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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

\[ \xymatrix@C =50pt@R=50pt{ & \operatorname{QCat}\ar [dr] & \\ \operatorname{\mathcal{C}}\ar [ur]^-{ \operatorname{sTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}} } \ar [rr]_-{ \operatorname{hTr}_{\operatorname{\mathcal{E}}/ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } } & & \mathrm{h} \mathit{\operatorname{QCat}} } \]

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 (Proposition 5.3.3.15). 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 5.1.6.1) 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

\[ \xymatrix@R =50pt@C=50pt{ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \ar [rr]^{\lambda } \ar [dr] & & \operatorname{\mathcal{E}}\ar [dl]_{U} \\ & \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}), & } \]

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

\[ \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \xleftarrow {\lambda } \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F} ) \xrightarrow {\lambda _{u}} \operatorname{\mathcal{E}}, \]

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 $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ and $V: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{D}}$ be morphisms of simplicial sets, and let $\operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}})$ denote the direct image of $\operatorname{\mathcal{E}}$ along $U$ (see Construction 4.5.9.1). In §5.3.6, we show that if $U$ is a cocartesian fibration and $V$ is a cartesian fibration, then the projection map $\operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{C}}$ 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 $\operatorname{\mathcal{A}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{B}}} \operatorname{\mathcal{B}}\rightarrow \operatorname{\mathcal{B}}$ (Proposition 5.3.7.1) which is, in some sense, freely generated by the $\infty $-category $\operatorname{\mathcal{A}}$ (Theorem 5.3.7.7).

Remark 5.3.0.1. There is a close analogy between the homotopy colimit construction (studied in §5.3.2) and the weighted nerve construction (studied in §5.3.3).

  • The formation of homotopy colimits determines a functor

    \[ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }}) \rightarrow ( \operatorname{Set_{\Delta }})_{ / \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \quad \quad \mathscr {F} \mapsto \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F} ). \]

    This functor has a right adjoint, which carries an object $\operatorname{\mathcal{E}}\in (\operatorname{Set_{\Delta }})_{ / \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) }$ to the diagram

    \[ \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}\quad \quad C \mapsto \operatorname{Fun}_{ / \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) }( \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{C/} ), \operatorname{\mathcal{E}}). \]

    See Corollary 5.3.2.24.

  • The formation of weighted nerves determines a functor

    \[ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }}) \rightarrow ( \operatorname{Set_{\Delta }})_{ / \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \quad \quad \mathscr {F} \mapsto \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}). \]

    This functor has a left adjoint, which carries an object $\operatorname{\mathcal{E}}\in (\operatorname{Set_{\Delta }})_{ / \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) }$ to the diagram

    \[ \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}\quad \quad C \mapsto \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}_{/C}) \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \operatorname{\mathcal{E}}. \]

    See Corollary 5.3.3.22.

Remark 5.3.0.2. After restricting to diagrams of Kan complexes, the results of this section supply a dictionary

\[ \xymatrix@R =50pt@C=50pt{ \{ \textnormal{Left fibrations $\operatorname{\mathcal{E}}\rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$} \} \ar@ <.8ex>[d]^-{ \operatorname{sTr}_{(-)/\operatorname{\mathcal{C}}}} \\ \{ \textnormal{Functors $\operatorname{\mathcal{C}}\rightarrow \operatorname{Kan}$} \} \ar@ <.8ex>[u]^-{ \operatorname{N}_{\bullet }^{(-)}(\operatorname{\mathcal{C}}) } } \]

This dictionary was formulated in work of Heuts and Moerdijk (using the language of model categories) which is closely related to the contents of this section. For more details, we refer the reader to [MR3318247].

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: Direct Image Fibrations
  • Subsection 5.3.7: Application: Path Fibrations