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5.6 Classification of Cocartesian Fibrations

Let $\operatorname{\mathcal{QC}}$ denote the $\infty $-category of (small) $\infty $-categories (Construction 5.4.4.1). For every functor of $\infty $-categories $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$, let $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ denote the $\infty $-category of elements of $\mathscr {F}$ (Definition 5.5.4.4). By virtue of Proposition 5.5.4.2, the forgetful functor $\int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$ is a cocartesian fibration of $\infty $-categories. Moreover, if two functors $\mathscr {F}, \mathscr {F}': \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$ 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 (Proposition 5.5.4.19). 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 $\operatorname{\mathcal{C}}$. Our goal in this section is to show that, modulo set-theoretic technicalities, this function is a bijection.

Definition 5.6.0.1. 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 5.6.0.2 (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 5.6.0.3. In the statement of Theorem 5.6.0.2, 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 5.5.4.18).

Remark 5.6.0.4. We can summarize Theorem 5.6.0.2 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 5.6.0.5. We will later show that the bijection of Theorem 5.6.0.2 can be upgraded to an equivalence of $\infty $-categories; see Theorem .

Corollary 5.6.0.6. 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.

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 5.6.2.1). In §5.6.2, we show this condition guarantees that $\mathscr {F}$ is a refinement of the homotopy transport representation introduced in §5.2 (see Remark 5.6.2.6). Theorem 5.6.0.2 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 §5.6.5, 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 5.6.5.1). We prove the contractibility of $\operatorname{TW}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}})$ in §5.6.6: 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 5.4.6.14).

Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of small $\infty $-categories. Theorem 5.6.0.2 guarantees that there exists an equivalence of $\infty $-categories $\alpha : \operatorname{\mathcal{E}}\rightarrow \int _{\operatorname{\mathcal{C}}} \mathscr {F}$ for some functor $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$. Beware that we generally cannot arrange that $\alpha $ is an isomorphism of simplicial sets, even in the special case $\operatorname{\mathcal{C}}= \Delta ^0$ (see Example 5.5.4.16). However, there is one case in which we can do better. In §5.6.3 we show that if $U$ is a cocartesian inner covering map, then there exists an isomorphism of simplicial sets $\operatorname{\mathcal{E}}\simeq \int _{\operatorname{\mathcal{C}}} \mathscr {F}$, where $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$ takes values in the full subcategory of $\operatorname{\mathcal{QC}}$ spanned by (the nerves of) ordinary categories (see Proposition 5.6.3.3). Specializing to the case where $\operatorname{\mathcal{C}}$ is (the nerve of) an ordinary category, this recovers a classical result of Grothendieck: every cocartesian fibration between ordinary categories can be obtained from the category of elements construction studied in §5.5.2 (Proposition 5.6.3.1). Another important special case occurs when $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is a left covering map of $\infty $-categories: in this case, $\operatorname{\mathcal{E}}$ can be recovered (up to isomorphism) as the $\infty $-category of elements of a set-valued functor $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }(\operatorname{Set})$ (Corollary 5.6.1.7). In §5.6.1 we show more generally that for any simplicial set $\operatorname{\mathcal{C}}$, the construction $\mathscr {F} \mapsto \int _{\operatorname{\mathcal{C}}} \mathscr {F}$ induces an equivalence of ordinary categories

\[ \operatorname{Fun}( \mathrm{h} \mathit{\operatorname{\mathcal{C}}}, \operatorname{Set}) \simeq \{ \textnormal{Left covering maps $\operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$} \} ; \]

see Corollary 5.6.1.6. When specialized to the case where $\operatorname{\mathcal{C}}$ is a Kan complex, this recovers the classical dictionary relating covering spaces with representations of the fundamental group(oid) (Corollary 5.6.1.10).

In the statement of Theorem 5.6.0.2, 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 §5.6.4, 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 5.6.4.3). From this, we deduce that every cocartesian fibration of simplicial sets is an isofibration (Corollary 5.6.4.5), and that the collection of categorical equivalences of simplicial sets is stable under the formation of pullback by cocartesian fibrations (Corollary 5.6.4.6).

Structure

  • Subsection 5.6.1: Covering Space Theory
  • Subsection 5.6.2: The Covariant Transport Representation
  • Subsection 5.6.3: Application: Fibrations of Ordinary Categories
  • Subsection 5.6.4: Application: Extending Cocartesian Fibrations
  • Subsection 5.6.5: Transport Witnesses
  • Subsection 5.6.6: Uniqueness of the Transport Representation