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5.6 Classification of Cocartesian Fibrations

Our goal in this section is to address the following:

In §5.2.7, we gave an answer to Question 5.6.0.1 under the assumption that $U$ is a left covering map. In this case, the construction $C \mapsto \operatorname{\mathcal{E}}_{C}$ determines a functor $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \operatorname{Set}$. Moreover, the $\infty $-category $\operatorname{\mathcal{E}}$ can be recovered (up to isomorphism) as the fiber product $\operatorname{\mathcal{C}}\times _{ \operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{\mathcal{C}}} ) } \int _{ \mathrm{h} \mathit{\operatorname{\mathcal{C}}} } \operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ (Proposition 5.2.7.2), where the second factor denotes the category of elements of the set-valued functor $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ (Construction 5.2.6.1).

In the setting of classical category theory, Grothendieck gave a complete answer to Question 5.6.0.1. Let $\operatorname{\mathcal{C}}$ be an ordinary category, and let $\mathbf{Cat}$ denote the (strict) $2$-category of small categories (Example 2.2.0.4), and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \mathbf{Cat}$ be a functor of $2$-categories. In §5.6.1, we introduce a category $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ whose objects are pairs $(C,X)$ where $C$ is an object of $\operatorname{\mathcal{C}}$ and $X$ is an object of the category $\mathscr {F}(C)$. We will refer to $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ as the category of elements of the functor $\mathscr {F}$ (Definition 5.6.1.1). The category $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ is equipped with a cocartesian fibration $U: \int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$, given on objects by the construction $(C,X) \mapsto C$. Moreover, the cocartesian fibration $U$ is essentially small: that is, for each object $C \in \operatorname{\mathcal{C}}$, the fiber $U^{-1} \{ C\} $ is an essentially small category (since it is equivalent to the small category $\mathscr {F}(C)$). In [MR0217088], Grothendieck showed that, up to isomorphism, every essentially small cocartesian fibration can be obtained in this way (Corollary 5.6.5.21).

In §5.6.2, we introduce an $\infty $-categorical counterpart of the preceding construction. Let $\operatorname{\mathcal{QC}}_{\operatorname{Obj}}$ denote the $\infty $-category of Construction 5.5.6.10, whose objects are pairs $(\operatorname{\mathcal{A}},X)$ where $\operatorname{\mathcal{A}}$ is a (small) $\infty $-category and $X$ is an object of $\operatorname{\mathcal{A}}$. For every morphism of simplicial sets $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$, we let $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ denote the fiber product $\operatorname{\mathcal{C}}\times _{ \operatorname{\mathcal{QC}}} \operatorname{\mathcal{QC}}_{\operatorname{Obj}}$. By construction, vertices of $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ can be identified with pairs $(C,X)$, where $C$ is a vertex of $\operatorname{\mathcal{C}}$ and $X$ is an object of the $\infty $-category $\mathscr {F}(C)$. Projection onto the first factor determines a cocartesian fibration of simplicial sets $U: \int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$, given on objects by the construction $(C,X) \mapsto C$ (Proposition 5.6.2.2). In particular, if $\operatorname{\mathcal{C}}$ is an $\infty $-category, then the simplicial set $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ is also an $\infty $-category, which we refer to as the $\infty $-category of elements of $\mathscr {F}$ (Definition 5.6.2.4). This construction has the following features:

• Let $\operatorname{\mathcal{C}}$ be an ordinary category and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \mathbf{Cat}$ be a functor of $2$-categories, so that the construction $C \mapsto \operatorname{N}_{\bullet }(\mathscr {F}(C) )$ determines a functor of $\infty $-categories $\operatorname{N}_{\bullet }(\mathscr {F}): \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{QC}}$. In §5.6.3, we construct a canonical isomorphism of simplicial sets

  \[ \int _{\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})} \operatorname{N}_{\bullet }(\mathscr {F}) \simeq \operatorname{N}_{\bullet }( \int _{\operatorname{\mathcal{C}}} \mathscr {F} ) \]

  where the left hand side is the $\infty $-category of elements of the functor $\operatorname{N}_{\bullet }(\mathscr {F})$ and the right hand side is the nerve of the ordinary category of elements of the functor $\mathscr {F}$ (Proposition 5.6.3.4). Consequently, we can view the $\infty $-category of elements construction as a generalization of the classical category of elements construction.

• Let $\operatorname{\mathcal{C}}$ be an ordinary category and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ be a functor of ordinary categories. Passing to the homotopy coherent nerve, we obtain a functor of $\infty $-categories $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\mathscr {F} ): \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{QC}}$. In §5.6.4, we construct a comparison map

  \[ \theta : \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \rightarrow \int _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \operatorname{N}_{\bullet }^{\operatorname{hc}}( \mathscr {F} ) \]

  and show that it is an equivalence of $\infty $-categories (Proposition 5.6.4.8). In other words, we can think of the $\infty $-category of elements as a variant of the weighted nerve construction, which can be applied to homotopy coherent diagrams which are not strictly commutative. Beware that $\theta $ is usually not an isomorphism of simplicial sets.

It is not difficult to show that if diagrams $\mathscr {F}, \mathscr {F}': \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$ are isomorphic (as objects of the $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{QC}})$), then the cocartesian fibrations

are equivalent (see Proposition 5.6.2.21). It follows that the construction $\mathscr {F} \mapsto \int _{\operatorname{\mathcal{C}}} \mathscr {F}$ determines a function from the collection of isomorphism classes in the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{QC}})$ to the collection of equivalence classes of cocartesian fibrations over $\operatorname{\mathcal{C}}$. We will show that, modulo set-theoretic technicalities, this function is a bijection.

Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets. We will say that a diagram $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$ is a covariant transport representation for $U$ if there exists an equivalence $\alpha : \operatorname{\mathcal{E}}\rightarrow \int _{\operatorname{\mathcal{C}}} \mathscr {F}$ of cocartesian fibrations over $\operatorname{\mathcal{C}}$ (Definition 5.6.5.1). Theorem 5.6.0.2 asserts that if $U$ is essentially small, then there exists a covariant transport representation for $U$ which is uniquely determined up to isomorphism (as an object of the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{QC}})$). In fact, we will prove something stronger: the covariant transport representation of $U$ is unique up to a contractible space of choices. In §5.6.8, we formulate this statement more precisely by introducing a Kan complex $\operatorname{TW}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}})$ whose vertices are pairs $(\mathscr {F}, \alpha )$ as above (see Notation 5.6.8.1). We prove the contractibility of $\operatorname{TW}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}})$ in §5.6.9: as we will see, it is a formal consequence of the fact that the homotopy transport representation of the cocartesian fibration $V: \operatorname{\mathcal{QC}}_{\operatorname{Obj}} \rightarrow \operatorname{\mathcal{QC}}$ determines an equivalence of $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched categories $\operatorname{hTr}_{ \operatorname{\mathcal{QC}}_{\operatorname{Obj}} / \operatorname{\mathcal{QC}}}: \mathrm{h} \mathit{\operatorname{\mathcal{QC}}} \rightarrow \mathrm{h} \mathit{\operatorname{QCat}}$ (Proposition 5.5.6.14).

Structure

• Subsection 5.6.1: Elements of Category-Valued Functors
• Subsection 5.6.2: Elements of $\operatorname{\mathcal{QC}}$-Valued Functors
• Subsection 5.6.3: Comparison with the Category of Elements
• Subsection 5.6.4: Comparison with the Weighted Nerve
• Subsection 5.6.5: The Universality Theorem
• Subsection 5.6.6: Application: Corepresentable Functors
• Subsection 5.6.7: Application: Extending Cocartesian Fibrations
• Subsection 5.6.8: Transport Witnesses
• Subsection 5.6.9: Proof of the Universality Theorem