Remark 5.1.0.1. For historical reasons, it is traditional to place more emphasis on the duals of the notions introduced above. We say that a functor of categories $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is a *cartesian fibration* (*fibration in sets*, *fibration in groupoids*) if the opposite functor $U^{\operatorname{op}}: \operatorname{\mathcal{E}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ is a cocartesian fibration (opfibration in sets, opfibration in groupoids). If $U$ is a cartesian fibration, then there exists a functor of $2$-categories $\mathscr {F}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \mathbf{Cat}$ and an isomorphism of $\operatorname{\mathcal{E}}$ with the (dual) Grothendieck construction $\int ^{\operatorname{\mathcal{C}}} \mathscr {F}$, characterized by the formula $(\int ^{\operatorname{\mathcal{C}}} \mathscr {F})^{\operatorname{op}} = \int _{\operatorname{\mathcal{C}}^{\operatorname{op}}} \mathscr {F}^{\operatorname{op}}$ (see Remark 5.1.3.4).

## 5.1 Fibered 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

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

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. In §5.1.3, we associate to every functor of $2$-categories $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \mathbf{Cat}$ a new category $\int _{\operatorname{\mathcal{C}}} X$, which we will refer to as the *Grothendieck construction on $\mathscr {F}$* (Variant 5.1.3.3). 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 $\mathscr {F}(C)$. Moreover, 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 $\mathscr {F}(C)$. We will see that the functor $\mathscr {F}$ can be recovered (up to isomorphism) from the category $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ and the forgetful functor $U$ (Proposition 5.1.5.16). Morevoer, passage from the functor $\mathscr {F}$ to its Grothendieck construction $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ has several concrete advantages:

The notion of a functor $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \mathbf{Cat}$ uses the formalism of $2$-categories. However, the Grothendieck construction $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ is an ordinary category which (together with the forgetful functor $U$) encodes essentially the same information.

In practice, it is often easier to describe the Grothendieck construction $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ than the original functor $\mathscr {F}$. 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 (see Examples 5.1.4.5 and 5.1.4.10):

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.

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 Grothendieck construction $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$.

Our goal in this section is to provide a detailed discussion of the Grothendieck construction $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ and its relationship to the functor $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \mathbf{Cat}$. We begin in §5.1.1 by studying the special case where the functor takes values in the category of sets (which we can identify with the full subcategory of $\mathbf{Cat}$ whose objects are categories having only identity morphisms). In this case, we will refer to $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ as the *category of elements* of the functor $\mathscr {F}$ (see Construction 5.1.1.1). Our main result is that the construction $\mathscr {F} \mapsto \int _{\operatorname{\mathcal{C}}} \mathscr {F}$ determines a fully faithful functor $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{Set}) \rightarrow \operatorname{Cat}_{ / \operatorname{\mathcal{C}}}$ (Proposition 5.1.1.8). In §5.1.2, we show that a functor $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ belongs to the essential image of this construction if and only if it satisfies the following lifting condition (Corollary 5.1.2.18):

For every morphism $\overline{f}: \overline{X} \rightarrow \overline{Y}$ in $\operatorname{\mathcal{C}}$ and every object $X \in \operatorname{\mathcal{E}}$ satisfying $U(X) = \overline{X}$, there is a unique morphism $f: X \rightarrow Y$ in $\operatorname{\mathcal{E}}$ satisfying $U(f) = \overline{f}$.

If this condition is satisfied, we say that the functor $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is an *opfibration in sets* (see Variant 5.1.2.2).

In §5.1.3, we define the Grothendieck construction $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ for an arbitrary category $\operatorname{\mathcal{C}}$ equipped with a functor $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \mathbf{Cat}$ (Variant 5.1.3.3). In general, the Grothendieck construction $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ is a category equipped with 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 $\mathscr {F}(C)$ (Example 5.1.3.12). The functor $U$ generally does not satisfy condition $(\ast )$. However, it satisfies a weaker lifting condition: given an object $X \in \int _{\operatorname{\mathcal{C}}} \mathscr {F}$ having image $\overline{X} = U(X)$ in $\operatorname{\mathcal{C}}$, every morphism $\overline{f}: \overline{X} \rightarrow \overline{Y}$ in $\operatorname{\mathcal{C}}$ has a *canonical* lift to a morphism $f: X \rightarrow Y$ in $\operatorname{\mathcal{C}}$, which is characterized up to isomorphism by the requirement that it is *$U$-cocartesian* (Variant 5.1.4.2). In §5.1.4, we axiomatize the situation by introducing the notion of a *cocartesian fibration* of categories $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$, and showing that the forgetful functor $\int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$ is always a cocartesian fibration (Proposition 5.1.4.13). In §5.1.5, we prove the converse: for every cocartesian fibration of categories $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$, there exists a functor $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \mathbf{Cat}$ and an isomorphism $\operatorname{\mathcal{E}}\simeq \int _{\operatorname{\mathcal{C}}} \mathscr {F}$ which is compatible with $U$ (Corollary 5.1.5.21).

Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of categories. In §5.1.6, we show that the following conditions are equivalent:

Every morphism of $\operatorname{\mathcal{E}}$ is $U$-cocartesian.

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 (Variant 5.1.6.8).

There exists an isomorphism of categories $\operatorname{\mathcal{E}}\simeq \int _{\operatorname{\mathcal{C}}} \mathscr {F}$ which is compatible with $U$, where $\mathscr {F}$ is a functor from $\operatorname{\mathcal{C}}$ to the $2$-category of groupoids (Variant 5.1.6.12).

The induced morphism $\operatorname{N}_{\bullet }(U): \operatorname{N}_{\bullet }(\operatorname{\mathcal{E}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is a left fibration of simplicial sets (Proposition 5.1.6.14).

If these equivalent conditions are satisfied, we say that $U$ is an *opfibration in groupoids* (Variant 5.1.6.4).

Warning 5.1.0.2. Our terminology 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 §5.2), where it is important to distinguish between several different notions of fibration.

Remark 5.1.0.3. The theory of fibered categories was introduced by Grothendieck in [MR0217088] (Exposé 6).