5 Fibrations of $\infty $-Categories

Let $\operatorname{ Ab }$ denote the category of abelian groups. For every commutative ring $A$, we let $\operatorname{Mod}_{A}(\operatorname{ Ab })$ denote the category of $A$-modules. Every homomorphism of commutative rings $u: A \rightarrow B$ determines a functor

\[ T_ u: \operatorname{Mod}_{A}( \operatorname{ Ab }) \rightarrow \operatorname{Mod}_{B}(\operatorname{ Ab }) \quad \quad T_ u(M) = B \otimes _{A} M, \]

which we will refer to as extension of scalars along $u$. One can summarize the situation informally by saying that there is a functor from commutative rings to (large) categories, which carries each commutative ring $A$ to the category $\operatorname{Mod}_{A}(\operatorname{ Ab })$ and each ring homomorphism $u: A \rightarrow B$ to the functor $T_ u$. However, we encounter the following subtleties:

$(1)$

Let $u: A \rightarrow B$ and $v: B \rightarrow C$ be homomorphisms of commutative rings. Then the diagram of categories

\[ \xymatrix@R =50pt@C=50pt{ & \operatorname{Mod}_{B}(\operatorname{ Ab }) \ar [dr]^{T_ v} & \\ \operatorname{Mod}_{A}(\operatorname{ Ab }) \ar [ur]^{T_ u} \ar [rr]^-{T_{vu}} & & \operatorname{Mod}_{C}(\operatorname{ Ab }) } \]

might not be strictly commutative. If $M$ is an $A$-module, one cannot reasonably expect $C \otimes _{A} M$ to be identical to the iterated tensor product $C \otimes _{B} (B \otimes _{A} M)$. Instead, there is a canonical isomorphism

\[ \mu _{v,u}(M): C \otimes _{B} (B \otimes _{A} M) \simeq C \otimes _{A} M, \]

which depends functorially on $M$, so that the collection $\{ \mu _{v,u}(M) \} _{M \in \operatorname{Mod}_{A}(\operatorname{ Ab }) }$ can be viewed as an isomorphism of functors $\mu _{v,u}: T_{v} \circ T_{u} \simeq T_{vu}$.

$(2)$

Let $A$ be a commutative ring, and let $\operatorname{id}_ A: A \rightarrow A$ be the identity map. Then the extension of scalars functor $T_{\operatorname{id}_ A}: \operatorname{Mod}_{A}(\operatorname{ Ab }) \rightarrow \operatorname{Mod}_{A}(\operatorname{ Ab })$ might not be equal to the identity functor $\operatorname{id}_{ \operatorname{Mod}_{A}(\operatorname{ Ab })}$. However, there is a natural isomorphism $\epsilon _{A}: \operatorname{id}_{ \operatorname{Mod}_{A}(\operatorname{ Ab }) } \simeq T_{ \operatorname{id}_{A} }$, which carries each $A$-module $M$ to the $A$-module isomorphism

\[ M \simeq A \otimes M \quad \quad x \mapsto 1 \otimes x. \]

Let $\operatorname{Cat}$ denote the ordinary category whose objects are categories (which, for the moment, we do not require to be small) and whose morphisms are functors. Because of the technical issues outlined above, the construction $A \mapsto \operatorname{Mod}_{A}(\operatorname{ Ab })$ cannot be viewed as a functor from the category of commutative rings to the category $\operatorname{Cat}$. However, this can be remedied using the language of $2$-categories. Recall that $\operatorname{Cat}$ can be realized as the underlying category of a (strict) $2$-category $\mathbf{Cat}$ (Example 2.2.0.4). The construction $A \mapsto \operatorname{Mod}_{A}(\operatorname{ Ab })$ can be promoted to a functor of $2$-categories

\[ \operatorname{Mod}_{\bullet }: \{ \textnormal{Commutative rings} \} \rightarrow \mathbf{Cat}, \]

whose composition and identity constraints are given by the natural isomorphisms $\mu _{v,u}: T_{v} \circ T_{u} \simeq T_{vu}$ and $\epsilon _{A}: \operatorname{id}_{ \operatorname{Mod}_{A}(\operatorname{ Ab }) } \simeq T_{ \operatorname{id}_{A} }$ described in $(1)$ and $(2)$ (see Definition 2.2.4.5).

It is often more convenient to encode the functoriality of the construction $A \mapsto \operatorname{Mod}_{A}(\operatorname{ Ab })$ in a different way. Let $\operatorname{\mathcal{C}}$ be an ordinary category. To every functor of $2$-categories $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \mathbf{Cat}$, one can associate a new category $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$, called the category of elements of $\mathscr {F}$ (Definition 5.6.1.1). By definition, objects of the category $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ are given by pairs $(C,X)$, where $C$ is an object of the category $\operatorname{\mathcal{C}}$ and $X$ is an object of the category $\mathscr {F}(C)$. The construction $(C,X) \mapsto C$ determines a forgetful functor $U: \int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$, whose fiber over an object $C \in \operatorname{\mathcal{C}}$ can be identified with the category $\mathscr {F}(C)$. Moreover, the functor $\mathscr {F}$ can be recovered (up to isomorphism) from the category $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ together with the functor $U$.

Passage from the data of the functor $\mathscr {F}$ to its category of elements $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ has several advantages. It can be somewhat cumbersome to specify a functor of $2$-categories $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \mathbf{Cat}$ explicitly: one must give not only the values of $\mathscr {F}$ on objects and morphisms of $\operatorname{\mathcal{C}}$, but also the composition and identity constraints of the functor $\mathscr {F}$ (see Definition 2.2.4.5). The same information is encoded implicitly in the composition law for morphisms in the category of elements $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$, in a way that is often easier to access in practice. For example, suppose that $\operatorname{\mathcal{C}}$ is the category of commutative rings and that $\mathscr {F}$ is the functor $A \mapsto \operatorname{Mod}_{A}(\operatorname{ Ab })$ described above. By definition, the functor $\mathscr {F}$ carries each ring homomorphism $u: A \rightarrow B$ to the extension of scalars functor

\[ T_{u}: \operatorname{Mod}_{A}( \operatorname{ Ab }) \rightarrow \operatorname{Mod}_{B}(\operatorname{ Ab }) \quad \quad T_ u(M) = B \otimes _{A} M. \]

Note that the construction of this functor requires certain choices, since the tensor product $B \otimes _{A} M$ is well-defined only up to (canonical) isomorphism. However, the category $\operatorname{Mod}(\operatorname{ Ab }) = \int _{\operatorname{\mathcal{C}}} \mathscr {F}$ has a more direct description which does not depend on these choices:

The objects of $\operatorname{Mod}(\operatorname{ Ab })$ are pairs $(A,M)$, where $A$ is a commutative ring and $M$ is an $A$-module.
A morphism from $(A,M)$ to $(B,N)$ in the category $\operatorname{Mod}(\operatorname{ Ab })$ is a pair $(u,f)$, where $u: A \rightarrow B$ is a homomorphism of commutative rings and $f: M \rightarrow N$ is a homomorphism of $A$-modules.

To characterize those categories which can be obtained as a category of elements $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$, it will be convenient to introduce some terminology.

Definition 5.0.0.1. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a functor between categories and let $f: X \rightarrow Y$ be a morphism in the category $\operatorname{\mathcal{E}}$.

We say that $f$ is $U$-cartesian if, for every object $W \in \operatorname{\mathcal{E}}$, the diagram of sets

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{\mathcal{E}}}(W,X) \ar [r]^-{f \circ } \ar [d]^-{U} & \operatorname{Hom}_{\operatorname{\mathcal{E}}}(W,Y) \ar [d]^{U} \\ \operatorname{Hom}_{\operatorname{\mathcal{C}}}( U(W), U(X) ) \ar [r]^-{U(f) \circ } & \operatorname{Hom}_{\operatorname{\mathcal{C}}}( U(W), U(Y) ) } \]

is a pullback square.
We say that $f$ is $U$-cocartesian if, for every object $Z \in \operatorname{\mathcal{E}}$, the diagram of sets

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{\mathcal{E}}}(Y,Z) \ar [r]^-{\circ f} \ar [d]^-{U} & \operatorname{Hom}_{\operatorname{\mathcal{E}}}(X,Z) \ar [d]^{U} \\ \operatorname{Hom}_{\operatorname{\mathcal{C}}}( U(Y), U(Z) ) \ar [r]^-{\circ U(f)} & \operatorname{Hom}_{\operatorname{\mathcal{C}}}( U(X), U(Z) ) } \]

is a pullback square.

Example 5.0.0.2. Let $\operatorname{Mod}(\operatorname{ Ab })$ be the category defined above and let $\operatorname{CAlg}(\operatorname{ Ab })$ denote the category of commutative rings, so that the construction $(A,M) \mapsto A$ determines a forgetful functor $U: \operatorname{Mod}(\operatorname{ Ab }) \rightarrow \operatorname{CAlg}(\operatorname{ Ab })$. Then:

A morphism $(u,f): (A,M) \rightarrow (B,N)$ in the category $\operatorname{Mod}(\operatorname{ Ab })$ is $U$-cartesian if and only if the underlying $A$-module homomorphism $f: M \rightarrow N$ is an isomorphism (so that the $A$-module $M$ is obtained from the $B$-module $N$ by restriction of scalars along the ring homomorphism $u$).
A morphism $(u,f): (A,M) \rightarrow (B,N)$ in the category $\operatorname{Mod}(\operatorname{ Ab })$ is $U$-cocartesian if and only if the underlying $A$-module homomorphism $f: M \rightarrow N$ induces a $B$-module isomorphism $B \otimes _{A} M \simeq N$ (so that the $B$-module $N$ is obtained from the $A$-module $M$ by extension of scalars along the ring homomorphism $u$).

Definition 5.0.0.3. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a functor between categories. We say that $U$ is a cartesian fibration if it satisfies the following condition:

For every object $Y$ of the category $\operatorname{\mathcal{E}}$ and every morphism $\overline{f}: \overline{X} \rightarrow U(Y)$ in the category $\operatorname{\mathcal{C}}$, there exists a pair $(X, f)$ where $X$ is an object of $\operatorname{\mathcal{E}}$ satisfying $U(X) = \overline{X}$ and $f: X \rightarrow Y$ is a $U$-cartesian morphism of $\operatorname{\mathcal{E}}$ satisfying $U(f) = \overline{f}$.

We say that $U$ is a cocartesian fibration if it satisfies the following dual condition:

For every object $X$ of the category $\operatorname{\mathcal{E}}$ and every morphism $\overline{f}: U(X) \rightarrow \overline{Y}$ in the category $\operatorname{\mathcal{C}}$, there exists a pair $(Y, f)$ where $Y$ is an object of $\operatorname{\mathcal{E}}$ satisfying $U(Y) = \overline{Y}$ and $f: X \rightarrow Y$ is a $U$-cocartesian morphism of $\operatorname{\mathcal{E}}$ satisfying $U(f) = \overline{f}$.

Warning 5.0.0.4. The terminology of Definition 5.0.0.3 is not standard. Many authors use the term fibration or Grothendieck fibration for what we refer to as a cartesian fibration of categories, and use the term opfibration or Grothendieck opfibration for what we refer to as a cocartesian fibration of categories. Our motivation is to be consistent with the terminology we will use for the analogous definitions in the $\infty $-categorical setting (see §5.1), where it is important to distinguish between several different notions of fibration.

Example 5.0.0.5. Let $\operatorname{Mod}(\operatorname{ Ab })$ be the category described in Example 5.0.0.2. Then the forgetful functor $U: \operatorname{Mod}(\operatorname{ Ab }) \rightarrow \operatorname{CAlg}(\operatorname{ Ab })$ is both a cartesian fibration and a cocartesian fibration.

Exercise 5.0.0.6. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a functor between categories. Show that the following conditions are equivalent:

The functor $U$ is a fibration in groupoids (Definition 4.2.2.1).
The functor $U$ is a cartesian fibration and every morphism of $\operatorname{\mathcal{E}}$ is $U$-cartesian.
The functor $U$ is a cartesian fibration and, for every object $C \in \operatorname{\mathcal{C}}$, the fiber $\operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is a groupoid.

For a more general statement, see Proposition 5.1.4.15.

Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a functor between categories. A classical theorem of Grothendieck ([MR0217088]) asserts that $U$ is a cocartesian fibration if $\operatorname{\mathcal{E}}$ can be realized as the category of elements of $\mathbf{Cat}$-valued functor on $\operatorname{\mathcal{C}}$: that is, if and only if there exists a functor of $2$-categories $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \mathbf{Cat}$ and an isomorphism of categories $\operatorname{\mathcal{E}}\simeq \int _{\operatorname{\mathcal{C}}} \mathscr {F}$ which carries $U$ to the forgetful functor $\int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$ (Corollary 5.6.5.21). Moreover, the functor $\mathscr {F}$ is uniquely determined up to isomorphism. Fixing the category $\operatorname{\mathcal{C}}$, the category of elements construction supplies a dictionary

5.1

\begin{equation} \label{equation:fibered-categories} \{ \textnormal{Functors $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \mathbf{Cat}$} \} \simeq \{ \textnormal{Cocartesian fibrations $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$} \} , \end{equation}

which is the starting point for the theory of fibered categories.

The goal of chapter is to introduce an $\infty $-categorical generalization of the correspondence (5.1). We begin in §5.1 by developing an $\infty $-categorical counterpart of the theory of (co)cartesian fibrations. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. We say that an edge $e$ of $\operatorname{\mathcal{E}}$ is $U$-cocartesian if every lifting problem

\[ \xymatrix { \Lambda ^{n}_{0} \ar [r]^{\sigma _0} \ar [d] & \operatorname{\mathcal{E}}\ar [d]^{U} \\ \Delta ^{n} \ar [r] \ar@ {-->}[ur] & \operatorname{\mathcal{C}}} \]

admits a solution, provided that $n \geq 2$ and the restriction $\sigma _0 |_{ \Delta ^1 }$ is equal to $e$ (Definition 5.1.1.1). We will be primarily interested in the situation where $U$ is an inner fibration of $\infty $-categories; in this case, we show that an edge $e \in \operatorname{\mathcal{E}}$ is $U$-cocartesian if and only if it satisfies a homotopy-theoretic counterpart of Definition 5.0.0.1 (Proposition 5.1.2.1). We say that a morphism of simplicial sets $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is a cocartesian fibration if it is an inner fibration having the property that, for every vertex $X \in \operatorname{\mathcal{E}}$ and every edge $\overline{e}: U(X) \rightarrow \overline{Y}$, there exists a $U$-cocartesian edge $e: X \rightarrow Y$ satisfying $U(e) = \overline{e}$ (Definition 5.1.4.1). This can be regarded as a generalization of Definition 5.0.0.3: a functor of ordinary categories $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is a cocartesian fibration if and only if the induced 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). It also generalizes the notion of left fibration introduced in §4.2: a morphism of simplicial sets $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is a left fibration if and only if it is a cocartesian fibration and every edge of $\operatorname{\mathcal{E}}$ is $U$-cocartesian (Proposition 5.1.4.15).

The remainder of this section is devoted to the problem of classifying cocartesian fibrations $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$, where $\operatorname{\mathcal{C}}$ is a fixed $\infty $-category. For each object $C \in \operatorname{\mathcal{C}}$, let $\operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ denote the corresponding fiber of $U$. We can then ask the following:

Question 5.0.0.7. What additional data is needed to reconstruct the $\infty $-category $\operatorname{\mathcal{E}}$ from the collection of $\infty $-categories $\{ \operatorname{\mathcal{E}}_{C} \} _{C \in \operatorname{\mathcal{C}}}$?

In §5.2, we give a partial answer to Question 5.0.0.7. Let $f: C \rightarrow D$ be a morphism in the $\infty $-category $\operatorname{\mathcal{C}}$. For each object $X \in \operatorname{\mathcal{E}}_{C}$, our assumption that $U$ is a cocartesian fibration guarantees that we can lift $f$ to a $U$-cocartesian morphism $\widetilde{f}: X \rightarrow Y$ of $\operatorname{\mathcal{E}}$. We will see that the construction $X \mapsto Y$ can be upgraded to a functor of $\infty $-categories $f_{!}: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}_{D}$, which we will refer to as the functor of covariant transport along $f$ (Definition 5.2.2.4). The construction of the functor $f_{!}$ requires some auxiliary choices, but its isomorphism class $[ f_{!} ]$ is uniquely determined (Proposition 5.2.2.8). Moreover, the construction $f \mapsto f_{!}$ is compatible with composition (Proposition 5.2.5.1), and therefore determines a functor of ordinary categories

\[ \operatorname{hTr}_{ \operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{QCat}} \quad \quad C \mapsto \operatorname{\mathcal{E}}_{C}; \]

here $\mathrm{h} \mathit{\operatorname{QCat}}$ denotes the homotopy category of $\infty $-categories (Construction 4.5.1.1). We will refer to $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ as the homotopy transport representation of the cocartesian fibration $U$ (Construction 5.2.5.2).

In some cases, the homotopy transport representation $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ provides an answer to Question 5.0.0.7:

If $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is a left covering map of simplicial sets, then we can regard $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ as a functor from the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ to the category of sets. In this case, we can reconstruct $\operatorname{\mathcal{E}}$ (up to isomorphism) as the fiber product

\[ \operatorname{\mathcal{C}}\times _{ \operatorname{N}_{\bullet }( \operatorname{Set}) } \operatorname{N}_{\bullet }( \operatorname{Set}_{\ast } ), \]

where $\operatorname{Set}_{\ast }$ denotes the category of pointed sets (Proposition 5.2.7.2). It follows that the the construction $\operatorname{\mathcal{E}}\mapsto \operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ defines an equivalence of categories

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

which we regard as a generalization of the classical theory of covering spaces (Corollary 5.2.7.3).
Suppose that $\operatorname{\mathcal{C}}= \Delta ^1$ is the standard $1$-simplex. In this case, the homotopy transport representation $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ records the data of the $\infty $-categories $\operatorname{\mathcal{E}}_{0}$ and $\operatorname{\mathcal{E}}_{1}$, together with (the isomorphism class of) the covariant transport functor $F: \operatorname{\mathcal{E}}_0 \rightarrow \operatorname{\mathcal{E}}_1$ associated to the nondegenerate edge of $\operatorname{\mathcal{C}}$. From this data, one can reconstruct the $\infty $-category $\operatorname{\mathcal{E}}$ up to equivalence. More precisely, we show that $\operatorname{\mathcal{E}}$ is categorically equivalent to the mapping cylinder $( \Delta ^1 \times \operatorname{\mathcal{E}}_0 ) {\coprod }_{ (\{ 1\} \times \operatorname{\mathcal{E}}_0) } \operatorname{\mathcal{E}}_1$; see Corollary 5.2.4.2.

In general, the homotopy transport representation of a cocartesian fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ does not contain enough information to reconstruct the $\infty $-category $\operatorname{\mathcal{E}}$, even up to equivalence. The essence of the problem is that the functor $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ encodes only the isomorphism classes of the covariant transport functors associated to the morphisms of $\operatorname{\mathcal{C}}$. To address Question 5.0.0.7, it is necessary to consider a refinement of $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ which witnesses the functoriality of the construction $C \mapsto \operatorname{\mathcal{E}}_{C}$ before passing to the homotopy category $\mathrm{h} \mathit{\operatorname{QCat}}$. In §5.3, we specialize to the situation where $\operatorname{\mathcal{C}}= \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}_0)$ is (the nerve of) an ordinary category $\operatorname{\mathcal{C}}_0$. In this case, we associate to each cocartesian fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ a functor of ordinary categories $\operatorname{sTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}_0}: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{QCat}$, which we refer to as the strict transport representation of $\operatorname{\mathcal{C}}$ (Construction 5.3.1.5). The strict transport representation is a refinement of the homotopy transport representation: more precisely, there is a canonical isomorphism of $\operatorname{hTr}_{\operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}}}$ with the composite functor

\[ \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \simeq \operatorname{\mathcal{C}}_0 \xrightarrow { \operatorname{sTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}_0} } \operatorname{QCat}\twoheadrightarrow \mathrm{h} \mathit{\operatorname{QCat}} \]

(Corollary 5.3.1.8). Moreover, we show that this refinement provides an answer to Question 5.0.0.7: according to Theorem 5.3.5.6, the construction $\operatorname{\mathcal{E}}\mapsto \operatorname{sTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}_0}$ induces a bijection

\[ \xymatrix@R =50pt@C=50pt{ \{ \textnormal{Cocartesian Fibrations $\operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$} \} / \textnormal{Equivalence} \ar [d] \\ \{ \textnormal{Functors $\operatorname{\mathcal{C}}_0 \rightarrow \operatorname{QCat}$} \} / \textnormal{Levelwise Equivalence}. } \]

Moreover, the inverse bijection admits an explicit description: it carries (the equivalence class of) a functor $\mathscr {F}: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{QCat}$ to (the equivalence class of) a cocartesian fibration $\operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}_0) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}_0)$. Here $\operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}_0)$ is an $\infty $-category which we refer to as the $\mathscr {F}$-weighted nerve of $\operatorname{\mathcal{C}}_0$ (Definition 5.3.3.1).

Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of $\infty $-categories. In general, it is not reasonable to expect that the homotopy transport representation $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{QCat}}$ can be promoted to a strictly commutative diagram in the category of simplicial sets. In other words, $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ generally cannot be lifted to a morphism from $\operatorname{\mathcal{C}}$ to the nerve $\operatorname{N}_{\bullet }( \operatorname{QCat})$. To address Question 5.0.0.7 in complete generality, we will instead contemplate homotopy coherent refinements of $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$, given by morphisms from $\operatorname{\mathcal{C}}$ to the homotopy coherent nerve $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{QCat})$. Here we regard $\operatorname{QCat}$ as a locally Kan simplicial category, with morphism spaces given by $\operatorname{Hom}_{\operatorname{QCat}}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{D}}' ) = \operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{D}}' )^{\simeq }$. The homotopy coherent nerve $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{QCat})$ is then an $\infty $-category which we will denote by $\operatorname{\mathcal{QC}}$ and refer to as the $\infty $-category of small $\infty $-categories (Construction 5.5.4.1). In §5.5, we study several variants of this construction. In particular, we introduce an $\infty $-category $\operatorname{\mathcal{QC}}_{\operatorname{Obj}}$ whose objects are pairs $(\operatorname{\mathcal{D}}, X)$, where $\operatorname{\mathcal{D}}$ is a small $\infty $-category and $X$ is an object of $\operatorname{\mathcal{D}}$, and whose morphisms are pairs $(F,u): (\operatorname{\mathcal{D}}, X) \rightarrow (\operatorname{\mathcal{D}}', X')$ where $F: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{D}}'$ is a functor of $\infty $-categories and $u: F(X) \rightarrow X'$ is a morphism in $\operatorname{\mathcal{D}}'$ (Definition 5.5.6.10).

The construction $(\operatorname{\mathcal{D}}, X) \mapsto \operatorname{\mathcal{D}}$ determines a forgetful functor $V: \operatorname{\mathcal{QC}}_{\operatorname{Obj}} \rightarrow \operatorname{\mathcal{QC}}$, which is a cocartesian fibration of $\infty $-categories (Proposition 5.5.6.11). In §5.0.0.7, we address Question 5.0.0.7 in general by showing that $V$ is a universal cocartesian fibration. For any functor of $\infty $-categories $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$, we let $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ denote the fiber product $\operatorname{\mathcal{C}}\times _{ \operatorname{\mathcal{QC}}} \operatorname{\mathcal{QC}}_{\operatorname{Obj}}$. We will refer to $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ as the $\infty $-category of elements of $\mathscr {F}$ (Definition 5.6.2.4); by construction, its objects are pairs $(C,X)$ where $C$ is an object of $\operatorname{\mathcal{C}}$ and $X$ is an object of the $\infty $-category $\mathscr {F}(C)$. Note that projection onto the first factor determines a forgetful functor $U: \int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$, which is a cocartesian fibration of $\infty $-categories (since it is a pullback of the cocartesian fibration $V$). Our main result is that the construction $\mathscr {F} \mapsto \int _{\operatorname{\mathcal{C}}} \mathscr {F}$ induces a bijection from the set of isomorphism classes in $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{QC}})$ to the set of equivalence classes of $\infty $-categories equipped with a cocartesian fibration to $\operatorname{\mathcal{C}}$ (Theorem 5.6.0.2). In particular, every cocartesian fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ fits into a categorical pullback square

\[ \xymatrix { \operatorname{\mathcal{E}}\ar [d]^{U} \ar [r] & \operatorname{\mathcal{QC}}_{\operatorname{Obj}} \ar [d] \\ \operatorname{\mathcal{C}}\ar [r]^{ \operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}} } & \operatorname{\mathcal{QC}}, } \]

where the functor $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$ is uniquely determined up to isomorphism. The functor $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ is an $\infty $-categorical refinement of the homotopy transport representation $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ (Remark 5.6.5.17), which we will refer to as the covariant transport representation of $U$ (Definition 5.6.5.1).

Remark 5.0.0.8. The classical theory of fibered categories was introduced by Grothendieck in [MR0217088] (Exposé 6).

Kerodon

5 Fibrations of $\infty $-Categories

Structure