# Kerodon

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### 5.6.2 The Covariant Transport Representation

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 5.4.6.10), 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 5.6.2.1. 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 5.6.2.2. 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 5.6.4.7).

Remark 5.6.2.3. 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 5.6.2.4 (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 5.6.2.1).

Example 5.6.2.5 (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 5.5.4.16 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 $\operatorname{\mathcal{E}}$.

Remark 5.6.2.6. 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. 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 5.5.4.22, $\alpha$ 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 5.5.4.20).

Remark 5.6.2.7. 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 5.5.4.19.

We now formulate a stronger version of Theorem 5.6.0.2:

Theorem 5.6.2.8 (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 5.6.2.8 in §5.6.5 (see Theorem 5.6.5.3), which we prove in §5.6.6.

Corollary 5.6.2.9. 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 5.6.2.10. 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 5.6.2.5). Applying Corollary 5.6.2.9, 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 5.6.2.11. 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 5.6.2.9, 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 5.6.2.6 (and Remark 5.2.7.5), 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 5.6.0.2. 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 5.6.2.8 in the special case $\operatorname{\mathcal{C}}_0 = \emptyset$, and the uniqueness follows from Corollary 5.6.2.11. $\square$

Notation 5.6.2.12 (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 5.6.2.10). 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 5.6.2.11, 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 5.6.2.7). 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 5.6.2.13. 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 5.6.2.10 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 $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$.