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5.2 Cartesian Fibrations

In §5.1, we studied the notions of cartesian and cocartesian fibration in the setting of classical category theory. Our goal in this section is to develop $\infty $-categorical counterparts of these notions. Our first step is to isolate an appropriate generalization for the notion of a cartesian morphism. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between ordinary categories. By definition, a morphism $g: Y \rightarrow Z$ in $\operatorname{\mathcal{C}}$ is $U$-cocartesian if and only if, for every morphism $h: X \rightarrow Z$ in $\operatorname{\mathcal{C}}$, every commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ & U(Y) \ar [dr]^{U(g)} & \\ U(X) \ar [ur]^{\overline{f} } \ar [rr]^{ U(h) } & & U(Z) } \]

in the category $\operatorname{\mathcal{D}}$ can be lifted uniquely to a commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ & Y \ar [dr]^{g} & \\ X \ar@ {-->}[ur]^{f} \ar [rr]^{h} & & Z } \]

in $\operatorname{\mathcal{C}}$. Equivalently, the morphism $g$ is $U$-cartesian if and only if every lifting problem

5.4
\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{C}}) \ar [d]^{ \operatorname{N}_{\bullet }(U)} \\ \Delta ^{2} \ar [r] \ar@ {-->}[ur]^{\sigma } & \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}}) } \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 $g$.

In the $\infty $-categorical setting, it is unreasonable to hope for the the lifting problem (5.4) 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 §5.2.1, we formalize this idea by considering analogues of (5.4) for higher-dimensional simplices. If $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is an arbitrary morphism of simplicial sets, we will say that an edge $g$ of $\operatorname{\mathcal{C}}$ is $U$-cartesian if every lifting problem

\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{n}_{n} \ar [r]^-{\sigma _0} \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{ U} \\ \Delta ^{n} \ar [r] \ar@ {-->}[ur]^{\sigma } & \operatorname{\mathcal{D}}} \]

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 $g$ (Definition 5.2.1.1). In the special case where $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are the nerves of ordinary categories, this reduces to the classical definition of cartesian morphism (Corollary 5.2.2.2).

The definition of $U$-cartesian edge makes sense for any morphism of simplicial sets $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$. However, it has poor formal properties in general. We will be primarily interested in the case where $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are $\infty $-categories and $U$ is an inner fibration. Assume that these conditions are satisfied and let $g: Y \rightarrow Z$ be a morphism of $\operatorname{\mathcal{C}}$, having image $\overline{g}: \overline{Y} \rightarrow \overline{Z}$ in $\operatorname{\mathcal{D}}$. For every object $X \in \operatorname{\mathcal{C}}$ having image $\overline{X} = U(X) \in \operatorname{\mathcal{D}}$, composition with the homotopy class $[g]$ determines a commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \ar [r]^-{[g] \circ } \ar [d] & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z) \ar [d] \\ \operatorname{Hom}_{\operatorname{\mathcal{D}}}( \overline{X}, \overline{Y} ) \ar [r]^-{[ \overline{g} ] \circ } & \operatorname{Hom}_{\operatorname{\mathcal{D}}}( \overline{X}, \overline{Z} ) } \]

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 §5.2.2, we show that $g$ is $U$-cartesian if and only if, for every object $X \in \operatorname{\mathcal{C}}$, the resulting lift is a homotopy pullback diagram of Kan complexes (Proposition 5.2.2.1). 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 5.2.2.5).

Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of simplicial sets. In §5.2.4 we study the situation where $\operatorname{\mathcal{C}}$ has “sufficiently many” $U$-cartesian edges in the following sense: for every vertex $Z \in \operatorname{\mathcal{C}}$, every edge $\overline{g}: \overline{Y} \rightarrow U(Z)$ of $\operatorname{\mathcal{D}}$ can be lifted to a $U$-cartesian edge $g: Y \rightarrow Z$ in $\operatorname{\mathcal{C}}$. 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 $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a cartesian fibration (in the sense of Definition 5.1.4.8) if and only if the induced functor of $\infty $-categories $\operatorname{N}_{\bullet }(U): \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}})$ is a cartesian fibration (Example 5.2.4.2).

  • Every right fibration of simplicial sets $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a cartesian fibration. Conversely, a cartesian fibration $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a right fibration if and only if every fiber of $U$ is a Kan complex (Proposition 5.2.4.12).

  • The collection of cartesian fibrations is closed under the formation of pullbacks (Remark 5.2.4.6) and composition (Proposition 5.2.4.11).

  • Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a cartesian fibration of simplicial sets and let $f: K \rightarrow \operatorname{\mathcal{C}}$ be any morphism of simplicial sets. Then the induced maps $\operatorname{\mathcal{C}}_{/f} \rightarrow \operatorname{\mathcal{D}}_{/(U \circ f)}$ and $\operatorname{\mathcal{C}}_{f/} \rightarrow \operatorname{\mathcal{D}}_{(U \circ f)/ }$ are cartesian fibrations (Propositions 5.2.4.14 and 5.2.4.16).

  • Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then the evaluation functor

    \[ \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \{ 0\} , \operatorname{\mathcal{C}}) \simeq \operatorname{\mathcal{C}} \]

    is a Cartesian fibration of $\infty $-categories (Example 5.2.5.2). This is a special case of a more general result about the comma construction (Proposition 5.2.5.1), which we prove in §5.2.5).

Suppose we are given a pullback diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}' \ar [d]^{U'} \ar [r]^-{F} & \operatorname{\mathcal{C}}\ar [d]^{U} \\ \operatorname{\mathcal{D}}' \ar [r] & \operatorname{\mathcal{D}}, } \]

and let $g$ be an edge of $\operatorname{\mathcal{C}}'$. It follows immediately from the definitions that if $F(g)$ is $U$-cartesian, then $g$ is $U'$-cartesian (Remark 5.2.1.9). The converse holds when $U$ is a cartesian fibration (Remark 5.2.4.6), but is false in general. In §5.2.3, we address this point by introducing the more general notion of a locally $U$-cartesian edge of a simplicial set $\operatorname{\mathcal{C}}$ equipped with a map $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ (Definition 5.2.3.1), which can be regarded as an $\infty $-categorical counterpart of Definition 5.1.4.14. In §5.2.6, we study inner fibrations $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ for which $\operatorname{\mathcal{C}}$ admits a sufficient supply of locally $U$-cartesian edges, which we will refer to as locally cartesian fibrations (Definition 5.2.6.1). Every cartesian fibration $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a locally cartesian fibration, and the converse holds if and only if the collection of locally $U$-cartesian edges of $\operatorname{\mathcal{C}}$ is closed under composition (Corollary 5.2.6.10).

Remark 5.2.0.1. The entirety of the preceding discussion can be dualized. If $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a morphism of simplicial sets, we will say that an edge $e$ of $\operatorname{\mathcal{C}}$ is (locally) $U$-cocartesian if it is (locally) $U^{\operatorname{op}}$-cartesian when viewed as an edge of the opposite simplicial set $\operatorname{\mathcal{C}}^{\operatorname{op}}$. We say that $U$ is a (locally) cocartesian fibration if the opposite functor $U^{\operatorname{op}}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{D}}^{\operatorname{op}}$ is a (locally) 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).

Structure

  • Subsection 5.2.1: Cartesian Edges of Simplicial Sets
  • Subsection 5.2.2: Cartesian Morphisms of $\infty $-Categories
  • Subsection 5.2.3: Locally Cartesian Morphisms
  • Subsection 5.2.4: Cartesian Fibrations
  • Subsection 5.2.5: Example: Path Fibrations
  • Subsection 5.2.6: Locally Cartesian Fibrations