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5.6.5 Application: Fibrations of Ordinary Categories

Let $\operatorname{\mathcal{C}}$ be a category, let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \mathbf{Cat}$ be a functor of $2$-categories, and let $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ denote the category of elements of $\mathscr {F}$ (Definition 5.5.2.1). It follows from Corollary 5.5.2.13 that the forgetful functor $\int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$ is a cocartesian fibration of categories. In this section, we will show that (up to isomorphism) every cocartesian fibration arises in this way:

Proposition 5.6.5.1 (Grothendieck). Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be functor between categories. The the following conditions are equivalent:

  • The functor $U$ is a cocartesian fibration and each fiber of $U$ is a small category.

  • There exists a functor of $2$-categories $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \mathbf{Cat}$ and an isomorphism $\int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{E}}$ whose composition with $U$ coincides with the forgetful functor $\int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$.

Let $\mathbf{Gpd} \subseteq \mathbf{Cat}$ denote the full subcategory spanned by the groupoids.

Corollary 5.6.5.2. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a functor between categories. The following conditions are equivalent:

  • The functor $U$ is an opfibration in groupoids (Variant 4.2.2.4) and each fiber of $U$ is a small groupoid.

  • There exists a functor of $2$-categories $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \mathbf{Gpd}$ and an isomorphism of categories $\int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{E}}$ which carries $U$ to the forgetful functor $\int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$.

We will deduce Proposition 5.6.5.1 from a more general assertion, which classifies cocartesian inner covering maps between simplicial sets.

Proposition 5.6.5.3. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. The following conditions are equivalent:

$(1)$

The morphism $U$ an inner covering map (Definition 4.1.5.1), a cocartesian fibration and each fiber of $U$ is small.

$(2)$

There exists morphism of simplicial sets $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }^{\operatorname{D}}( \operatorname{Pith}(\mathbf{Cat}) ) \subseteq \operatorname{\mathcal{QC}}$ and an isomorphism $G: \operatorname{\mathcal{E}}\simeq \int _{\operatorname{\mathcal{C}}} \mathscr {F}$ in the category $(\operatorname{Set_{\Delta }})_{/\operatorname{\mathcal{C}}}$.

Proof. The implication $(2) \Rightarrow (1)$ follows from Corollary 5.5.5.4 and Proposition 5.5.4.2. For each vertex $C \in \operatorname{\mathcal{C}}$, our assumption that $U$ is an inner covering map guarantees that the fiber $\{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is isomorphic to the nerve of a (small) category $\mathscr {F}_0(C)$ (Example 4.1.5.3). Let $\operatorname{\mathcal{C}}_0$ be the $0$-skeleton of $\operatorname{\mathcal{C}}$, so that the construction $C \mapsto \mathscr {F}_0(C)$ determines a morphism of simplicial sets $\mathscr {F}_0: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{N}_{\bullet }^{\operatorname{D}}( \operatorname{Pith}(\mathbf{Cat}) )$. Let $\operatorname{\mathcal{E}}_0$ denote the inverse image $\operatorname{\mathcal{C}}_0 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$, so that Proposition 5.5.5.3 supplies an isomorphism of simplicial sets $G_0: \operatorname{\mathcal{E}}_0 \simeq \int _{\operatorname{\mathcal{C}}_0} \mathscr {F}_0$. In particular, $G_0$ is an equivalence of cocartesian fibrations over $\operatorname{\mathcal{C}}_0$. Invoking Theorem 5.6.4.9, we can extend $\mathscr {F}_0$ to a diagram $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }^{\operatorname{D}}( \operatorname{Pith}( \mathbf{Cat} ) )$ and $G_0$ to a morphism of simplicial sets $G: \operatorname{\mathcal{E}}\rightarrow \int _{\operatorname{\mathcal{C}}} \mathscr {F}$ which is an equivalence of cocartesian fibrations over $\operatorname{\mathcal{C}}$. We will complete the proof by showing that $G$ is an isomorphism of simplicial sets. To prove this, it will suffice to show that for every simplex $G_{\sigma }: \Delta ^{n} \rightarrow \operatorname{\mathcal{C}}$, the induced map

\[ G_{\sigma }: \Delta ^{n} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \Delta ^{n} \times _{\operatorname{\mathcal{C}}} \int _{\operatorname{\mathcal{C}}} \mathscr {F} \]

is an isomorphism of simplicial sets. Replacing $U$ by the projection map $\Delta ^{n} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \Delta ^ n$, we are reduced to proving that $G$ is an isomorphism under the additional assumption that $\operatorname{\mathcal{C}}= \Delta ^ n$ is a standard simplex. Since $U$ and the projection map $\int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$ are inner covering maps, the simplicial sets $\operatorname{\mathcal{E}}$ and $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ are isomorphic to the nerves of their homotopy categories $\mathrm{h} \mathit{\operatorname{\mathcal{E}}}$ and $\mathrm{h} \mathit{ \int _{\operatorname{\mathcal{C}}} \mathscr {F} }$, respectively; it will therefore suffice to show that the functor of ordinary categories $\mathrm{h} \mathit{G}: \mathrm{h} \mathit{\operatorname{\mathcal{E}}} \rightarrow \mathrm{h} \mathit{ \int _{\operatorname{\mathcal{C}}} \mathscr {F} }$ is an isomorphism. Our assumption that $G$ is an equivalence of cocartesian fibrations over $\operatorname{\mathcal{C}}= \Delta ^ n$ guarantees that it is an equivalence of $\infty $-categories (Corollary 5.6.2.8), so that $\mathrm{h} \mathit{G}$ is an equivalence of ordinary categories. It will therefore suffice to show that the functor $\mathrm{h} \mathit{G}$ is bijective on objects: that is, that the morphism $G$ is bijective on vertices. This is clear, since the morphism $G_0 = G|_{\operatorname{\mathcal{E}}_0}$ is an isomorphism. $\square$

Proof of Proposition 5.6.5.1. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of categories whose fibers are small; we will show that there exists a functor of $2$-categories $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \mathbf{Cat}$ and an isomorphism of categories $\int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{E}}$ which is compatible with $U$ (the converse implication follows from Corollary 5.5.2.13). Note that the map $\operatorname{N}_{\bullet }(U): \operatorname{N}_{\bullet }(\operatorname{\mathcal{E}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is a cocartesian fibration of simplicial sets (Example 5.1.4.2) and an inner covering map (Proposition 4.1.5.10). By virtue of Proposition 5.6.5.3, there exists a morphism of simplicial sets $\mathscr {F}': \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{D}}( \operatorname{Pith}( \mathbf{Cat}) )$ and an isomorphism of simplicial sets $V: \int _{\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})} \mathscr {F} \simeq \operatorname{N}_{\bullet }(\operatorname{\mathcal{E}})$ which is compatible with $\operatorname{N}_{\bullet }(U)$. By virtue of Theorem 2.3.4.1 (and Corollary 2.3.4.5), we have $\mathscr {F}' = \operatorname{N}_{\bullet }^{\operatorname{D}}( \mathscr {F} )$ for a unique functor of $2$-categories $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \mathbf{Cat}$. In this case, we can use Proposition 5.5.5.3 to identify $\int _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})} \mathscr {F}'$ with the nerve of the ordinary category of elements $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$. Under this identification, $V$ corresponds to the nerve of an isomorphism of categories $\int _{\operatorname{\mathcal{C}}} \mathscr {F}' \simeq \operatorname{\mathcal{E}}$ which is compatible with $U$. $\square$