Kerodon

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

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 $w$.

Proposition 5.2.3.1 is a special case of a more general statement concerning parametrized covariant transport (Proposition 5.2.3.7), which we prove at the end of this section.

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 of Example 5.2.2.6 (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: Contravariant 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 Example 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 5.2.2.15, 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.

Proposition 5.2.3.1 is a special case of the following more general result:

Proposition 5.2.3.7. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets. Let $C_0$, $C_1$, and $C_2$ be vertices of $\operatorname{\mathcal{C}}$ and suppose we are given a morphism of simplicial sets $f: K \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C_0, C_1, C_2)$ (see Construction 4.6.7.9). For $0 \leq i \leq j \leq 2$, let $f_{ij}$ denote the composite map

\[ K \xrightarrow {f} \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C_0, C_1, C_2) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C_ i, C_ j ). \]

Let $F_{01}: K \times \operatorname{\mathcal{E}}_{C_0} \rightarrow \operatorname{\mathcal{E}}_{C_1}$ and $F_{12}: K \times \operatorname{\mathcal{E}}_{C_1} \rightarrow \operatorname{\mathcal{E}}_{C_2}$ be given by covariant transport along $f_{01}$ and $f_{12}$, respectively, and let $F_{02}$ denote the composite map

\[ K \times \operatorname{\mathcal{E}}_{C_0} \hookrightarrow K \times K \times \operatorname{\mathcal{E}}_{C_0} \xrightarrow { \operatorname{id}_ K \times F_{10} } K \times \operatorname{\mathcal{E}}_{C_1} \xrightarrow { F_{21} } \operatorname{\mathcal{E}}_{C_2}. \]

Then $F_{02}$ is given by covariant transport along $f_{02}$.

Proof of Proposition 5.2.3.1. Apply Proposition 5.2.3.7 in the special case $K = \Delta ^0$. $\square$

Proof of Proposition 5.2.3.7. Choose morphisms

\[ \widetilde{F}_{01}: \operatorname{N}_{\bullet }( \{ 0 < 1 \} ) \times K \times \operatorname{\mathcal{E}}_{C_0} \rightarrow \operatorname{\mathcal{E}} \]

\[ \widetilde{F}_{12}: \operatorname{N}_{\bullet }( \{ 1 < 2 \} ) \times K \times \operatorname{\mathcal{E}}_{C_1} \rightarrow \operatorname{\mathcal{E}} \]

which witness $F_{01}$ and $F_{12}$ as given by covariant transport along $f_{01}$ and $f_{12}$, respectively. Let $H$ denote the composite map

\[ \Delta ^2 \times K \times \operatorname{\mathcal{E}}_{C_0} \rightarrow \Delta ^2 \times K \xrightarrow {\operatorname{id}\times f} \Delta ^2 \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C_0, C_1, C_2) \rightarrow \operatorname{\mathcal{C}}. \]

Amalgamating $\widetilde{F}_{01}$ with the composite map

\[ \operatorname{N}_{\bullet }( \{ 1 < 2 \} ) \times K \times \operatorname{\mathcal{E}}_{C_0} \xrightarrow {\operatorname{id}\times F_{01} } \operatorname{N}_{\bullet }( \{ 1 < 2 \} ) \times K \times \operatorname{\mathcal{E}}_{C_1} \xrightarrow { \widetilde{F}_{12} } \operatorname{\mathcal{E}}_{C_2}, \]

we obtain a morphism of simplicial sets $\widetilde{F}': \Lambda ^{2}_{1} \times K \times \operatorname{\mathcal{E}}_{C_0} \rightarrow \operatorname{\mathcal{E}}$ which fits into a commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{2}_{1} \times K \times \operatorname{\mathcal{E}}_{C_0} \ar [d] \ar [r]^-{\widetilde{F}'} & \operatorname{\mathcal{E}}\ar [d]^{U} \\ \Delta ^2 \times K \times \operatorname{\mathcal{E}}_{C_0} \ar@ {-->}[ur]^{\widetilde{F} } \ar [r]^-{H} & \operatorname{\mathcal{C}}. } \]

Since the left vertical map is inner anodyne (Lemma 1.4.6.9), this lifting problem admits a solution $\widetilde{F}: \Delta ^2 \times K \times \operatorname{\mathcal{E}}_{C_0} \rightarrow \operatorname{\mathcal{E}}$. For every vertex $v \in K$ and every object $X \in \operatorname{\mathcal{E}}_{C_0}$, the restriction of $\widetilde{F}$ to $\Delta ^2 \times \{ v\} \times \{ X \} $ determines a $2$-simplex $\widetilde{F}(v,X)$ of $\operatorname{\mathcal{E}}$, which we can depict as a diagram

\[ \xymatrix@R =50pt@C=50pt{ & F_{01}(v,X) \ar [dr] & \\ X \ar [ur] \ar [rr] & & F_{12}(v, F_{01}(v,X) ). } \]

By construction, the diagonal edges in this diagram are locally $U$-cocartesian, and therefore $U$-cocartesian (Remark 5.1.4.5). Applying Proposition 5.1.4.12, we deduce that the horizontal edge is also $U$-cocartesian. It follows that $\widetilde{F}_{02}$ witnesses $F_{02}$ as given by covariant transport along $f_{02}$, in the sense of Definition 5.2.2.1. $\square$

Corollary 5.2.3.8. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of $\infty $-categories. Let $C$, $D$, and $E$ be objects of $\operatorname{\mathcal{C}}$, and let

\[ F_{C,D}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D) \times \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}_{D} \quad \quad F_{D,E}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(D,E) \times \operatorname{\mathcal{E}}_{D} \rightarrow \operatorname{\mathcal{E}}_{E} \]
\[ F_{C,E}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,E) \times \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}_{E} \]

be the parametrized covariant transport functors of Example 5.2.2.7. Then the diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(D,E) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D) \times \operatorname{\mathcal{E}}_{C} \ar [r]^-{ \operatorname{id}\times F_{C,D} } \ar [d] & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(D,E) \times \operatorname{\mathcal{E}}_{D} \ar [d]^{ F_{D,E} } \\ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,E) \times \operatorname{\mathcal{E}}_{C} \ar [r]^-{ F_{C,E } } & \operatorname{\mathcal{E}}_{E} } \]

commutes in the homotopy category $\mathrm{h} \mathit{\operatorname{QCat}}$; here the left vertical map is given by the composition law of Construction 4.6.7.9.

Using Corollary 5.2.3.8, we obtain the following refinement of Construction 5.2.3.2:

Construction 5.2.3.9 (Enriched Homotopy Transport: Covariant Case). Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of $\infty $-categories and let us regard the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ as enriched over the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$ of Kan complexes (Construction 4.6.7.13). It follows from Corollary 5.2.3.8 (and Example 5.2.2.2) that there is a unique $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched 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 object $C$ of the $\infty $-category $\operatorname{\mathcal{C}}$, $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}(C)$ 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 every pair of objects $C,D \in \operatorname{\mathcal{C}}$, the induced map

    \[ \operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{E}}_{C}, \operatorname{\mathcal{E}}_{D} )^{\simeq } \]

    in $\mathrm{h} \mathit{\operatorname{Kan}}$ corresponds to the parametrized covariant transport functor $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D) \times \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}_{D}$ of Example 5.2.2.7 (which is well-defined up to homotopy).

We will refer to $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ as the enriched homotopy transport representation of the cocartesian fibration $U$. Note that the underlying functor of ordinary categories $\mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{QCat}}$ coincides with homotopy transport representation of Construction 5.2.3.2.

Variant 5.2.3.10 (Enriched Homotopy Transport: Contravariant Case). Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cartesian fibration of $\infty $-categories. Applying Construction 5.2.3.9 to the opposite functor $U^{\operatorname{op}}$, we deduce that there is a unique $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functor $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}}^{\operatorname{op}} \rightarrow \mathrm{h} \mathit{\operatorname{QCat}}$ with the following properties:

  • For each object $C$ of the $\infty $-category $\operatorname{\mathcal{C}}$, $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}(C)$ 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 every pair of objects $C,D \in \operatorname{\mathcal{C}}$, the induced map

    \[ \operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{E}}_{D}, \operatorname{\mathcal{E}}_{C} )^{\simeq } \]

    is given by the parametrized contravariant transport functor $\operatorname{\mathcal{E}}_{D} \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(D,C) \rightarrow \operatorname{\mathcal{E}}_{C}$ of Example 5.2.2.12.

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

Example 5.2.3.11. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ denote its homotopy category, which we regard as enriched over the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$ of Kan complexes. Applying Proposition 5.2.2.13, we obtain the following:

  • For every object $C \in \operatorname{\mathcal{C}}$, the corepresentable $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functor

    \[ \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{Kan}} \quad \quad D \mapsto \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D) \]

    is the enriched homotopy transport representation for the left fibration $\{ C\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$.

  • For every object $D \in \operatorname{\mathcal{C}}$, the representable $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functor

    \[ \mathrm{h} \mathit{\operatorname{\mathcal{C}}}^{\operatorname{op}} \rightarrow \mathrm{h} \mathit{\operatorname{Kan}} \quad \quad C \mapsto \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D) \]

    is the enriched homotopy transport representation for the right fibration $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \{ D\} \rightarrow \operatorname{\mathcal{C}}$.