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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. The primary goal in this section is to show that every cocartesian fibration between small $\infty $-categories can be obtained in this way. To formulate this result more precisely, it will be convenient to introduce some terminology.

Definition 5.6.0.1. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of $\infty $-categories. We will say that a functor $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$ is a transport representation of $U$ if there exists an equivalence of $\infty $-categories $\operatorname{\mathcal{E}}\rightarrow \int _{\operatorname{\mathcal{C}}} \mathscr {F}$ for which the composition $\operatorname{\mathcal{E}}\rightarrow \int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$ is equal to $U$.

We can formulate a preliminary version of our main result as follows:

Theorem 5.6.0.2 (Universality Theorem: Preliminary Version). Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration between $\infty $-categories. Assume that, 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 (Definition 5.6.3.11). Then $U$ admits a covariant transport representation $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$, which is uniquely determined up to isomorphism (as an object of the functor $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{QC}})$).

Warning 5.6.0.3. In the statement of Theorem 5.6.0.2, the essential smallness assumption cannot be omitted. If a cocartesian fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ admits a covariant transport representation $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$, then each fiber $\operatorname{\mathcal{E}}_{C}$ is equivalent to the small $\infty $-category $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}(C) \in \operatorname{\mathcal{QC}}$.

Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of $\infty $-categories, and let $U': \operatorname{\mathcal{E}}' \rightarrow \operatorname{\mathcal{C}}$ be another cocartesian fibration having the same target. We will say that $U$ and $U'$ are equivalent as cocartesian fibrations over $\operatorname{\mathcal{C}}$ if there exists an equivalence of $\infty $-categories $F: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{E}}'$ satisfying $U = U' \circ F$ (this is a special case of the more general notion of equivalence of inner fibrations, which we study in §5.6.2). It is not difficult to show that the construction $\mathscr {F} \mapsto \int _{\operatorname{\mathcal{C}}} \mathscr {F}$ carries isomorphic objects of $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{QC}})$ to equivalent inner fibrations over $\operatorname{\mathcal{C}}$ (Corollary 5.6.3.3). In §5.6.3, we reformulate a slightly stronger version of Theorem 5.6.0.2, which characterizes the $\infty $-category $\operatorname{\mathcal{QC}}$ up to equivalence: for every $\infty $-category $\operatorname{\mathcal{C}}$, the construction $\mathscr {F} \mapsto \int _{\operatorname{\mathcal{C}}} \mathscr {F}$ induces a bijection from $\pi _0( \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{QC}})^{\simeq } )$ to the collection of equivalence classes of cocartesian fibrations $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ having essentially small fibers (Corollary 5.6.3.12). We will deduce this from an even stronger relative version of Theorem 5.6.0.2 (Theorem 5.6.4.9), which we formulate in §5.6.4 and prove in §5.6.7.

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 $G: \operatorname{\mathcal{E}}\rightarrow \int _{\operatorname{\mathcal{C}}} \mathscr {F}$ for some functor $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$. Beware that we generally cannot arrange that $G$ is an isomorphism of simplicial sets, even in the special case $\operatorname{\mathcal{C}}= \Delta ^0$ (see Example 5.5.4.17). However, there is one case in which we can do better. In §5.6.5 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 5.6.5.3). 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 5.6.5.1). 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 of with representations of the fundamental group(oid) (Corollary 5.6.1.10).

In the statement of Theorem 5.6.0.2, we have assumed for the sake of simplicity that $\operatorname{\mathcal{C}}$ is an $\infty $-category. However, this is not necessary: if $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is any cocartesian fibration of simplicial sets (having essentially small fibers), then there exists a diagram $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$ and an equivalence $G: \operatorname{\mathcal{E}}\rightarrow \int _{\operatorname{\mathcal{C}}} \mathscr {F}$ of cocartesian fibrations over $\operatorname{\mathcal{C}}$. This additional generality will play an essential role in our proof (which will require us to analyze the restriction of $U$ to simplicial subsets of $\operatorname{\mathcal{C}}$). However, it also has a number of pleasant consequence: 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 §5.6.6, 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 5.6.6.3). From this, we deduce that every cocartesian fibration of simplicial sets is an isofibration (Corollary 5.6.6.4), and that the collection of categorical equivalences of simplicial sets is stable under the formation of pullback by cocartesian fibrations (Corollary 5.6.6.5).

Structure

  • Subsection 5.6.1: Covering Space Theory
  • Subsection 5.6.2: Equivalence of Fibrations
  • Subsection 5.6.3: Universal Fibrations
  • Subsection 5.6.4: The Covariant Transport Representation
  • Subsection 5.6.5: Application: Fibrations of Ordinary Categories
  • Subsection 5.6.6: Application: Extending Cocartesian Fibrations
  • Subsection 5.6.7: Proof of the Universality Theorem