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Section 5.1: Cartesian Fibrations (cite)
Subsection 5.1.1: Cartesian Edges of Simplicial Sets
Subsection 5.1.2: Cartesian Morphisms of $\infty $-Categories
Subsection 5.1.3: Locally Cartesian Edges
Subsection 5.1.4: Cartesian Fibrations
Subsection 5.1.5: Fiberwise Equivalence
Subsection 5.1.6: Equivalence of Inner Fibrations
5.1 Cartesian Fibrations
The goal in this section is to extend the theory of (co)cartesian fibrations to the setting of $\infty $-categories. The first step is to introduce an $\infty $-categorical analogue of Definition 5.0.0.1. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a functor between categories, and let $f: X \rightarrow Y$ be a morphism in $\operatorname{\mathcal{E}}$. By definition, $f$ is $U$-cartesian if and only if, for every morphism $h: W \rightarrow Y$ in $\operatorname{\mathcal{E}}$, every commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ & U(X) \ar [dr]_{U(f)} & \\ U(W) \ar [ur]^{\overline{g} } \ar [rr]^-{ U(h) } & & U(Y) } \]
in the category $\operatorname{\mathcal{C}}$ can be lifted uniquely to a commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ & X \ar [dr]_{f} & \\ W \ar@ {-->}[ur]^{g} \ar [rr]^-{h} & & Y } \]
in the category $\operatorname{\mathcal{E}}$. Equivalently, the morphism $f$ is $U$-cartesian if and only if every lifting problem
\begin{equation} \label{equation:lifting-problem-cartesian-2-simplex} \begin{gathered}\xymatrix@R =50pt@C=50pt{ \Lambda ^{2}_{2} \ar [r]^-{\sigma _0} \ar [d] & \operatorname{N}_{\bullet }(\operatorname{\mathcal{E}}) \ar [d]^-{ \operatorname{N}_{\bullet }(U)} \\ \Delta ^{2} \ar [r] \ar@ {-->}[ur]^{\sigma } & \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \end{gathered} \end{equation}
has a unique solution, assuming that $\sigma _0$ carries the "final edge" $\operatorname{N}_{\bullet }( \{ 1 < 2 \} ) \subseteq \Lambda ^{2}_{2}$ to the morphism $f$.
In the $\infty $-categorical setting, it is unreasonable to ask for the lifting problem (5.2) to admit a unique solution. Instead, we should require that the collection of possible choices for $\sigma $ are, in some sense, parametrized by a contractible space. In §5.1.1, we formalize this idea by considering analogues of (5.2) for higher-dimensional simplices. If $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is an arbitrary morphism of simplicial sets, we will say that an edge $f$ of $\operatorname{\mathcal{E}}$ is $U$-cartesian if every lifting problem
\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{n}_{n} \ar [r]^-{\sigma _0} \ar [d] & \operatorname{\mathcal{E}}\ar [d]^-{ U} \\ \Delta ^{n} \ar [r] \ar@ {-->}[ur]^{\sigma } & \operatorname{\mathcal{C}}} \]
admits a solution, provided that $n \geq 2$ and $\sigma _0$ carries the "final edge" $\operatorname{N}_{\bullet }( \{ n-1 < n \} ) \subseteq \Lambda ^{n}_{n}$ to $f$ (Definition 5.1.1.1). In the special case where $\operatorname{\mathcal{E}}$ and $\operatorname{\mathcal{C}}$ are the nerves of ordinary categories, this reduces to the classical definition of cartesian morphism (Corollary 5.1.2.2).
The definition of $U$-cartesian edge makes sense for any morphism of simplicial sets $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$. However, it has poor formal properties in general. We will be primarily interested in the case where $\operatorname{\mathcal{E}}$ and $\operatorname{\mathcal{C}}$ are $\infty $-categories and $U$ is an inner fibration. Assume that these conditions are satisfied and let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{E}}$, having image $\overline{f}: \overline{X} \rightarrow \overline{Y}$ in $\operatorname{\mathcal{D}}$. For every object $W \in \operatorname{\mathcal{E}}$ having image $\overline{X} = U(X) \in \operatorname{\mathcal{C}}$, composition with the homotopy class $[f]$ determines a commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{\mathcal{E}}}(W,X) \ar [r]^-{[f] \circ } \ar [d] & \operatorname{Hom}_{\operatorname{\mathcal{E}}}(W,Y) \ar [d] \\ \operatorname{Hom}_{\operatorname{\mathcal{C}}}( \overline{W}, \overline{X} ) \ar [r]^-{[ \overline{f} ] \circ } & \operatorname{Hom}_{\operatorname{\mathcal{C}}}( \overline{W}, \overline{Y} ) } \]
in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$, which (after suitable modifications on the left hand side) can be lifted to a commutative diagram in the category of simplicial sets. In §5.1.2, we show that $f$ is $U$-cartesian if and only if, for every object $W \in \operatorname{\mathcal{E}}$, the resulting lift is a homotopy pullback diagram of Kan complexes (Proposition 5.1.2.1). This has a number of pleasant consequences: for example, it implies that the collection of $U$-cartesian morphisms is closed under composition (for a stronger statement, see Corollary 5.1.2.4).
Suppose we are given a pullback diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}' \ar [d]^-{U'} \ar [r]^-{F} & \operatorname{\mathcal{E}}\ar [d]^-{U} \\ \operatorname{\mathcal{C}}' \ar [r] & \operatorname{\mathcal{C}}, } \]
and let $f$ be an edge of $\operatorname{\mathcal{E}}'$. It follows immediately from the definitions that if $F(f)$ is $U$-cartesian, then $f$ is $U'$-cartesian (Remark 5.1.1.11). The converse holds when $U$ is a cartesian fibration (Remark 5.1.4.6), but is false in general. In §5.1.3, we address this point by introducing the more general notion of a locally $U$-cartesian edge of a simplicial set $\operatorname{\mathcal{E}}$ equipped with a map $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ (Definition 5.1.3.1).
Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of simplicial sets. In §5.1.4 we study the situation where $\operatorname{\mathcal{E}}$ has "sufficiently many" $U$-cartesian edges in the following sense: for every vertex $Y \in \operatorname{\mathcal{E}}$, every edge $\overline{f}: \overline{X} \rightarrow U(Y)$ of $\operatorname{\mathcal{C}}$ can be lifted to a $U$-cartesian edge $f: X \rightarrow Y$ of $\operatorname{\mathcal{C}}$. If this condition is satisfied, we say that $U$ is a cartesian fibration of simplicial sets. This definition has the following features:
A functor of ordinary categories $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is a cartesian fibration (in the sense of Definition 5.0.0.3) if and only if the induced functor of $\infty $-categories $\operatorname{N}_{\bullet }(U): \operatorname{N}_{\bullet }(\operatorname{\mathcal{E}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is a cartesian fibration (Example 5.1.4.2).
Every right fibration of simplicial sets $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is a cartesian fibration. Conversely, a cartesian fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is a right fibration if and only if every fiber of $U$ is a Kan complex (Proposition 5.1.4.14).
The collection of cartesian fibrations is closed under the formation of pullbacks (Remark 5.1.4.6) and composition (Proposition 5.1.4.13).
Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cartesian fibration of simplicial sets and let $f: K \rightarrow \operatorname{\mathcal{E}}$ be any morphism of simplicial sets. Then the induced maps $\operatorname{\mathcal{E}}_{/f} \rightarrow \operatorname{\mathcal{C}}_{/(U \circ f)}$ and $\operatorname{\mathcal{E}}_{f/} \rightarrow \operatorname{\mathcal{C}}_{(U \circ f)/ }$ are cartesian fibrations (Propositions 5.1.4.17 and 5.1.4.19).
Suppose we are given a commutative diagram of $\infty $-categories
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [rr]^-{F} \ar [dr]^{U} & & \operatorname{\mathcal{D}}\ar [dl]_{V} \\ & \operatorname{\mathcal{E}}, & } \]
where $U$ and $V$ are isofibrations. Recall that, if $F$ is an equivalence of $\infty $-categories, then the induced map of fibers $F_{E}: \operatorname{\mathcal{C}}_{E} \rightarrow \operatorname{\mathcal{D}}_{E}$ is also an equivalence of $\infty $-categories for every object $E \in \operatorname{\mathcal{E}}$ (Corollary 4.5.2.25). The converse is false in general (Warning 4.5.2.26). Nevertheless, in §5.1.5 we show that the converse is true if we assume that $U$ is a cartesian fibration and that $F$ carries $U$-cartesian morphisms of $\operatorname{\mathcal{C}}$ to $V$-cartesian morphisms of $\operatorname{\mathcal{D}}$ (Theorem 5.1.5.1). In §5.1.6, we prove a counterpart of this result in the case where $\operatorname{\mathcal{E}}$ is not assumed to be an $\infty $-category (Proposition 5.1.6.14): in this case, $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ need not be $\infty $-categories, but it is still possible to show that $F$ is an equivalence of inner fibrations over $\operatorname{\mathcal{E}}$ (see Definition 5.1.6.1).
Remark 5.1.0.1. The entirety of the preceding discussion can be dualized. If $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is a morphism of simplicial sets, we will say that an edge $f$ of $\operatorname{\mathcal{E}}$ is $U$-cocartesian if it is $U^{\operatorname{op}}$-cartesian when viewed as an edge of the opposite simplicial set $\operatorname{\mathcal{E}}^{\operatorname{op}}$. We say that $U$ is a cocartesian fibration if the opposite functor $U^{\operatorname{op}}: \operatorname{\mathcal{E}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ is a cartesian fibration. For the sake of brevity, we will sometimes state our results only for cartesian fibrations (in which case there is always a counterpart for cocartesian fibrations, which can be obtained by passing to opposite simplicial sets).