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5.6 Classification of Cocartesian Fibrations

Our goal in this section is to address the following:

Question 5.6.0.1. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of $\infty $-categories. To what extent can $U$ be recovered from the collection of $\infty $-categories $\{ \operatorname{\mathcal{E}}_{C} \} _{C \in \operatorname{\mathcal{C}}}$?

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 $(C,X) \mapsto C$. In [MR0217088], Grothendieck showed that, up to isomorphism, every cocartesian fibration between (small) categories can be obtained in this way (Corollary 5.6.5.19).

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 §5.6.4.8, 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

\[ \int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}\quad \quad \int _{\operatorname{\mathcal{C}}} \mathscr {F}' \rightarrow \operatorname{\mathcal{C}} \]

are equivalent (see Proposition 5.6.2.19). 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.

Definition 5.6.0.2. Let $\operatorname{\mathcal{E}}$ be an $\infty $-category. We say that $\operatorname{\mathcal{E}}$ is essentially small if there exists an equivalence of $\infty $-categories $\operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{E}}'$, where $\operatorname{\mathcal{E}}'$ is a small $\infty $-category.

Theorem 5.6.0.3 (Universality Theorem). Let $\operatorname{\mathcal{C}}$ be a simplicial set. Then the construction

\[ ( \mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}) \mapsto (\int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}) \]

induces a bijection from $\pi _0( \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{QC}})^{\simeq } )$ to the set of equivalence classes of cocartesian fibrations $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ having the following property: for every object $C \in \operatorname{\mathcal{C}}$, the $\infty $-category $\operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is essentially small.

Warning 5.6.0.4. In the statement of Theorem 5.6.0.3, the essential smallness assumption cannot be omitted: if the cocartesian fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is equivalent to $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ for some diagram $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$, then each fiber $\operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is equivalent to the small $\infty $-category $\mathscr {F}(C)$ (see Example 5.6.2.18).

Remark 5.6.0.5. We can summarize Theorem 5.6.0.3 more informally by saying that the projection map $V: \operatorname{\mathcal{QC}}_{\operatorname{Obj}} \rightarrow \operatorname{\mathcal{QC}}$ is universal among cocartesian fibrations having essentially small fibers. Note that this property characterizes the $\infty $-category $\operatorname{\mathcal{QC}}$ (and the cocartesian fibration $V$) up to equivalence.

Remark 5.6.0.6. We will later show that the bijection of Theorem 5.6.0.3 can be upgraded to an equivalence of $\infty $-categories; see Theorem .

Corollary 5.6.0.7. Let $\operatorname{\mathcal{C}}$ be a simplicial set. Then the construction

\[ ( \mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}) \mapsto (\int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}) \]

induces a bijection from $\pi _0( \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}})^{\simeq } )$ to the set of equivalence classes of left fibrations $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ having the following property: for every object $C \in \operatorname{\mathcal{C}}$, the Kan complex $\operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is essentially small.

Example 5.6.0.8. Let $\operatorname{\mathcal{C}}$ be a small $\infty $-category and let $X$ be an object of $\operatorname{\mathcal{C}}$. It follows from Corollary 5.6.0.7 that there is an essentially unique functor $h^{X}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ for $\int _{\operatorname{\mathcal{C}}} h^{X}$ is equivalent to $\operatorname{\mathcal{C}}_{X/}$ as left fibrations over $\operatorname{\mathcal{C}}$. We will refer to $h^{X}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ as the functor corepresented by $X$. For every object $Y \in \operatorname{\mathcal{C}}$, we have isomorphisms

\[ h^{X}(Y) \simeq \{ Y\} \times _{\operatorname{\mathcal{C}}} \int _{\operatorname{\mathcal{C}}} h^{X} \simeq \{ Y\} \times _{ \operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{X/} = \operatorname{Hom}^{\mathrm{L}}_{\operatorname{\mathcal{C}}}(X,Y) \simeq \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \]

in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$, depending functorially on $Y$. In §5.6.6, we will show that this property characterizes the functor $h^{X}$ up to isomorphism (Theorem 5.6.6.12).

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 5.6.0.3 asserts that if the cocartesian fibration $U$ has essentially small fibers, 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).

Remark 5.6.0.9. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration between (small) simplicial sets. We will denote the covariant transport representation of $U$ by $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$; it can be regarded as a homotopy coherent refinement of the homotopy transport representation $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ introduced in Construction 5.2.5.2 (see Remark 5.6.5.8 for a precise statement). We can summarize the situation with the following informal answer to Question 5.6.0.1:

  • For every cocartesian fibration between (small) $\infty $-categories $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$, the construction $C \mapsto \operatorname{\mathcal{E}}_{C}$ determines a functor of $\infty $-categories $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$. Moreover, we can recover $\operatorname{\mathcal{E}}$ (up to equivalence) as the $\infty $-category of elements $\int _{\operatorname{\mathcal{C}}} \operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$.

Remark 5.6.0.10. In the statement of Theorem 5.6.0.3, it is not necessary to assume that the simplicial set $\operatorname{\mathcal{C}}$ is an $\infty $-category. This additional generality will play an essential role in our proof (which will require us to analyze the restriction of the cocartesian fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ to simplicial subsets of $\operatorname{\mathcal{C}}$). Moreover, it has a number of pleasant consequences: since $\operatorname{\mathcal{QC}}$ is an $\infty $-category, it guarantees that every cocartesian fibration of simplicial sets is equivalent to the pullback of a cocartesian fibration between $\infty $-categories. In §5.6.7, we use this to prove a sharper statement: every cocartesian fibration of simplicial sets is isomorphic to the pullback of a cocartesian fibration between $\infty $-categories (Corollary 5.6.7.3). From this, we deduce that every cocartesian fibration of simplicial sets is an isofibration (Corollary 5.6.7.5), and that the collection of categorical equivalences of simplicial sets is stable under the formation of pullback by cocartesian fibrations (Corollary 5.6.7.6).

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