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5.5 The Category of Elements

Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. To every cocartesian fibration of $\infty $-categories $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$, Construction 5.2.3.2 supplies a homotopy transport representation

\[ \operatorname{hTr}_{U}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{QCat}} \quad \quad C \mapsto \operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}. \]

If $\operatorname{\mathcal{C}}= \Delta ^ n$ is a standard simplex, then every functor $\mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{QCat}}$ arises in this way (Exercise 5.2.4.180. In general, this need not be true: for a functor $F: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{QCat}}$ to be (isomorphic to) the homotopy transport representation of a cocartesian fibration $\operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$, it is necessary and sufficient that $F$ can be lifted to a functor of $\infty $-categories $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$. We will prove the necessity of this condition in §5.6 (Remark 5.6.4.14). Our goal in this section is to show that it is sufficient. To every functor of $\infty $-categories $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$, we will explicitly construct a cocartesian fibration of $\infty $-categories $U: \int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$ whose homotopy transport representation $\operatorname{hTr}_{U}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{QCat}}$ is isomorphic to the functor $\mathrm{h} \mathit{\mathscr {F}}$ (Corollary 5.5.6.10).

We begin in §5.5.1 with an easy special case. Suppose that $\operatorname{\mathcal{C}}$ is an ordinary category and that $\mathscr {F}$ is a functor from $\operatorname{\mathcal{C}}$ to the category of sets. To this data, we can associate a new category whose objects are pairs $(C,x)$, where $C$ is an object of $\operatorname{\mathcal{C}}$ and $x$ is an element of the set $\mathscr {F}(C)$. We will denote this category by $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$, and refer to it as the category of elements of the functor $\mathscr {F}$ (Construction 5.5.1.1). 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 set $\mathscr {F}(C)$ (regarded as a discrete category). If we regard the category $\operatorname{\mathcal{C}}$ as fixed, then the functor $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set}$ and the functor $U: \int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$ are equivalent data: either can be functorially recovered from the other (Proposition 5.5.1.8).

In §5.5.2, we study a generalization of Construction 5.5.1.1 which still lies within purview of classical category theory. Let $\operatorname{\mathcal{C}}$ be an ordinary category and let $\mathscr {F}$ be a functor from $\operatorname{\mathcal{C}}$ to the $2$-category $\mathbf{Cat}$ of categories. In this case, we can form a new 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 $\mathscr {F}(C)$, which we will again refer to as the category of elements of $\mathscr {F}$ (Definition 5.5.2.1). As before, 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 $\mathscr {F}(C)$. The forgetful functor $U$ is a cocartesian fibration of categories (Corollary 5.5.2.13), which implicitly encodes the structure of the original functor $\mathscr {F}$.

Remark 5.5.0.1. The category of elements $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ was originally introduced by Gorhtendieck in [MR0217088]. For this reason, many authors refer to the category $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ as the Grothendieck construction on the functor $\mathscr {F}$.

In §5.5.3, we introduce a simplicial variant of the category of elements construction. Let $\operatorname{\mathcal{C}}$ be an ordinary category and let $\operatorname{Set_{\Delta }}$ denote the category of simplicial sets. To every functor $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$, we associate a simplicial set $\int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F}$ which we refer to as the $\mathscr {F}$-weighted nerve of $\operatorname{\mathcal{C}}$ (Definition 5.5.3.1). This construction has the following features:

  • The weighted nerve $\int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F}$ is equipped with a projection map $U: \int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F} \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$, whose fiber over an object $C \in \operatorname{\mathcal{C}}$ can be identified with the simplicial set $\mathscr {F}(C)$ (Example 5.5.3.7).

  • Suppose that, for every object $C \in \operatorname{\mathcal{C}}$, the simplicial set $\mathscr {F}(C)$ is an $\infty $-category. Then the projection map $U: \int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F} \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is a cocartesian fibration of $\infty $-categories (Proposition 5.5.3.13), whose homotopy transport representation $\operatorname{hTr}_{U}: \operatorname{\mathcal{C}}\rightarrow \mathrm{h} \mathit{\operatorname{QCat}}$ is induced by the functor $\mathscr {F}$ (Remark 5.5.3.15).

  • Suppose that, for every object $C \in \operatorname{\mathcal{C}}$, the simplicial set $\mathscr {F}(C)$ is the nerve of an ordinary category $\mathscr {F}_0(C)$, so that the construction $C \mapsto \mathscr {F}_0(C)$ determines a functor $\mathscr {F}_0: \operatorname{\mathcal{C}}\rightarrow \operatorname{Cat}$ (see Proposition 1.2.2.1). In this case, the weighted nerve $\int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F}$ can be identified with the nerve of the category of elements $\int _{\operatorname{\mathcal{C}}} \mathscr {F}_0$ (Example 5.5.3.4).

For many applications, the weighted nerve construction is not sufficiently flexible: it requires $\operatorname{\mathcal{C}}$ to be an ordinary category and $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ to be a strictly commutative diagram of simplicial sets. In §5.5.4, we address this point by considering a more general situation. Let $\operatorname{\mathcal{C}}$ be an arbitrary simplicial set, and let $\mathscr {F}$ be a morphism from $\operatorname{\mathcal{C}}$ to the homotopy coherent nerve $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{Set_{\Delta }})$ (which we can view as a homotopy coherent diagram of simplicial sets indexed by $\operatorname{\mathcal{C}}$). To this data, we associate a new simplicial set

\[ \int _{\operatorname{\mathcal{C}}} \mathscr {F} = \operatorname{\mathcal{C}}\times _{ \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{Set_{\Delta }})} \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{Set_{\Delta }})_{ \Delta ^{0} / }. \]

We will be particularly interested in the case where $\operatorname{\mathcal{C}}$ is an $\infty $-category and $\mathscr {F}$ takes values in the $\infty $-category $\operatorname{\mathcal{QC}}\subset \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{Set_{\Delta }})$. In this case, the simplicial set $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ is also an $\infty $-category (Corollary 5.5.4.3), which we will refer to as the $\infty $-category of elements of $\operatorname{\mathcal{C}}$ (Definition 5.5.4.4). 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 §5.5.5, 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 $\mathscr {F}$ (Proposition 5.5.5.3). 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 §5.5.6.8, we construct a comparison map

    \[ \theta : \int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F} \rightarrow \int _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \operatorname{N}_{\bullet }^{\operatorname{hc}}( \mathscr {F} ) \]

    and show that it is an equivalence of $\infty $-categories (Proposition 5.5.6.8). Beware that $\theta $ is usually not an isomorphism of simplicial sets.

  • Let $\operatorname{\mathcal{C}}$ be an arbitrary simplicial set and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$ be a morphism of simplicial sets. Then the projection map $U: \int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$ is a cocartesian fibration (Proposition 5.5.4.2), whose homotopy transport representation $\operatorname{hTr}_{U}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{QCat}}$ is isomorphic to $\mathrm{h} \mathit{\mathscr {F}}$ (Corollary 5.5.6.10). In particular, for each object $C \in \operatorname{\mathcal{C}}$, the $\infty $-category $\mathscr {F}(C)$ is canonically isomorphic to $\{ C \} \times _{\operatorname{\mathcal{C}}} \int _{\operatorname{\mathcal{C}}} \mathscr {F}$ as an object of the homotopy category $\mathrm{h} \mathit{\operatorname{QCat}}$. Beware that $\mathscr {F}(C)$ and $\{ C \} \times _{\operatorname{\mathcal{C}}} \int _{\operatorname{\mathcal{C}}} \mathscr {F}$ are usually not isomorphic in the category of simplicial sets (see Example 5.5.4.19).

Structure

  • Subsection 5.5.1: Elements of Set-Valued Functors
  • Subsection 5.5.2: Elements of Category-Valued Functors
  • Subsection 5.5.3: The Weighted Nerve
  • Subsection 5.5.4: Elements of $\operatorname{\mathcal{QC}}$-Valued Functors
  • Subsection 5.5.5: Comparison with the Category of Elements
  • Subsection 5.5.6: Comparison with the Weighted Nerve