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9.10.3 Classical Stuff

Definition 9.10.3.1. Let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a functor between categories and let $f: X \rightarrow Y$ be a morphism in $\operatorname{\mathcal{D}}$ having image $\overline{f}: \overline{X} \rightarrow \overline{Y}$ in the category $\operatorname{\mathcal{C}}$.

  • We say that $f$ is locally $U$-cartesian if, for every object $W$ of the fiber category $\operatorname{\mathcal{D}}_{ \overline{X} } = \{ \overline{X} \} \times _{ \operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}$, postcomposition with $f$ induces a bijection

    \[ \operatorname{Hom}_{ \operatorname{\mathcal{D}}_{\overline{X}} }( W, X) \xrightarrow {f \circ } \{ f' \in \operatorname{Hom}_{\operatorname{\mathcal{D}}}(X,Y): U(f') = U(f) \} . \]
  • We say that $f$ is locally $U$-cocartesian if, for every object $Z$ of the fiber category $\operatorname{\mathcal{D}}_{ \overline{Y} } = \{ \overline{Y} \} \times _{ \operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}$, precomposition with $f$ induces a bijection

    \[ \operatorname{Hom}_{ \operatorname{\mathcal{D}}_{\overline{Y}} }( Y, Z) \xrightarrow {\circ f} \{ f' \in \operatorname{Hom}_{\operatorname{\mathcal{D}}}(X,Z): U(f') = U(f) \} . \]

Remark 9.10.3.2. Let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a functor between categories and let $U^{\operatorname{op}}: \operatorname{\mathcal{D}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ be the induced functor of opposite categories. Let $f: X \rightarrow Y$ be a morphism in the category $\operatorname{\mathcal{D}}$, which we identify with a morphism $f^{\operatorname{op}}: Y \rightarrow X$ in the opposite category $\operatorname{\mathcal{D}}^{\operatorname{op}}$. Then $f$ is locally $U$-cartesian if and only if $f^{\operatorname{op}}$ is locally $U^{\operatorname{op}}$-cocartesian.

Example 9.10.3.3. 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 5.1.2.2 that $f$ is locally $U$-cartesian (in the sense of Definition 9.10.3.1) 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 5.1.3.1). Similarly, $f$ is locally $U$-cocartesian if and only if it is locally $\operatorname{N}_{\bullet }(U)$-cocartesian.

Remark 9.10.3.4. Let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a functor between categories and let $f: X \rightarrow Y$ be a morphism in the category $\operatorname{\mathcal{D}}$. Then the image $U(f)$ can be identified with a functor from the partially ordered set $[1] = \{ 0 < 1 \} $ to the category $\operatorname{\mathcal{C}}$ (carrying $0$ to the object $U(X)$, and $1$ to the object $U(Y)$). Form a pullback diagram of categories

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{D}}' \ar [r] \ar [d]^-{U'} & \operatorname{\mathcal{D}}\ar [d]^-{U} \\ {[1]} \ar [r]^-{U(f)} & \operatorname{\mathcal{C}}. } \]

By construction, $f$ can be lifted uniquely to a morphism $f': X' \rightarrow Y'$ in the category $\operatorname{\mathcal{D}}'$ satisfying $U'(X') = 0$ and $U'(Y') = 1$. Then:

  • The morphism $f$ is locally $U$-cartesian (in the sense of Definition 9.10.3.1) if and only if $f'$ is $U'$-cartesian (in the sense of Definition 5.0.0.1).

  • The morphism $f$ is locally $U$-cocartesian (in the sense of Definition 9.10.3.1) if and only if $f'$ is $U'$-cocartesian.

Remark 9.10.3.5. Let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a functor between categories and let $f: X \rightarrow Y$ be a morphism in the category $\operatorname{\mathcal{D}}$. If $f$ is $U$-cartesian, then it is locally $U$-cartesian. If $f$ is $U$-cocartesian, then it is locally $U$-cocartesian. This follows from Remark 9.10.3.4.

Remark 9.10.3.6 (Uniqueness). Let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a functor of categories. Suppose we are given a pair of morphisms $f: X \rightarrow Y$ and $f': X' \rightarrow Y$ in the category $\operatorname{\mathcal{D}}$ having the same target, with $U(X') = U(X)$ and $U(f') = U(f)$. If the morphism $f$ is locally $U$-cartesian, then the morphism $f'$ factors uniquely as a composition $f' = f \circ e$, where $e: X' \rightarrow X$ is a morphism satisfying $U(e) = \operatorname{id}_{ U(X)}$. In this case, the morphism $f'$ is locally $U$-cartesian if and only if $e$ is an isomorphism.

We can summarize the situation more informally as follows: if $Y$ is an object of the category $\operatorname{\mathcal{D}}$ and $\overline{f}: \overline{X} \rightarrow Y$ is a morphism in $\operatorname{\mathcal{C}}$ which can be lifted to a locally cartesian morphism $f: X \rightarrow Y$ in $\operatorname{\mathcal{D}}$, then the morphism $f$ is uniquely determined up to isomorphism.

Example 9.10.3.7. Let $\operatorname{\mathcal{C}}$ be a category, let $\mathscr {F}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \mathbf{Cat}$ be unitary lax functor, let $\int ^{\operatorname{\mathcal{C}}} \mathscr {F}$ denote the Grothendieck construction on $\mathscr {F}$ (Construction 9.5.0.52), and let $U: \int ^{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$ be the forgetful functor. Suppose we are given a morphism $(f,u): (C,X) \rightarrow (D,Y)$ in the category $\int ^{\operatorname{\mathcal{C}}} \mathscr {F}$, so that $f: C \rightarrow D$ is a morphism in the category $\operatorname{\mathcal{C}}$ and $u: X \rightarrow \mathscr {F}(f)(Y)$ is a morphism in the category $\mathscr {F}(C)$. For any other object $X' \in \mathscr {F}(C)$, postcomposition with $(f,u)$ determines a function

\[ \theta : \operatorname{Hom}_{ (\int ^{\operatorname{\mathcal{C}}} \mathscr {F})_{C} }( X', X ) \rightarrow \operatorname{Hom}_{ \int ^{\operatorname{\mathcal{C}}} \mathscr {F} }( (C,X'), (D,Y) ) \times _{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D) } \{ f\} . \]

Using the assumption that $\mathscr {F}$ is unitary (together with Remark 5.6.1.12), we can identify $\theta $ with the map $\operatorname{Hom}_{\mathscr {F}(C)}( X', X) \rightarrow \operatorname{Hom}_{\mathscr {F}(C) }( X', \mathscr {F}(f)(Y) )$ given by postcomposition with $u$. It follows that $(f,u)$ is a locally $U$-cartesian morphism of the category $\int ^{\operatorname{\mathcal{C}}} \mathscr {F}$ if and only if $u$ is an isomorphism in the category $\mathscr {F}(C)$.

Definition 9.10.3.8. Let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a functor between categories. We say that $U$ is a locally cartesian fibration if it satisfies the following condition:

  • For every object $Y$ of the category $\operatorname{\mathcal{D}}$ and every morphism $\overline{f}: \overline{X} \rightarrow U(Y)$ in the category $\operatorname{\mathcal{C}}$, there exists a pair $(X, f)$ where $X$ is an object of $\operatorname{\mathcal{D}}$ satisfying $U(X) = \overline{X}$ and $f: X \rightarrow Y$ is a locally $U$-cartesian morphism of $\operatorname{\mathcal{D}}$ satisfying $U(f) = \overline{f}$.

We say that $U$ is a locally cocartesian fibration if it satisfies the following dual condition:

  • For every object $X$ of the category $\operatorname{\mathcal{D}}$ and every morphism $\overline{f}: U(X) \rightarrow \overline{Y}$ in the category $\operatorname{\mathcal{C}}$, there exists a pair $(Y, f)$ where $X$ is an object of $\operatorname{\mathcal{D}}$ satisfying $U(X) = \overline{X}$ and $f: X \rightarrow Y$ is a locally $U$-cocartesian morphism of $\operatorname{\mathcal{D}}$ satisfying $U(f) = \overline{f}$.

Example 9.10.3.9. Let $q: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between ordinary categories. Then $q$ is a locally cartesian fibration (in the sense of Definition 9.10.3.8) if and only if the induced morphism of simplicial sets $\operatorname{N}_{\bullet }(q): \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}})$ is a locally cartesian fibration (in the sense of Definition 9.10.4.2). Similarly, $q$ is a locally cocartesian fibration if and only if $\operatorname{N}_{\bullet }(q)$ is a cocartesian fibration of simplicial sets. See Corollary 5.1.2.2.

Example 9.10.3.10. Let $\operatorname{\mathcal{C}}$ be a category and let $\mathscr {F}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \mathbf{Cat}$ be a unitary lax functor. Then the forgetful functor $\int ^{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$ is a locally cartesian fibration (see Example 9.10.3.7). Similarly, if $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \mathbf{Cat}^{\operatorname{c}}$ is a unitary lax functor, then the forgetful functor $\int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$ is a locally cocartesian fibration.

Example 9.10.3.11. Let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a functor between categories. If $U$ is a cartesian fibration, then it is a locally cartesian fibration. If $U$ is a cocartesian fibration, then it is a locally cocartesian fibration.

We now prove a more refined version of Corollary 5.6.1.16.

Proposition 9.10.3.12. Let $\operatorname{\mathcal{C}}$ be a category. Then:

  • For every unitary lax functor of $2$-categories $\mathscr {F}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \mathbf{Cat}$, the forgetful functor $U: \int ^{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$ is a locally cartesian fibration of categories. It is a cartesian fibration if and only if $\mathscr {F}$ is a functor.

  • For every unitary lax functor of $2$-categories $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \mathbf{Cat}^{\operatorname{c}}$, the forgetful functor $U: \int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$ is a locally cocartesian fibration of categories. It is a cocartesian fibration if and only if $\mathscr {F}$ is a functor.

Proof. We will prove the first assertion; the proof of the second is similar. Let $\mathscr {F}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \mathbf{Cat}$ be a unitary lax functor of $2$-categories. Suppose we are given an object $(D,Y)$ of the Grothendieck construction $\int ^{\operatorname{\mathcal{C}}} \mathscr {F}$, so that $D$ is an object of the category $\operatorname{\mathcal{C}}$ and $Y$ is an object of the category $\mathscr {F}(D)$. For every morphism $f: C \rightarrow D$ in the category $\operatorname{\mathcal{C}}$, we can choose an isomorphism $u: X \rightarrow \mathscr {F}(f)(Y)$ in the category $\mathscr {F}(C)$ (for example, we can take $X = \mathscr {F}(f)(Y)$ and $u$ to be the identity morphism). Then $(f,u)$ is a morphism from $(C,X)$ to $(D,Y)$ in the category $\int ^{\operatorname{\mathcal{C}}} \mathscr {F}$, and our assumption that $\mathscr {F}$ is unitary guarantees that $(f,u)$ is locally $U$-cartesian (Example 9.10.3.7). This proves that $U$ is a locally cartesian fibration of categories. Suppose that we are given another locally $U$-cartesian morphism $(g, v): (D,Y) \rightarrow (E,Z)$ in $\int ^{\operatorname{\mathcal{C}}} \mathscr {F}$, so that $g: D \rightarrow E$ is a morphism in $\operatorname{\mathcal{C}}$ and $v: Y \simeq \mathscr {F}(g)(Z)$ is an isomorphism in the category $\mathscr {F}(D)$ (Example 9.10.3.7). The composition $(g,v) \circ (f,u)$ can then be identified with $(g \circ f, w)$, where $w$ is given by the composition

\[ X \xrightarrow {u} \mathscr {F}(f)(Y) \xrightarrow { \mathscr {F}(f)(v) } (\mathscr {F}(f) \circ \mathscr {F}(g))(Z) \xrightarrow { \mu _{f,g}(Z)} \mathscr {F}(g \circ f)(Z); \]

here $\mu _{f,g}$ denotes the composition constraint for the lax functor $\mathscr {F}$. Invoking Example 9.10.3.7 again, we conclude that $(g,v) \circ (f,u)$ is locally $U$-cartesian if and only if $\mu _{f,g}(Z)$ is an isomorphism. Allowing $f$, $g$, and $Z$ to vary (and invoking the criterion of Proposition 9.10.3.13), we conclude that $U$ is a cartesian fibration if and only if each of the composition constraints $\mu _{f,g}$ is an isomorphism: that is, if and only if the lax functor $\mathscr {F}$ is a functor. $\square$

Proposition 9.10.3.13. Let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a functor between categories. The following conditions are equivalent:

$(1)$

The functor $U$ is a cartesian fibration of categories.

$(2)$

The functor $U$ is a locally cartesian fibration of categories and every locally $U$-cartesian morphism of $\operatorname{\mathcal{D}}$ is $U$-cartesian.

$(3)$

The functor $U$ is a locally cartesian fibration of categories and the collection of locally $U$-cartesian morphisms in $\operatorname{\mathcal{D}}$ is closed under composition.

Proof. The implication $(2) \Rightarrow (1)$ is immediate from the definition. We now show that $(1) \Rightarrow (2)$. Assume that $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ is a cartesian fibration of categories. Then $U$ is also a locally cartesian fibration (Example 9.10.3.11). Let $f: X \rightarrow Y$ be a locally $U$-cartesian morphism in $\operatorname{\mathcal{D}}$. Since $U$ is a cartesian fibration, we can choose a $U$-cartesian morphism $f': X' \rightarrow Y$ satisfying $U(X') = U(X)$ and $U(f') = U(f)$. Using our assumption that $f$ is locally $U$-cartesian, we deduce that $f'$ factors uniquely as a composition $f \circ e$, where $e: X' \rightarrow X$ has the property that $U(e) = \operatorname{id}_{ U(X)}$. Since $f'$ is locally $U$-cartesian, the morphism $e$ is an isomorphism (Remark 9.10.3.6). Applying Remark ***, we deduce that $f = f' \circ e^{-1}$ is also $U$-cartesian.

The implication $(2) \Rightarrow (3)$ follows from Remark ***. We will complete the proof by showing that $(3) \Rightarrow (2)$. Assume that $U$ is a locally cartesian fibration of categories and that the collection of locally $U$-cartesian morphisms of $\operatorname{\mathcal{D}}$ is closed under composition. Let $g: X \rightarrow Y$ be a locally $U$-cartesian morphism in $\operatorname{\mathcal{D}}$; we wish to show that $g$ is $U$-cartesian. Fix an object $W \in \operatorname{\mathcal{D}}$ and a morphism $\overline{f}: U(W) \rightarrow U(X)$, so that postcomposition with $g$ induces a function

\[ \theta : \{ f \in \operatorname{Hom}_{\operatorname{\mathcal{D}}}(W,X): U(f) = \overline{f} \} \rightarrow \{ h \in \operatorname{Hom}_{\operatorname{\mathcal{D}}}(W,Y): U(h) = U(g) \circ \overline{f} \} . \]

We wish to show that $\theta $ is a bijection. To prove this, we invoke our assumption that $U$ is a locally cartesian fibration to choose a locally $U$-cartesian morphism $f': W' \rightarrow X$ in $\operatorname{\mathcal{D}}$ satisfying $U(W') = U(W)$ and $U(f') = \overline{f}$. We then have a commutative diagram

\[ \xymatrix@R =50pt@C=-50pt{ & \{ e \in \operatorname{Hom}_{\operatorname{\mathcal{D}}}(W,W'): U(e) = \operatorname{id}_{U(W)} \} \ar [dl]^{f' \circ } \ar [dr]_{g \circ f' \circ } & \\ \{ f \in \operatorname{Hom}_{\operatorname{\mathcal{D}}}(W,X): U(f) = \overline{f} \} \ar [rr]^-{\theta } & & \{ h \in \operatorname{Hom}_{\operatorname{\mathcal{D}}}(W,Y): U(h) = U(g) \circ \overline{f} \} . } \]

Since $f'$ and $g \circ f'$ are locally $U$-cartesian, the vertical maps in this diagram are bijections. It follows that $\theta $ is also a bijection, as desired. $\square$