Kerodon

5.3 Fibrations over Ordinary Categories

Let $\operatorname{Set_{\Delta }}$ denote the category of simplicial sets, let $\operatorname{QCat}\subset \operatorname{Set_{\Delta }}$ denote the full subcategory spanned by the $\infty $-categories, and let $\mathrm{h} \mathit{\operatorname{QCat}}$ denote its homotopy category (Construction 4.5.1.1). In §5.2.5, we associated to every cocartesian fibration of simplicial sets $U: \operatorname{\mathcal{E}}\rightarrow S$ a functor $\operatorname{hTr}_{\operatorname{\mathcal{E}}/S}: \mathrm{h} \mathit{S} \rightarrow \mathrm{h} \mathit{\operatorname{QCat}}$ called the homotopy transport representation of $U$, given on objects by the formula $\operatorname{hTr}_{\operatorname{\mathcal{E}}/S}(s) = \{ s\} \times _{S} \operatorname{\mathcal{E}}$ (Construction 5.2.5.2). In §5.3.1, we specialize to the situation where $S = \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is the nerve of an ordinary category $\operatorname{\mathcal{C}}$. In this case, we show that $\operatorname{hTr}_{\operatorname{\mathcal{E}}/ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})}$ can be lifted to a functor taking values in the category $\operatorname{QCat}$. More precisely, we introduce a functor $\operatorname{sTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ which we refer to as the strict transport representation of $U$ (Construction 5.3.1.5), and show that the diagram

commutes up to canonical isomorphism (Corollary 5.3.1.8).

Our primary goal in this section is to show that a cocartesian fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ can be recovered, up to equivalence, from its strict transport representation $\operatorname{sTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$. To formulate this precisely, we need another construction. In §5.3.3, we associate to every diagram $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ a simplicial set $\operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$, which we will refer to as the $\mathscr {F}$-weighted nerve of $\operatorname{\mathcal{C}}$ (Definition 5.3.3.1). The weighted nerve is equipped with a projection map $V: \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$, whose fiber over an object $C \in \operatorname{\mathcal{C}}$ can be identified with the simplicial set $\mathscr {F}(C)$ (Example 5.3.3.8). If each of these simplicial sets is an $\infty $-category, then $V$ is a cocartesian fibration of $\infty $-categories (Corollary 5.3.3.16). Our main results can be summarized as follows:

$(1)$

Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ be a diagram of $\infty $-categories having weighted nerve $\operatorname{\mathcal{E}}= \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$. Then there is a natural transformation from $\mathscr {F}$ to the strict transport representation $\operatorname{sTr}_{\operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}}}$, which carries each object $C \in \operatorname{\mathcal{C}}$ to an equivalence of $\infty $-categories $\mathscr {F}(C) \rightarrow \operatorname{sTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}(C)$ (Corollary 5.3.4.19).

$(2)$

Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ be a cocartesian fibration of $\infty $-categories having strict transport representation $\mathscr {F} = \operatorname{sTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$. Then $U$ is equivalent (in the sense of Definition 5.1.7.1) to the cocartesian fibration $\operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ (Theorem 5.3.5.6).

The proof of $(1)$ is relatively straightforward. However, the proof of $(2)$ is somewhat more difficult: given a cocartesian fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ there is no obvious comparison map between the simplicial sets $\operatorname{\mathcal{E}}$ and $\operatorname{N}_{\bullet }^{\operatorname{sTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}}(\operatorname{\mathcal{C}})$. To relate them, we need an auxiliary construction. In §5.3.2, we associate to every diagram $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ a simplicial set $ \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F})$, which we refer to as the homotopy colimit of $\mathscr {F}$ (Construction 5.3.2.1). The formation of homotopy colimits plays an important role in the classical homotopy theory of simplicial sets: it can be regarded as a replacement for the usual notion of colimit (see Remark 5.3.2.9) which is compatible with weak homotopy equivalence (Proposition 5.3.2.18). Beware that the homotopy colimit $ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} )$ is generally not an $\infty $-category (even in the special case where $\mathscr {F}$ is a diagram of $\infty $-categories). Nevertheless, it is equipped with a projection map $ \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F} ) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$, whose fiber over each object $C \in \operatorname{\mathcal{C}}$ can be identified with the simplicial set $\mathscr {F}(C)$, and which behaves in certain respects like a cocartesian fibration. In §5.3.4, we make this heuristic precise by introducing the notion of a scaffold. If $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is a cocartesian fibration of $\infty $-categories, we define a scaffold of $U$ to be a commutative diagram

where $\lambda $ restricts to a categorical equivalence $\mathscr {F}(C) \rightarrow \operatorname{\mathcal{E}}_{C}$ for each $C \in \operatorname{\mathcal{C}}$ and behaves well with respect to the collection of $U$-cocartesian morphisms of $\operatorname{\mathcal{E}}$ (Definition 5.3.4.2). We are primarily interested in two examples:

• To any cocartesian fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$, we associate a universal scaffold $\lambda _{u}: \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \rightarrow \operatorname{\mathcal{E}}$, where $\mathscr {F} = \operatorname{sTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ is the strict transport representation of $U$ (see Construction 5.3.4.7 and Proposition 5.3.4.8).

• To any diagram of $\infty $-categories $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$, we associate a taut scaffold $\lambda : \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F} ) \rightarrow \operatorname{\mathcal{E}}$, where $\operatorname{\mathcal{E}}= \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$ is the $\mathscr {F}$-weighted nerve of $\operatorname{\mathcal{C}}$ (see Construction 5.3.4.11 and Proposition 5.3.4.17).

In §5.3.5, we show that every scaffold $ \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F} ) \rightarrow \operatorname{\mathcal{E}}$ is a categorical equivalence of simplicial sets (Theorem 5.3.5.7). In particular, if $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is a cocartesian fibration with strict transport representation $\mathscr {F} = \operatorname{sTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$, then we can exploit the taut and universal scaffolds

to deduce the existence of an equivalence of $\infty $-categories $\operatorname{\mathcal{E}}\simeq \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$ (compatible with the projection $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$), thereby obtaining a proof of $(2)$ (see Theorem 5.3.5.6).

We close this section by describing some other applications of our theory of scaffolds. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{B}}$ be a morphism of simplicial sets, let $\operatorname{\mathcal{D}}$ be an $\infty $-category, and let $\operatorname{Fun}( \operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{D}})$ be the relative exponential of Construction 4.5.9.1. In §5.3.6, we show that if $U$ is a cocartesian fibration, then the projection map $\operatorname{Fun}( \operatorname{\mathcal{C}}/ \operatorname{\mathcal{B}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{B}}$ is a cartesian fibration. More generally, for every cartesian fibration of simplicial sets $V: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$, the induced map

is also a cartesian fibration (Proposition 5.3.6.6). In §5.3.7, we apply this result to study the oriented fiber product of Definition 4.6.4.1. For any functor of $\infty $-categories $F: \operatorname{\mathcal{A}}\rightarrow \operatorname{\mathcal{B}}$, projection onto the second factor determines a cocartesian fibration $\operatorname{\mathcal{A}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{B}}} \operatorname{\mathcal{B}}\rightarrow \operatorname{\mathcal{B}}$ (Corollary 5.3.7.3) which is, in some sense, freely generated by the $\infty $-category $\operatorname{\mathcal{A}}$ (Theorem 5.3.7.7).

Structure

• Subsection 5.3.1: The Strict Transport Representation
• Subsection 5.3.2: Homotopy Colimits of Simplicial Sets
• Subsection 5.3.3: The Weighted Nerve
• Subsection 5.3.4: Scaffolds of Cocartesian Fibrations
• Subsection 5.3.5: Application: Classification of Cocartesian Fibrations
• Subsection 5.3.6: Application: Relative Exponentials
• Subsection 5.3.7: Application: Path Fibrations
• Subsection 5.3.8: Digression: Minimal Fibrations