow S$ be a morphism of simplicial sets and let $e$ be an edge of $X$ having image $\overline{e} = q(e)$ in $S$. Form a pullback diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ X_{e} \ar [r] \ar [d]^{q'} & X \ar [d]^{q} \\ \Delta ^1 \ar [r]^-{ \overline{e} } & S, } \]

so that $e$ lifts uniquely to an edge $\widetilde{e}$ of $X_{e}$ having nondegenerate image in $\Delta ^1$. We say that $e$ is locally $q$-cartesian if $\widetilde{e}$ is a $q'$-cartesian edge of the simplicial set $X_{e}$. We say that $e$ is locally $q$-cocartesian if $\widetilde{e}$ is a $q'$-cocartesian edge of the simplicial set $X_{e}$.

Remark 5.2.3.2. Let $q: X \rightarrow S$ be a morphism of simplicial sets and $q^{\operatorname{op}}: X^{\operatorname{op}} \rightarrow S^{\operatorname{op}}$ be the opposite morphism. Then an edge $e$ of $X$ is locally $q$-cartesian if and only if it is locally $q^{\operatorname{op}}$-cocartesian.

Example 5.2.3.3. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a functor between categories and let $\operatorname{N}_{\bullet }(U): \operatorname{N}_{\bullet }(\operatorname{\mathcal{E}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ be the induced morphism of simplicial sets. Let $f: X \rightarrow Y$ be a morphism in the category $\operatorname{\mathcal{E}}$. It follows from Corollary 5.2.2.2 that $f$ is locally $U$-cartesian (in the sense of Definition 5.1.4.14) if and only if it is locally $\operatorname{N}_{\bullet }(U)$-cartesian when viewed as an edge of the simplicial set $\operatorname{N}_{\bullet }(\operatorname{\mathcal{E}})$ (in the sense of Definition 5.2.3.1). Similarly, $f$ is locally $U$-cocartesian if and only if it is locally $\operatorname{N}_{\bullet }(U)$-cocartesian.

Remark 5.2.3.4. Let $q: X \rightarrow S$ be a morphism of simplicial sets. Then every $q$-cartesian edge of $X$ is locally $q$-cartesian, and every $q$-cocartesian edge of $X$ is locally $q$-cocartesian (see Remark 5.2.1.9). Beware that the converse is false in general (see Example 5.1.4.23).

Remark 5.2.3.5. Let $q: X \rightarrow S$ be a morphism of simplicial sets and let $e: x \rightarrow y$ be an edge of $X$. Suppose that $S$ is isomorphic to a left cone $K^{\triangleleft }$ and that $q$ carries the vertex $x \in X$ to the cone point of $K^{\triangleleft }$. Then $e$ is $q$-cartesian if and only if it is locally $q$-cartesian. Similarly, if $S$ is isomorphic to a right cone $L^{\triangleright }$ and $q$ carries the vertex $y \in X$ to the cone point of $L^{\triangleright }$, then $e$ is $q$-cocartesian if and only if it is locally $q$-cocartesian.

Remark 5.2.3.6. Suppose we are given a pullback diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ X' \ar [d]^{q'} \ar [r]^-{f} & X \ar [d]^{q} \\ S' \ar [r] & S. } \]

Let $e'$ be an edge of the simplicial set $X'$, having image $e = f(e')$ in $X$. Then $e$ is locally $q$-cartesian if and only if $e'$ is locally $q'$-cartesian. Similarly, $e$ is locally $q$-cocartesian if and only if $e'$ is locally $q'$-cocartesian.

Example 5.2.3.7. Let $q: X \rightarrow S$ be a morphism of simplicial sets and let $e$ be an edge of $X$ such that $q(e) = \operatorname{id}_{s}$ is a degenerate edge of $S$. Suppose that the fiber $X_{s} = \{ s\} \times _{S} X$ is an $\infty $-category (this condition is satisfied, for example, if $q$ is an inner fibration). The following conditions are equivalent:

  • The edge $e$ is locally $q$-cartesian.

  • The edge $e$ is locally $q$-cocartesian.

  • The edge $e$ is an isomorphism in the $\infty $-category $X_{s}$.

To prove this, we can use Remark 5.2.3.6 to reduce to the situation where $S = \{ s\} $ consists of a single vertex. In this case, the edge $e$ is locally $q$-cartesian if and only if it is $q$-cartesian, and locally $q$-cocartesian if and only if it is $q$-cocartesian (Remark 5.2.3.5). The desired result now follows from Example 5.2.1.4.

Proposition 5.2.3.8. Let $q: X \rightarrow S$ be an inner fibration of simplicial sets and let $\sigma : \Delta ^2 \rightarrow X$ be a $2$-simplex of $X$, which we will depict as a diagram

\[ \xymatrix@R =50pt@C=50pt{ & y \ar [dr]^{g} & \\ x \ar [ur]^{f} \ar [rr]^{h} & & z. } \]

  • Suppose that $g$ is $q$-cartesian. Then $f$ is locally $q$-cartesian if and only if $h$ is locally $q$-cartesian.

  • Suppose that $f$ is $q$-cocartesian. Then $g$ is locally $q$-cocartesian if and only if $h$ is locally $q$-cocartesian.

Proof. We will prove the first assertion; the proof of the second is similar. Using Remarks 5.2.1.9 and 5.2.3.6, we can replace $q$ by the projection map $\Delta ^2 \times _{S} X \rightarrow \Delta ^2$, and thereby reduce to the case where $S = \Delta ^2$ and $q(\sigma )$ is the identity morphism $\operatorname{id}_{ \Delta ^2 }$. In this case, both $X$ and $S$ are $\infty $-categories and the morphisms $f$ and $h$ are locally $q$-cartesian if and only if they are $q$-cartesian (Remark 5.2.3.5). The desired result now follows from Corollary 5.2.2.5. $\square$

Remark 5.2.3.9 (Uniqueness of Locally Cartesian Lifts). Let $q: X \rightarrow S$ be an inner fibration of simplicial sets and let $g: y \rightarrow z$ be a locally $q$-cartesian edge of $X$. Suppose that $h: x \rightarrow z$ is another edge of $X$ satisfying $q(h) = q(g)$. Set $s = q(x) = q(y)$, and let $X_{s} = \{ s\} \times _{S} X$ denote the fiber of $q$ over the vertex $s$. Our assumption that $g$ is $q$-cartesian then guarantees that we can choose a $2$-simplex $\sigma $ of $X$ satisfying

\[ d_0(\sigma ) = g \quad \quad d_1(\sigma ) = h \quad \quad q(\sigma ) = s_0( q(g) ), \]

which we display informally as a diagram

\[ \xymatrix@R =50pt@C=50pt{ & y \ar [dr]^{g} & \\ x \ar [ur]^{f} \ar [rr]^{h} & & z; } \]

here $f = d_2(\sigma )$ is a morphism in the $\infty $-category $X_{s}$. In this case, the following conditions are equivalent:

$(1)$

The morphism $f$ is an isomorphism in the $\infty $-category $X_{s}$.

$(2)$

The morphism $h$ is locally $q$-cartesian.

To see this, we can replace $q$ by the projection map $\Delta ^1 \times _{S} X \rightarrow X$, and thereby reduce to the case where $g$ and $h$ are both lifts of the unique nondegenerate edge of $S = \Delta ^1$. In this case, the morphism $g$ is $q$-cartesian, and $(1)$ is equivalent to the assertion that $f$ is locally $q$-cartesian (Example 5.2.3.7). The equivalence of $(1)$ and $(2)$ is now a special case of Proposition 5.2.3.8.

Corollary 5.2.3.10. Let $q: X \rightarrow S$ be an inner fibration of simplicial sets, let $z$ be a vertex of $X$, and let $e: s \rightarrow q(z)$ be an edge of $S$. Suppose that there exists a $q$-cartesian edge $g: y \rightarrow z$ of $X$ satisfying $q(g) = e$. Then any locally $q$-cartesian edge $h: x \rightarrow z$ satisfying $q(h) = e$ is $q$-cartesian.

Proof. By virtue of Remark 5.2.1.10, we may assume without loss of generality that $S$ is an $\infty $-category (or even a simplex). Applying Remark 5.2.3.9, we deduce that there is a $2$-simplex of $X$ as depicted in the diagram

\[ \xymatrix@R =50pt@C=50pt{ & y \ar [dr]^{g} & \\ x \ar [ur]^{f} \ar [rr]^{h} & & z, } \]

where $f$ is an isomorphism in the $\infty $-category $X$. Then $f$ is also $q$-cartesian (Proposition 5.2.1.7), so Corollary 5.2.2.5 guarantees that $h$ is $q$-cartesian. $\square$

We now record an analogue of Proposition 5.2.2.1 for detecting locally cartesian edges.

Notation 5.2.3.11. Let $q: X \rightarrow S$ be a morphism of simplicial sets, let $y$ and $z$ be vertices of $X$ having images $s = q(y)$ and $t = q(z)$, and let $\overline{e}: s \rightarrow t$ be an edge of $S$. Recall that the relative morphism space $\operatorname{Hom}_{X}(y,z)_{\overline{e}}$ is defined to be the fiber product $\operatorname{Hom}_{X}(y,z) \times _{ \operatorname{Hom}_{S}( s,t )} \{ \overline{e} \} $ (Construction 4.6.1.13).

Let $x$ be another vertex of $X$ satisfying $q(x) = s$, and let $\sigma $ denote the image of $\overline{e}$ under the degeneracy map $\operatorname{Hom}_{S}(s,t) \rightarrow \operatorname{Hom}_{S}(s,s,t)$ (see Notation 4.6.3.1). It follows from Proposition 4.6.3.4 (and Example 4.6.1.15) that restriction along the inclusion $\Lambda ^{2}_{1} \hookrightarrow \Delta ^2$ induces a trivial Kan fibration of simplicial sets

\[ \theta : \operatorname{Hom}_{X}(x,y,z) \times _{\operatorname{Hom}_{S}( s,s,t ) } \{ \sigma \} \rightarrow \operatorname{Hom}_{X}(y,z)_{ \overline{e} } \times \operatorname{Hom}_{ X_{s} }(x,y), \]

where $X_{\overline{y}}$ denotes the $\infty $-category given by the fiber $\{ \overline{y} \} \times _{S} X$. In particular, the homotopy class $[\theta ]$ is an isomorphism in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$. Combining the inverse isomorphism $[\theta ]^{-1}$ with the restriction map $\operatorname{Hom}_{X}(x,y,z) \times _{\operatorname{Hom}_{S}( s,s,t ) } \{ \sigma \} \rightarrow \operatorname{Hom}_{X}(x,z)_{ \overline{e} }$, we obtain a composition law

\[ \circ : \operatorname{Hom}_{X}(y,z)_{ \overline{e} } \times \operatorname{Hom}_{ X_{s} }(x,y) \rightarrow \operatorname{Hom}_{X}(x,z)_{ \overline{e} }. \]

If $e: y \rightarrow z$ is an edge of $X$ satisfying $q(e) = \overline{e}$, then the restriction of this composition law to $\{ e \} \times \operatorname{Hom}_{X_ s}(x,y)$ determines a morphism of Kan complexes $\operatorname{Hom}_{X_ s}( x,y) \xrightarrow { [e] \circ } \operatorname{Hom}_{X}(x,z)_{ \overline{e} }$, which is well-defined up to homotopy.

Proposition 5.2.3.12. Let $q: X \rightarrow S$ be an inner fibration of simplicial sets, and let $e: y \rightarrow z$ be an edge of the simplicial set $X$ having image $\overline{e}: s \rightarrow t$ in $S$. Then $e$ is locally $q$-cartesian if and only if, for every object $x$ of the $\infty $-category $X_{s}$, the composition map

\[ \operatorname{Hom}_{X_ s}( x,y ) \xrightarrow { [e] \circ } \operatorname{Hom}_{X}(x,z)_{\overline{e} } \]

of Notation 5.2.3.11 is an isomorphism in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$.

Proof. Without loss of generality, we can replace $q: X \rightarrow S$ by the projection map $X \times _{S} \Delta ^1 \rightarrow \Delta ^1$ and thereby reduce to the case where $S = \Delta ^1$ and $\overline{e}$ is the unique nondegenerate edge of $\Delta ^1$. In this case, the edge $e$ is locally $q$-cartesian if and only if it is $q$-cartesian, and the desired result is a special case of Corollary 5.2.2.4. $\square$