Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

5.6.4 The Covariant Transport Representation

Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets. Corollary 5.6.3.12 asserts that, if each fiber $\operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is essentially small, then $U$ is equivalent (as a cocartesian fibration) to the projection map $\int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$ for some diagram $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$, which is uniquely determined up to isomorphism. We will prove this by applying the structural analysis of §5.2.6 to the projection map $\Delta ^{n} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \Delta ^ n$, for each simplex $\sigma : \Delta ^ n \rightarrow \operatorname{\mathcal{C}}$. To carry out this strategy, we need to formulate a stronger version of Corollary 5.6.3.12.

We begin by elaborating on Definition 5.6.0.1. In what follows, we let $\operatorname{\mathcal{QC}}_{\ast }^{\operatorname{lax}}$ 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 5.4.6.10), and we let $V: \operatorname{\mathcal{QC}}_{\ast }^{\operatorname{lax}} \rightarrow \operatorname{\mathcal{QC}}$ denote the forgetful functor (given on objects by the formula $V(\operatorname{\mathcal{C}},C) = \operatorname{\mathcal{C}}$).

Definition 5.6.4.1. 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

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

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

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

is an equivalence of cocartesian fibrations over $\operatorname{\mathcal{C}}$, in the sense of Definition 5.6.2.1. 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 exhibits $\mathscr {F}$ as a covariant transport representation of $U$.

Remark 5.6.4.2. 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 5.6.2.5, a functor $G: \operatorname{\mathcal{E}}\rightarrow \int _{\operatorname{\mathcal{C}}} \mathscr {F}$ exhibits $\mathscr {F}$ as a covariant transport representation of $U$ if and only if it is an equivalence of $\infty $-categories for which the composite map $\operatorname{\mathcal{E}}\xrightarrow {G} \int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$ is equal to $U$. In particular, $\mathscr {F}$ is a covariant transport representation for $U$ in the sense of Definition 5.6.4.1 if and only if it is a covariant transport representation for $U$ in the sense of Definition 5.6.0.1. We will later extend this observation to the case where $\operatorname{\mathcal{C}}$ is a general simplicial set (Corollary 5.6.6.6).

Remark 5.6.4.3. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets. A commutative diagram

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

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

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

  • For every object $C \in \operatorname{\mathcal{C}}$, 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.

See Proposition 5.6.2.11.

Remark 5.6.4.4. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$ be a morphism of simplicial sets. Then $\mathscr {F}$ is a covariant transport representation of $U$ if and only if there exists a morphism of simplicial sets $G: \operatorname{\mathcal{E}}\rightarrow \int _{\operatorname{\mathcal{C}}} \mathscr {F}$ which is an equivalence of cocartesian fibrations over $\operatorname{\mathcal{C}}$ (so that, in particular, the composite map $\operatorname{\mathcal{E}}\xrightarrow {G} \int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$ is equal to $U$). In this case, we will say that the morphism $G$ exhibits $\mathscr {F}$ as a covariant transport representation of $U$.

Example 5.6.4.5 (Left Covering Maps). Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a left covering map of simplicial sets and let $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \operatorname{Set}$ be the functor of Construction 5.6.1.2, which we identify with a morphism of simplicial sets $\operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }(\operatorname{Set}) \subset \operatorname{\mathcal{QC}}$. Then Proposition 5.6.1.5 supplies an 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 5.6.4.1).

Example 5.6.4.6 (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 5.5.4.17 supplies an equivalence of $\infty $-categories $\operatorname{\mathcal{E}}\rightarrow \int _{\Delta ^0} \mathscr {F}$ which exhibits $\mathscr {F}$ as a covariant transport representation of $U$. More generally, a morphism $\Delta ^0 \rightarrow \operatorname{\mathcal{QC}}$ is a covariant transport representation of $U$ if and only if corresponds to an $\infty $-category which is equivalent to $\operatorname{\mathcal{E}}$.

Remark 5.6.4.7. 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 5.2.3.2). 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. Suppose that $G: \operatorname{\mathcal{E}}\rightarrow \int _{\operatorname{\mathcal{C}}} \mathscr {F}$ exhibits $\mathscr {F}$ as a covariant transport representation of $U$. By virtue of Corollary 5.5.6.10, $G$ 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 $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{QC}})$.

Remark 5.6.4.8. 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 Corollary 5.6.3.3.

We can now formulate our main result.

Theorem 5.6.4.9 (Relative Universality Theorem). Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets having essentially small fibers, let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be a simplicial subset having inverse image $\operatorname{\mathcal{E}}' = \operatorname{\mathcal{C}}' \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\subseteq \operatorname{\mathcal{E}}$, and let $U': \operatorname{\mathcal{E}}' \rightarrow \operatorname{\mathcal{C}}'$ be the restriction $U|_{\operatorname{\mathcal{E}}'}$. Suppose we are given a commutative diagram of simplicial sets

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

which exhibits $\mathscr {F}'$ as a covariant transport representation of $U'$. Then there exists a commutative diagram of simplicial sets

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

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

We will give the proof of Theorem 5.6.4.9 in §5.6.7.

Corollary 5.6.4.10. 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 of 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 5.6.4.11. Let $\operatorname{\mathcal{Q}}$ be a full subcategory of $\operatorname{\mathcal{QC}}$ and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a $\operatorname{\mathcal{Q}}$-small cocartesian fibration of simplicial sets (Definition 5.6.3.4). 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 5.6.4.6). Applying Corollary 5.6.4.10, 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 5.6.4.12. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets and let $\mathscr {F}_0, \mathscr {F}_1: S \rightarrow \operatorname{\mathcal{QC}}$ 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 5.6.4.10, 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 5.6.4.7 (and Remark 5.2.2.7), we see that the isomorphism class $[u_ C]$ is isomorphic (as an object of the arrow category $\operatorname{Fun}( [1], \mathrm{h} \mathit{\operatorname{QCat}} )$) to the isomorphism 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 isomorphism class of the identity functor $\operatorname{id}_{ \operatorname{\mathcal{E}}_{C} }$. $\square$

Proof of Theorem 5.6.3.6. Let $\operatorname{\mathcal{Q}}$ be a full subcategory of $\operatorname{\mathcal{QC}}$ and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a $\operatorname{\mathcal{Q}}$-small cocartesian fibration. We wish to show that $U$ admits a covariant transport representation $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{Q}}$, which is uniquely determined up to isomorphism (as an object of the functor $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{Q}})$). The existence statement follows from Corollary 5.6.4.11, and the uniqueness from Corollary 5.6.4.12. $\square$

Notation 5.6.4.13 (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 5.6.4.11). 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 5.6.4.12, 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 5.6.4.8). 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 5.6.4.14. Let $\operatorname{\mathcal{C}}$ be a simplicial set equipped with a functor $\overline{\mathscr {F}}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{QCat}}$. Combining Corollary 5.6.4.11, we see that $\overline{\mathscr {F}}$ 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 lifted to a diagram $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$.