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5.6.5 The Universality Theorem

Throughout this section, we let $\operatorname{\mathcal{QC}}_{\operatorname{Obj}}$ denote the $\infty $-category of pairs $(\operatorname{\mathcal{C}}, C)$, where $\operatorname{\mathcal{C}}$ is a small $\infty $-category and $C$ is an object of $\operatorname{\mathcal{C}}$ (Definition 5.5.6.10), and we let $V: \operatorname{\mathcal{QC}}_{\operatorname{Obj}} \rightarrow \operatorname{\mathcal{QC}}$ denote the forgetful functor (given on objects by the formula $V(\operatorname{\mathcal{C}},C) = \operatorname{\mathcal{C}}$).

Definition 5.6.5.1. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets. We will say that a commutative diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [d]^{U} \ar [r]^-{ \widetilde{\mathscr {F}} } & \operatorname{\mathcal{QC}}_{\operatorname{Obj}} \ar [d]^{V} \\ \operatorname{\mathcal{C}}\ar [r]^-{ \mathscr {F} } & \operatorname{\mathcal{QC}}} \]

witnesses $\mathscr {F}$ as a covariant transport representation of $U$ if the induced map

\[ \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{QC}}} \operatorname{\mathcal{QC}}_{\operatorname{Obj}} = \int _{\operatorname{\mathcal{C}}} \mathscr {F} \]

is an equivalence of cocartesian fibrations over $\operatorname{\mathcal{C}}$, in the sense of Definition 5.1.6.1. We say that $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$ is a covariant transport representation of $U$ if there exists a diagram which witnesses $\mathscr {F}$ as a covariant transport representation of $U$.

Remark 5.6.5.2. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of $\infty $-categories and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$ be a functor. By virtue of Proposition 5.1.6.5, a diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [d]^{U} \ar [r]^-{ \widetilde{\mathscr {F}} } & \operatorname{\mathcal{QC}}_{\operatorname{Obj}} \ar [d]^{V} \\ \operatorname{\mathcal{C}}\ar [r]^-{ \mathscr {F} } & \operatorname{\mathcal{QC}}} \]

witnesses $\mathscr {F}$ as a covariant transport representation for $U$ if and only if the induced map $\operatorname{\mathcal{E}}\rightarrow \int _{\operatorname{\mathcal{C}}} \mathscr {F}$ is an equivalence of $\infty $-categories. We will later extend this observation to the case where $\operatorname{\mathcal{C}}$ is a general simplicial set (Corollary 5.6.7.7).

Remark 5.6.5.3. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets. A commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [d]^{U} \ar [r]^-{ \widetilde{\mathscr {F}} } & \operatorname{\mathcal{QC}}_{\operatorname{Obj}} \ar [d]^{V} \\ \operatorname{\mathcal{C}}\ar [r]^-{ \mathscr {F} } & \operatorname{\mathcal{QC}}} \]

witnesses $\mathscr {F}$ as a covariant transport representation of $U$ if and only if it satisfies the following pair of conditions:

$(a)$

For every vertex $C \in \operatorname{\mathcal{C}}$, the map of fibers

\[ \widetilde{\mathscr {F}}_{C}: \operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \{ C\} \times _{\operatorname{\mathcal{C}}} \int _{\operatorname{\mathcal{C}}} \mathscr {F} \]

is an equivalence of $\infty $-categories.

$(b)$

The morphism $\widetilde{\mathscr {F}}$ carries $U$-cocartesian edges of $\operatorname{\mathcal{E}}$ to $V$-cocartesian edges of $\operatorname{\mathcal{QC}}_{\operatorname{Obj}}$.

See Proposition 5.1.6.14. Moreover, we can replace $(b)$ by the following a priori weaker condition (see Remark 5.1.5.8):

$(b')$

For every vertex $X \in \operatorname{\mathcal{E}}$ and every edge $\overline{e}: U(X) \rightarrow \overline{Y}$ in $\operatorname{\mathcal{C}}$, there exists a $U$-cocartesian edge $e: X \rightarrow Y$ of $\operatorname{\mathcal{E}}$ for which $U( e) = \overline{e}$ and and $\widetilde{\mathscr {F}}(e)$ is a $V$-cocartesian edge of $\operatorname{\mathcal{QC}}_{\operatorname{Obj}}$.

Example 5.6.5.4 (Left Covering Maps). Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a left covering map of simplicial sets and let $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \operatorname{Set}$ be the homotopy transport representation of $U$ (Example 5.2.5.3), so that $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ can be identified with a morphism of simplicial sets $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }(\operatorname{Set})$. Combining Proposition 5.2.7.2 with Example 5.6.2.8, we obtain a canonical isomorphism of simplicial sets $\operatorname{\mathcal{E}}\simeq \int _{\operatorname{\mathcal{C}}} \operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$, which exhibits $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ as a covariant transport representation of $U$ (in the sense of Definition 5.6.5.1).

Example 5.6.5.5 (Fibrations over a Point). Let $\operatorname{\mathcal{E}}$ be a small $\infty $-category, which we identify with a morphism $\mathscr {F}: \Delta ^0 \rightarrow \operatorname{\mathcal{QC}}$. Then $\mathscr {F}$ is a covariant transport representation of the projection map $U: \operatorname{\mathcal{E}}\rightarrow \Delta ^0$. More precisely, Example 5.6.2.16 supplies an equivalence of $\infty $-categories $\operatorname{\mathcal{E}}\rightarrow \int _{\Delta ^0} \mathscr {F}$ which witnesses $\mathscr {F}$ as a covariant transport representation of $U$. More generally, a functor $\Delta ^0 \rightarrow \operatorname{\mathcal{QC}}$ is a covariant transport representation of $U$ if and only if corresponds to an $\infty $-category which is equivalent to $\operatorname{\mathcal{E}}$.

Example 5.6.5.6 (Weighted Nerves). Let $\operatorname{\mathcal{C}}$ be an ordinary category, let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ be a functor, and let $\operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$ be the weighted nerve of Definition 5.3.3.1. Then the projection map $U: \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is a cocartesian fibration (Proposition 5.3.3.15). Moreover, the equivalence

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

of Proposition 5.6.4.8 exhibits $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\mathscr {F})$ as a covariant transport representation for $U$.

Example 5.6.5.7 (Strict Transport). Let $\operatorname{\mathcal{C}}$ be an ordinary category, let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ be a cocartesian fibration of $\infty $-categories, and let $\operatorname{sTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ be the strict transport representation of $U$ (Construction 5.3.1.5). Then the functor

\[ \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{sTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}} ): \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{QCat}) = \operatorname{\mathcal{QC}} \]

is a covariant transport representation for $U$ (in the sense of Definition 5.6.5.1). In other words, $U$ is equivalent to the cocartesian fibration $U': \int _{\operatorname{\mathcal{C}}} \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{sTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}} ) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$. To see this, we observe that both $U$ and $U'$ are equivalent to the cocartesian fibration $\operatorname{N}_{\bullet }^{ \operatorname{sTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}} }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$: this follows from Theorem 5.3.5.6 and Proposition 5.6.4.8.

Remark 5.6.5.8. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets, and let $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ be the homotopy transport representation of $U$ (Construction 5.2.5.2). Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$ be a morphism of simplicial sets and let $\mathrm{h} \mathit{\mathscr {F}}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{QCat}}$ be the induced functor between homotopy categories. Let $\alpha : \operatorname{\mathcal{E}}\rightarrow \int _{\operatorname{\mathcal{C}}} \mathscr {F}$ be an equivalence of cocartesian fibrations over $\operatorname{\mathcal{C}}$. By virtue of Corollary 5.6.2.22, $\alpha $ induces an isomorphism from $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ to $\mathrm{h} \mathit{\mathscr {F}}$ in the functor category $\operatorname{Fun}( \mathrm{h} \mathit{\operatorname{\mathcal{C}}}, \mathrm{h} \mathit{\operatorname{QCat}})$. Stated more informally, any covariant transport representation of $U$ provides a lifting of the homotopy transport representation $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ from the ordinary category $\operatorname{Fun}( \mathrm{h} \mathit{\operatorname{\mathcal{C}}}, \mathrm{h} \mathit{\operatorname{QCat}} )$ to the $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{QC}})$. Moreover, if the simplicial set $\operatorname{\mathcal{C}}$ is an $\infty $-category, then the identification $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}} \simeq \mathrm{h} \mathit{\mathscr {F}}$ is an isomorphism of $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functors (Proposition 5.6.2.20).

Remark 5.6.5.9. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets and let $\mathscr {F}, \mathscr {F}': \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$ be morphisms which are isomorphic as objects of the diagram $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{QC}})$. Then $\mathscr {F}$ is a covariant transport representation of $U$ if and only if $\mathscr {F}'$ is a covariant transport representation of $U$. This follows immediately from Proposition 5.6.2.19.

We now formulate a stronger version of Theorem 5.6.0.3:

Theorem 5.6.5.10 (Relative Universality Theorem). Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets having essentially small fibers, let $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ be a simplicial subset having inverse image $\operatorname{\mathcal{E}}_0 = \operatorname{\mathcal{C}}_0 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\subseteq \operatorname{\mathcal{E}}$, and let $U_0: \operatorname{\mathcal{E}}_0 \rightarrow \operatorname{\mathcal{C}}_0$ be the restriction $U|_{\operatorname{\mathcal{E}}_0}$. Suppose we are given a commutative diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}_0 \ar [d]^{U_0} \ar [r]^-{ \widetilde{\mathscr {F}}_0 } & \operatorname{\mathcal{QC}}_{\operatorname{Obj}} \ar [d]^{V} \\ \operatorname{\mathcal{C}}_0 \ar [r]^-{ \mathscr {F}_0 } & \operatorname{\mathcal{QC}}} \]

which witnesses $\mathscr {F}_0$ as a covariant transport representation of $U_0$. Then there exists a commutative diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [d]^{U} \ar [r]^-{ \widetilde{\mathscr {F}} } & \operatorname{\mathcal{QC}}_{\operatorname{Obj}} \ar [d]^{V} \\ \operatorname{\mathcal{C}}\ar [r]^-{ \mathscr {F} } & \operatorname{\mathcal{QC}}} \]

which witnesses $\mathscr {F}$ as a covariant transport representation of $U$, where $\mathscr {F}_0 = \mathscr {F}|_{\operatorname{\mathcal{C}}_0}$ and $\widetilde{\mathscr {F}}_0 = \widetilde{\mathscr {F}}|_{\operatorname{\mathcal{E}}_0}$.

We will give a reformulation of Theorem 5.6.5.10 in §5.6.8 (see Theorem 5.6.8.3), which we prove in §5.6.9.

Corollary 5.6.5.11. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets having essentially small fibers, let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be a simplicial subset, and let $\mathscr {F}': \operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{QC}}$ be a covariant transport representation for the projection map $\operatorname{\mathcal{C}}' \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}'$. Then there exists a morphism $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$ satisfying $\mathscr {F}' = \mathscr {F}|_{\operatorname{\mathcal{C}}'}$ which is a covariant transport representation of $U$.

Corollary 5.6.5.12. Let $\operatorname{\mathcal{Q}}$ be a full subcategory of $\operatorname{\mathcal{QC}}$ and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets having the property that, for each vertex $C \in \operatorname{\mathcal{C}}$, the fiber $\operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is equivalent to an $\infty $-category which belongs to $\operatorname{\mathcal{Q}}$. Then there exists a morphism $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{Q}}\subseteq \operatorname{\mathcal{QC}}$ which is a covariant transport representation of $U$.

Proof. For each vertex $C \in \operatorname{\mathcal{C}}$, choose an $\infty $-category $\mathscr {F}'(C) \in \operatorname{\mathcal{Q}}$ which is equivalent to the fiber $\operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$. The construction $C \mapsto \mathscr {F}'(C)$ determines a morphism of simplicial sets $\mathscr {F}': \operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{Q}}$, where $\operatorname{\mathcal{C}}' = \operatorname{sk}_0(\operatorname{\mathcal{C}})$ is the $0$-skeleton of $\operatorname{\mathcal{C}}$, which is a covariant transport representation of the projection map $\operatorname{\mathcal{C}}' \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}'$ (see Example 5.6.5.5). Applying Corollary 5.6.5.11, we can extend $\mathscr {F}'$ to a morphism $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$ which is a covariant transport representation of $U$. By construction, the morphism $\mathscr {F}$ takes values in the full subcategory $\operatorname{\mathcal{Q}}\subseteq \operatorname{\mathcal{QC}}$. $\square$

Corollary 5.6.5.13. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets and let $\mathscr {F}_0, \mathscr {F}_1: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$ be covariant transport representations for $U$. Then $\mathscr {F}_0$ and $\mathscr {F}_1$ are isomorphic as objects of the $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{QC}})$.

Proof. Let $U_{\Delta ^1}: \Delta ^1 \times \operatorname{\mathcal{E}}\rightarrow \Delta ^1 \times \operatorname{\mathcal{C}}$ be the product of $U$ with the identity map $\operatorname{id}_{ \Delta ^1}$, and define $U_{\operatorname{\partial \Delta }^1}: \operatorname{\partial \Delta }^1 \times \operatorname{\mathcal{E}}\rightarrow \operatorname{\partial \Delta }^1 \times \operatorname{\mathcal{C}}$ similarly. Note that the map $( \mathscr {F}_0, \mathscr {F}_1 ): \operatorname{\partial \Delta }^1 \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$ is a covariant transport representation of $U_{\operatorname{\partial \Delta }^1}$. Applying Corollary 5.6.5.11, we deduce that $U_{\Delta ^1}$ admits a covariant transport representation $\mathscr {F}: \Delta ^1 \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$ which satisfies $\mathscr {F}|_{ \{ 0\} \times S} = \mathscr {F}_0$ and $\mathscr {F}|_{ \{ 1\} \times S} = \mathscr {F}_1$. Let us identify $\mathscr {F}$ with a morphism $u: \mathscr {F}_{0} \rightarrow \mathscr {F}_{1}$ in the $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{QC}})$. We will complete the proof by showing that $u$ is an isomorphism. By virtue of Theorem 4.4.4.4, it will suffice to show that for each vertex $C \in \operatorname{\mathcal{C}}$, the induced map $u_ C: \mathscr {F}_0(C) \rightarrow \mathscr {F}_1(C)$ is an isomorphism in $\operatorname{\mathcal{QC}}$. Using Remark 5.6.5.8 (and Remark 5.2.8.5), we see that the homotopy class $[u_ C]$ is isomorphic (as an object of the arrow category $\operatorname{Fun}( [1], \mathrm{h} \mathit{\operatorname{QCat}} )$) to the homotopy class of the functor $\operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}_{C}$ given by covariant transport along the degenerate edge $\operatorname{id}_{C}$ of $\operatorname{\mathcal{C}}$: that is, the homotopy class of the identity functor $\operatorname{id}_{ \operatorname{\mathcal{E}}_{C} }$. $\square$

Proof of Theorem 5.6.0.3. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibrations of simplicial sets and suppose that, for each vertex $C \in \operatorname{\mathcal{C}}$, the fiber $\operatorname{\mathcal{E}}_{C} = \{ C \} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is essentially small. We wish to show that $U$ admits a covariant transport representation $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$, which is uniquely determined up to isomorphism (as an object of the functor $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{QC}})$). The existence statement follows by applying Theorem 5.6.5.10 in the special case $\operatorname{\mathcal{C}}_0 = \emptyset $, and the uniqueness follows from Corollary 5.6.5.13. $\square$

Notation 5.6.5.14 (The Covariant Transport Representation). Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets having essentially small fibers. We let $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ denote a covariant transport representation of $U$, regarded as an object of the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{QC}})$ (which exists by virtue of Corollary 5.6.5.12). We write $[ \operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}} ]$ for the isomorphism class of the diagram $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$, regarded as an object of the set $\pi _0( \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{QC}})^{\simeq } )$. By virtue of Corollary 5.6.5.13, the isomorphism class $[ \operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}} ]$ is well-defined: that is, it depends only on the cocartesian fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$. Beware that $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ is not unique determined: in fact, any diagram isomorphic to $\operatorname{Tr}_{\operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}}}$ is also a covariant transport representation of $U$ (Remark 5.6.5.9). Nevertheless, it will be convenient to abuse terminology and refer to $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ as the covariant transport representation of $U$, with the caveat that it is well-defined only up to isomorphism.

Remark 5.6.5.15. Let $\operatorname{\mathcal{C}}$ be a simplicial set equipped with a functor $\overline{\mathscr {F}}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{QCat}}$. It follows from Corollary 5.6.5.12 that the functor $\overline{\mathscr {F}}$ is isomorphic to the homotopy transport representation of a cocartesian fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ if and only if it can be promoted to a diagram $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$.

Corollary 5.6.5.16. Let $\operatorname{\mathcal{C}}$ be a small category. Then passage to the homotopy coherent nerve induces a bijection

\[ \xymatrix { \{ \textnormal{Functors of ordinary categories $\operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$} \} / \textnormal{Levelwise equivalence} \ar [d] \\ \{ \textnormal{Functors of $\infty $-categories $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{QC}}$} \} / \textnormal{Isomorphism}. } \]

Remark 5.6.5.17 (Rectification). Corollary 5.6.5.16 is a prototypical example of a rectification result. If $\operatorname{\mathcal{C}}$ is an ordinary category, then a functor of $\infty $-categories $\mathscr {F}: \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{QC}}$ can be viewed as a homotopy coherent diagram in the simplicial category $\operatorname{QCat}$:

  • To every object $X$ of the category $\operatorname{\mathcal{C}}$, the functor $\mathscr {F}$ associates an $\infty $-category $\mathscr {F}(X)$.

  • To every morphism $u: X \rightarrow Y$ of the category $\operatorname{\mathcal{C}}$, the functor $\mathscr {F}$ associates a functor of $\infty $-categories $\mathscr {F}(u): \mathscr {F}(X) \rightarrow \mathscr {F}(Y)$.

  • To every pair of composable morphisms $u: X \rightarrow Y$ and $v: Y \rightarrow Z$ in the category $\operatorname{\mathcal{C}}$, the functor $\mathscr {F}$ associates an isomorphism of functors $\alpha _{u,v}: \mathscr {F}(v) \circ \mathscr {F}(u) \rightarrow \mathscr {F}(v \circ u)$.

  • When applied to higher-dimensional simplices of $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$, the functor $\mathscr {F}$ provides additional data which encode coherence laws satisfied by the isomorphisms $\alpha _{u,v}$.

Corollary 5.6.5.16 asserts that we can always find a strictly commutative diagram $\mathscr {G}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ which is isomorphic to $\mathscr {F}$ in the $\infty $-category $\operatorname{Fun}( \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}), \operatorname{\mathcal{QC}})$. In particular, the diagram $\mathscr {G}$ carries each object $X \in \operatorname{\mathcal{C}}$ to an $\infty $-category $\mathscr {G}(X)$ which is equivalent to $\mathscr {F}(X)$ (beware that we generally cannot arrange that $\mathscr {G}(X)$ is isomorphic to $\mathscr {F}(X)$ as a simplicial set).

In §, we will prove a more refined version of this result, which allows us to describe the entire $\infty $-category $\operatorname{Fun}( \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}), \operatorname{\mathcal{QC}})$ in terms of strictly commutative diagrams indexed by $\operatorname{\mathcal{C}}$ (Proposition ).

Using Theorem 5.6.5.10, we obtain the following converse of Corollary 5.6.3.5.

Proposition 5.6.5.18. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. The following conditions are equivalent:

$(1)$

The morphism $U$ an inner covering map (Definition 4.1.5.1), a cocartesian fibration, and each fiber of $U$ is small.

$(2)$

There exists morphism of simplicial sets $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }^{\operatorname{D}}( \operatorname{Pith}(\mathbf{Cat}) ) \subseteq \operatorname{\mathcal{QC}}$ and an isomorphism $G: \operatorname{\mathcal{E}}\simeq \int _{\operatorname{\mathcal{C}}} \mathscr {F}$ in the category $(\operatorname{Set_{\Delta }})_{/\operatorname{\mathcal{C}}}$.

Proof. The implication $(2) \Rightarrow (1)$ follows from Corollary 5.6.3.5 and Proposition 5.6.2.2. For each vertex $C \in \operatorname{\mathcal{C}}$, our assumption that $U$ is an inner covering map guarantees that the fiber $\{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is isomorphic to the nerve of a (small) category $\mathscr {F}_0(C)$ (Example 4.1.5.3). Let $\operatorname{\mathcal{C}}_0$ be the $0$-skeleton of $\operatorname{\mathcal{C}}$, so that the construction $C \mapsto \mathscr {F}_0(C)$ determines a morphism of simplicial sets $\mathscr {F}_0: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{N}_{\bullet }^{\operatorname{D}}( \operatorname{Pith}(\mathbf{Cat}) )$. Let $\operatorname{\mathcal{E}}_0$ denote the inverse image $\operatorname{\mathcal{C}}_0 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$, so that Proposition 5.6.3.4 supplies an isomorphism of simplicial sets $G_0: \operatorname{\mathcal{E}}_0 \simeq \int _{\operatorname{\mathcal{C}}_0} \mathscr {F}_0$. In particular, $G_0$ is an equivalence of cocartesian fibrations over $\operatorname{\mathcal{C}}_0$. Invoking Theorem 5.6.5.10, we can extend $\mathscr {F}_0$ to a diagram $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }^{\operatorname{D}}( \operatorname{Pith}( \mathbf{Cat} ) )$ and $G_0$ to a morphism of simplicial sets $G: \operatorname{\mathcal{E}}\rightarrow \int _{\operatorname{\mathcal{C}}} \mathscr {F}$ which is an equivalence of cocartesian fibrations over $\operatorname{\mathcal{C}}$. We will complete the proof by showing that $G$ is an isomorphism of simplicial sets. To prove this, it will suffice to show that for every simplex $G_{\sigma }: \Delta ^{n} \rightarrow \operatorname{\mathcal{C}}$, the induced map

\[ G_{\sigma }: \Delta ^{n} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \Delta ^{n} \times _{\operatorname{\mathcal{C}}} \int _{\operatorname{\mathcal{C}}} \mathscr {F} \]

is an isomorphism of simplicial sets. Replacing $U$ by the projection map $\Delta ^{n} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \Delta ^ n$, we are reduced to proving that $G$ is an isomorphism under the additional assumption that $\operatorname{\mathcal{C}}= \Delta ^ n$ is a standard simplex. Since $U$ and the projection map $\int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$ are inner covering maps, the simplicial sets $\operatorname{\mathcal{E}}$ and $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ are isomorphic to the nerves of their homotopy categories $\mathrm{h} \mathit{\operatorname{\mathcal{E}}}$ and $\mathrm{h} \mathit{ \int _{\operatorname{\mathcal{C}}} \mathscr {F} }$, respectively; it will therefore suffice to show that the functor of ordinary categories $\mathrm{h} \mathit{G}: \mathrm{h} \mathit{\operatorname{\mathcal{E}}} \rightarrow \mathrm{h} \mathit{ \int _{\operatorname{\mathcal{C}}} \mathscr {F} }$ is an isomorphism. Our assumption that $G$ is an equivalence of cocartesian fibrations over $\operatorname{\mathcal{C}}= \Delta ^ n$ guarantees that it is an equivalence of $\infty $-categories (Corollary 5.1.6.8), so that $\mathrm{h} \mathit{G}$ is an equivalence of ordinary categories. It will therefore suffice to show that the functor $\mathrm{h} \mathit{G}$ is bijective on objects: that is, that the morphism $G$ is bijective on vertices. This is clear, since the morphism $G_0 = G|_{\operatorname{\mathcal{E}}_0}$ is an isomorphism. $\square$

Corollary 5.6.5.19 (Grothendieck). Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be functor between categories. The following conditions are equivalent:

$(1)$

The functor $U$ is a cocartesian fibration and each fiber of $U$ is a small category.

$(2)$

There exists a functor of $2$-categories $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \mathbf{Cat}$ and an isomorphism $\int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{E}}$ whose composition with $U$ coincides with the forgetful functor $\int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$.

Proof. We will show that $(1) \Rightarrow (2)$; the reverse implication follows from Corollary 5.6.1.16. Note that the 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) and an inner covering map (Proposition 4.1.5.10). By virtue of Proposition 5.6.5.18, there exists a morphism of simplicial sets $\mathscr {F}': \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{D}}( \operatorname{Pith}( \mathbf{Cat}) )$ and an isomorphism of simplicial sets $V: \int _{\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})} \mathscr {F} \simeq \operatorname{N}_{\bullet }(\operatorname{\mathcal{E}})$ which is compatible with $\operatorname{N}_{\bullet }(U)$. By virtue of Theorem 2.3.4.1 (and Corollary 2.3.4.5), we have $\mathscr {F}' = \operatorname{N}_{\bullet }^{\operatorname{D}}( \mathscr {F} )$ for a unique functor of $2$-categories $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \mathbf{Cat}$. In this case, we can use Proposition 5.6.3.4 to identify $\int _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})} \mathscr {F}'$ with the nerve of the ordinary category of elements $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$. Under this identification, $V$ corresponds to the nerve of an isomorphism $\int _{\operatorname{\mathcal{C}}} \mathscr {F}' \simeq \operatorname{\mathcal{E}}$ which is compatible with $U$. $\square$

Let $\mathbf{Gpd} \subseteq \mathbf{Cat}$ denote the full subcategory spanned by the groupoids.

Corollary 5.6.5.20. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a functor between categories. The following conditions are equivalent:

  • The functor $U$ is an opfibration in groupoids (Variant 4.2.2.4) and each fiber of $U$ is a small groupoid.

  • There exists a functor of $2$-categories $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \mathbf{Gpd}$ and an isomorphism of categories $\int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{E}}$ which carries $U$ to the forgetful functor $\int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$.