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5.2.3 Transitivity of Covariant Transport

We now study the behavior of the transport functors of ยง5.2.2 with respect to composition.

Proposition 5.2.3.1 (Transitivity). Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets and let $\sigma $ be a $2$-simplex of $\operatorname{\mathcal{C}}$, which we display as a diagram

\[ \xymatrix@R =50pt@C=50pt{ & D \ar [dr]^{g} & \\ C \ar [ur]^{f} \ar [rr]^{h} & & E. } \]

Let $f_!: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}_{D}$ and $g_!: \operatorname{\mathcal{E}}_{D} \rightarrow \operatorname{\mathcal{E}}_{E}$ be functors which are given by covariant transport along $f$ and $g$, respectively. Then the composite functor $(g_{!} \circ f_{!}): \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}_{E}$ is given by covariant transport along $w$.

Proof. Without loss of generality, we may replace $U$ by the projection map $\Delta ^{2} \times _{ \operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \Delta ^2$, and thereby reduce to the case where $\operatorname{\mathcal{C}}= \Delta ^2$ and $\sigma $ is the unique nondegenerate $2$-simplex of $\operatorname{\mathcal{C}}$. In this case, $\operatorname{\mathcal{E}}$ is an $\infty $-category. Let $\widetilde{f}: \Delta ^1 \times \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}$ and $\widetilde{g}: \Delta ^1 \times \operatorname{\mathcal{E}}_{D} \rightarrow \operatorname{\mathcal{E}}$ be natural transformations which witness $f_!$ and $g_!$ as given by covariant transport along $f$ and $g$, respectively. Let $\widetilde{g}'$ denote the composition

\[ \Delta ^{1} \times \operatorname{\mathcal{E}}_{C} \xrightarrow { \operatorname{id}\times f_! } \Delta ^1 \times \operatorname{\mathcal{E}}_{D} \xrightarrow {\widetilde{g}} \operatorname{\mathcal{E}}, \]

which we view as a morphism from $f_{!}$ to $g_{!} \circ f_{!}$ in the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{E}}_{C}, \operatorname{\mathcal{E}})$. Let $\widetilde{h}: \operatorname{id}_{\operatorname{\mathcal{E}}_ C} \rightarrow (g_! \circ f_!)$ be any composition of $\widetilde{f}$ and $\widetilde{g}'$ in the functor $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{E}}_{C}, \operatorname{\mathcal{E}})$. For every object $X \in \operatorname{\mathcal{E}}_{C}$, the edge $\widetilde{h}_{X}: X \rightarrow (g_! \circ f_!)(X)$ can be realized as a composition the $U$-cocartesian morphisms $\widetilde{f}_{X}: X \rightarrow f_{!}(X)$ and $\widetilde{g}_{f_{!}(X)}: f_{!}(X) \rightarrow (g_! \circ f_!)(X)$, and is therefore also $U$-cocartesian (Corollary 5.1.2.5). It follows that $\widetilde{h}$ witnesses $g_{!} \circ f_{!}$ as given by covariant transport along $h$. $\square$

Construction 5.2.3.2 (The Homotopy Transport Representation: Covariant Case). Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets and let $\mathrm{h} \mathit{\operatorname{QCat}}$ denote the homotopy category of $\infty $-categories. It follows from Proposition 5.2.3.1 and Example 5.2.2.2 that there is a unique functor $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{QCat}}$ with the following properties:

  • For each vertex $C$ of the simplicial set $\operatorname{\mathcal{C}}$, $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}(C)$ is the $\infty $-category $\operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ (regarded as an object of $\mathrm{h} \mathit{\operatorname{QCat}}$).

  • For each edge $f: C \rightarrow D$ of the simplicial set $\operatorname{\mathcal{C}}$ representing a morphism $[f] \in \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(C,D)$, we have $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}( [f] ) = [ f_{!} ]$, where $[f_{!}]$ denotes the isomorphism class of the covariant transport functor defined in Notation 5.2.2.5 (regarded as an element of $\operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{QCat}} }( \operatorname{\mathcal{E}}_{C}, \operatorname{\mathcal{E}}_{D} ) = \pi _0( \operatorname{Fun}( \operatorname{\mathcal{E}}_{C}, \operatorname{\mathcal{E}}_{D})^{\simeq } )$).

We will refer to $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ as the homotopy transport representation of the cocartesian fibration $U$.

Remark 5.2.3.3. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets, and let $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{QCat}}$ be the homotopy transport representation of Construction 5.2.3.2. It follows from Proposition 5.1.4.14 that $U$ is a left fibration if and only if the functor $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ factors through the full subcategory $\mathrm{h} \mathit{\operatorname{Kan}} \subseteq \mathrm{h} \mathit{\operatorname{QCat}}$. In particular, if $U$ is a left fibration, then Construction 5.2.3.2 determines a functor $\mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{Kan}}$ which we will also refer to as the homotopy transport representation of the left fibration $U$.

Construction 5.2.3.2 has an analogue for cartesian fibrations:

Construction 5.2.3.4 (The Homotopy Transport Representation: Covariant Case). Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cartesian fibration of simplicial sets and let $\mathrm{h} \mathit{\operatorname{QCat}}$ denote the homotopy category of $\infty $-categories (Construction 4.5.1.1). It follows from Proposition 5.2.3.1 and Example 5.2.2.2 that there is a unique functor $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}}^{\operatorname{op}} \rightarrow \mathrm{h} \mathit{\operatorname{QCat}}$ satisfying the following conditions:

  • For each vertex $C$ of the simplicial set $\operatorname{\mathcal{C}}$, $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}(C)$ is the $\infty $-category $\operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ (regarded as an object of $\mathrm{h} \mathit{\operatorname{QCat}}$).

  • For each edge $f: C \rightarrow D$ of the simplicial set $\operatorname{\mathcal{C}}$ representing a morphism $[f] \in \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(C,D)$, we have $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}( [f] ) = [ f^{\ast } ]$, where $[f^{\ast }]$ denotes the isomorphism class of the contravariant transport functor defined in Notation 5.2.2.11.

We will refer to $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ as the homotopy transport representation of the cartesian fibration $U$.

Warning 5.2.3.5. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets which is both a cartesian fibration and a cocartesian fibration. Then Constructions 5.2.3.2 and 5.2.3.4 supply functors $\mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{QCat}}$ and $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}^{\operatorname{op}} \rightarrow \mathrm{h} \mathit{\operatorname{QCat}}$ respectively, which are both referred to as the homotopy transport representation of $U$ and both denoted by $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$. We will see later that these two functors are interchangeable data: either can be recovered from the other (see Proposition 6.2.3.5).

Example 5.2.3.6. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. Combining Remark 5.2.3.3 with Theorem 5.2.2.13, we deduce that the following conditions are equivalent:

  • The morphism $U$ is a Kan fibration.

  • The morphism $U$ is a cocartesian fibration and the homotopy transport representation $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{QCat}}$ of Construction 5.2.3.2 factors through the subcategory $\mathrm{h} \mathit{\operatorname{Kan}}^{\simeq } \subseteq \mathrm{h} \mathit{\operatorname{QCat}}$.

  • The morphism $U$ is a cartesian fibration and the homotopy transport representation $\operatorname{hTr}'_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}}^{\operatorname{op}} \rightarrow \mathrm{h} \mathit{\operatorname{QCat}}$ of Construction 5.2.3.4 factors through the subcategory $\mathrm{h} \mathit{\operatorname{Kan}}^{\simeq } \subseteq \mathrm{h} \mathit{\operatorname{QCat}}$.

If these conditions are satisfied, then $\operatorname{hTr}'_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ is given by the composition

\[ \mathrm{h} \mathit{\operatorname{\mathcal{C}}}^{\operatorname{op}} \xrightarrow { \operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}^{\operatorname{op}} } ( \mathrm{h} \mathit{\operatorname{Kan}}^{\simeq } )^{\operatorname{op}} \xrightarrow {\iota } \mathrm{h} \mathit{\operatorname{Kan}}^{\simeq }, \]

where $\iota $ is the isomorphism which carries each morphism in $\mathrm{h} \mathit{\operatorname{Kan}}^{\simeq }$ to its inverse.