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5.3.3 Homotopy Transport for Cartesian Fibrations

We now study the behavior of the transport functors of §5.3.2 with respect to composition.

Proposition 5.3.3.1 (Transitivity). Let $q: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a cartesian fibration of simplicial sets and let $\sigma $ be a $2$-simplex of $\operatorname{\mathcal{D}}$, which we display as a diagram

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

Let $f^{\ast }: \operatorname{\mathcal{C}}_{Y} \rightarrow \operatorname{\mathcal{C}}_{X}$ and $g^{\ast }: \operatorname{\mathcal{C}}_{Z} \rightarrow \operatorname{\mathcal{C}}_{Y}$ be functors which are given by contravariant transport along $f$ and $g$, respectively. Then the composite functor $f^{\ast } \circ g^{\ast }: \operatorname{\mathcal{C}}_{Z} \rightarrow \operatorname{\mathcal{C}}_{X}$ is given by contravariant transport along $h$.

Proof. Without loss of generality, we may replace $q$ by the projection map $\Delta ^{2} \times _{ \operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}\rightarrow \Delta ^2$, and thereby reduce to the case where $\operatorname{\mathcal{D}}= \Delta ^2$ and $\sigma $ is the unique nondegenerate $2$-simplex of $\operatorname{\mathcal{D}}$. In this case, $\operatorname{\mathcal{C}}$ is an $\infty $-category. Let $h: \Delta ^1 \times \operatorname{\mathcal{C}}_{Y} \rightarrow \operatorname{\mathcal{C}}$ and $h': \Delta ^1 \times \operatorname{\mathcal{C}}_{Z} \rightarrow \operatorname{\mathcal{C}}$ be morphisms which witness $f^{\ast }$ and $g^{\ast }$ as given by contravariant transport along $f$ and $g$, respectively. Then the composite map

\[ \Delta ^{1} \times \operatorname{\mathcal{C}}_{Z} \xrightarrow { \operatorname{id}\times g^{\ast } } \Delta ^1 \times \operatorname{\mathcal{C}}_{Y} \xrightarrow {h} \operatorname{\mathcal{C}} \]

can be identified with a morphism $\alpha $ from $f^{\ast } \circ g^{\ast }$ to $g^{\ast }$ in the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}_{Z}, \operatorname{\mathcal{C}})$. Similarly, $h'$ can be identified with a morphism $\beta $ from $g^{\ast }$ to $\operatorname{id}_{\operatorname{\mathcal{C}}_{Z}}$ in the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}_{Z}, \operatorname{\mathcal{C}})$. Note that for each object $C \in \operatorname{\mathcal{C}}_{Z}$, the induced maps

\[ \alpha _{C}: (f^{\ast } \circ g^{\ast })(C) \rightarrow g^{\ast }(C) \quad \quad \beta _{C}: g^{\ast }(C) \rightarrow C \]

are $q$-cartesian. Let $\gamma : f^{\ast } \circ g^{\ast } \rightarrow \operatorname{id}_{\operatorname{\mathcal{C}}_{Z}}$ be a composition of $\alpha $ with $\beta $. Then, for each object $C \in \operatorname{\mathcal{C}}_{Z}$, the morphism $\gamma _{C}: (f^{\ast } \circ g^{\ast }(C) \rightarrow C$ is also $q$-cartesian (Corollary 5.2.2.5). It follows that $\gamma $ can be identified with a morphism of simplicial sets $\Delta ^1 \times \operatorname{\mathcal{C}}_{Z} \rightarrow \operatorname{\mathcal{C}}$ which witnesses $f^{\ast } \circ g^{\ast }$ as given by contravariant transport along $h$.. $\square$

Warning 5.3.3.2. The conclusion of Proposition 5.3.3.1 is generally not satisfied if $q: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is only assumed to be a locally cartesian fibration of simplicial sets. We will return to this point in § (see Proposition ).

Construction 5.3.3.3 (The Homotopy Transport Representation: Cartesian Case). Let $q: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a cartesian fibration of simplicial sets and let $\mathrm{h} \mathit{\operatorname{Cat}_{\infty }}$ denote the homotopy category of $\infty $-categories (Construction 4.5.1.1). It follows from Proposition 5.3.3.1 and Example 5.3.2.6 that there is a unique morphism of simplicial sets $\operatorname{hTr}_{q}: \operatorname{\mathcal{D}}^{\operatorname{op}} \rightarrow \operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{Cat}_{\infty } } )$ with the following properties:

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

  • For each edge $e: X \rightarrow Y$ in the simplicial set $\operatorname{\mathcal{D}}$, $\operatorname{hTr}_{q}(e)$ is the (isomorphism class of) the contravariant transport functor $[e^{\ast }]$ of Notation 5.3.2.5, regarded as an element of $\operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{Cat}_{\infty }} }( \operatorname{\mathcal{C}}_{Y}, \operatorname{\mathcal{C}}_{X} ) = \pi _0( \operatorname{Fun}( \operatorname{\mathcal{C}}_{Y}, \operatorname{\mathcal{C}}_{X})^{\simeq } )$.

Let $\mathrm{h} \mathit{\operatorname{\mathcal{D}}}$ denote the homotopy category of the simplicial set $\operatorname{\mathcal{D}}$ (Notation 1.2.5.3). Then the morphism $\operatorname{hTr}_{q}$ determines a functor of ordinary categories $\mathrm{h} \mathit{\operatorname{\mathcal{D}}}^{\operatorname{op}} \rightarrow \mathrm{h} \mathit{\operatorname{Cat}_{\infty }}$, which we will denote also by $\operatorname{hTr}_{q}$ and will refer to as the homotopy transport representation of the cartesian fibration $q$.

Example 5.3.3.4. Let $q: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a cartesian fibration of categories (Definition 5.1.4.8), so that the induced map $\operatorname{N}_{\bullet }(q): \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}})$ is a cartesian fibration of $\infty $-categories (Example 5.2.4.2). Then the homotopy transport representation $\operatorname{hTr}_{\operatorname{N}_{\bullet }(q)}: \operatorname{\mathcal{D}}^{\operatorname{op}} \rightarrow \mathrm{h} \mathit{\operatorname{Cat}_{\infty }}$ is given by the composition

\[ \operatorname{\mathcal{D}}^{\operatorname{op}} \xrightarrow { \chi _{q} } \operatorname{Pith}(\mathbf{Cat}) \rightarrow \mathrm{h} \mathit{\operatorname{Cat}} \xrightarrow { \operatorname{N}_{\bullet } } \mathrm{h} \mathit{\operatorname{Cat}_{\infty }}. \]

Here $\chi _{q}$ denotes the transport representation of Construction 5.1.5.10 (with respect to any cleavage of the fibration $q$), the second functor is the truncation map of Remark 2.3.2.12, and $\operatorname{N}_{\bullet }$ is the fully faithful functor of Remark 4.5.1.3. Stated more informally, the homotopy transport representation $\operatorname{hTr}_{ \operatorname{N}_{\bullet }(q)}$ of Construction 5.3.3.3 can be obtained from the transport representation $\chi _{ \operatorname{N}_{\bullet }(q)}$ of Construction 5.1.5.10 by passing from the $2$-category $\mathbf{Cat}$ to its homotopy category.

Example 5.3.3.5. Let $q: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between ordinary categories which is a fibration in sets (Definition 5.1.2.1), so that the induced map $\operatorname{N}_{\bullet }(q): \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}})$ is a right fibration, and in particular a cartesian fibration. Then the homotopy transport representation $\operatorname{hTr}_{\operatorname{N}_{\bullet }(q)}$ of Construction 5.3.3.3 is given by the composition

\[ \operatorname{\mathcal{D}}^{\operatorname{op}} \xrightarrow { \chi _{q} } \operatorname{Set}\hookrightarrow \mathrm{h} \mathit{\operatorname{Cat}_{\infty } }, \]

where $\chi _{q}$ is the transport representation of Construction 5.1.2.14 and $\operatorname{Set}\hookrightarrow \mathrm{h} \mathit{\operatorname{Cat}_{\infty }}$ is the fully faithful embedding which associates to each set $X$ the associated discrete simplicial set, regarded as an $\infty $-category.

Remark 5.3.3.6. Let $q: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a cartesian fibration of simplicial sets, and let $\operatorname{hTr}_{q}: \mathrm{h} \mathit{\operatorname{\mathcal{D}}}^{\operatorname{op}} \rightarrow \mathrm{h} \mathit{\operatorname{Cat}_{\infty }}$ be the homotopy transport representation of Construction 5.3.3.3. It follows from Proposition 5.2.4.12 that $q$ is a right fibration if and only if the functor $\operatorname{hTr}_{q}$ factors through the full subcategory $\mathrm{h} \mathit{\operatorname{Kan}} \subseteq \mathrm{h} \mathit{\operatorname{Cat}_{\infty }}$. In particular, if $q$ is a right fibration, then Construction 5.3.3.3 determines a functor $\mathrm{h} \mathit{\operatorname{\mathcal{D}}}^{\operatorname{op}} \rightarrow \mathrm{h} \mathit{\operatorname{Kan}}$ which we will also refer to as the homotopy transport representation of $q$.

For later reference, we record a dual version of Construction 5.3.3.3:

Construction 5.3.3.7 (The Covariant Transport Functor). Let $q: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a cocartesian fibration of simplicial sets and let $\mathrm{h} \mathit{\operatorname{Cat}_{\infty }}$ denote the homotopy category of $\infty $-categories. Then there is a unique morphism of simplicial sets $\operatorname{hTr}_{q}: \operatorname{\mathcal{D}}\rightarrow \operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{Cat}_{\infty } } )$ with the following properties:

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

  • For each edge $e: X \rightarrow Y$ in the simplicial set $\operatorname{\mathcal{D}}$, $\operatorname{hTr}_{q}(e)$ is the (isomorphism class of) the covariant transport functor $[e_!]$ of Notation 5.3.2.12, regarded as an element of $\operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{Cat}_{\infty }} }( \operatorname{\mathcal{C}}_{X}, \operatorname{\mathcal{C}}_{Y} ) = \pi _0( \operatorname{Fun}( \operatorname{\mathcal{C}}_{X}, \operatorname{\mathcal{C}}_{Y})^{\simeq } )$.

Let $\mathrm{h} \mathit{\operatorname{\mathcal{D}}}$ denote the homotopy category of the simplicial set $\operatorname{\mathcal{D}}$ (Notation 1.2.5.3). Then the morphism $\operatorname{hTr}_{q}$ determines a functor of ordinary categories $\mathrm{h} \mathit{\operatorname{\mathcal{D}}} \rightarrow \mathrm{h} \mathit{\operatorname{Cat}_{\infty }}$, which we will denote also by $\operatorname{hTr}_{q}$ and will refer to as the homotopy transport representation of the cocartesian fibration $q$.

Warning 5.3.3.8. Let $q: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets which is both a cartesian fibration and a cocartesian fibration. Then Constructions 5.3.3.7 and 5.3.3.3 supply functors $\mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{Cat}_{\infty }}$ and $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}^{\operatorname{op}} \rightarrow \mathrm{h} \mathit{\operatorname{Cat}_{\infty }}$ respectively, which are both referred to as the homotopy transport representation of $q$ and denoted by $\operatorname{hTr}_{q}$. We will see later that these two functors are interchangeable data: either can be recovered from the other (see Proposition ).

Example 5.3.3.9. Let $q: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets. Combining Remark 5.3.3.6 with Theorem 5.3.2.14, we deduce that the following conditions are equivalent:

  • The morphism $q$ is a Kan fibration.

  • The morphism $q$ is a cartesian fibration and the homotopy transport representation $\operatorname{hTr}_{q}: \mathrm{h} \mathit{\operatorname{\mathcal{D}}}^{\operatorname{op}} \rightarrow \mathrm{h} \mathit{\operatorname{Cat}_{\infty }}$ of Construction 5.3.3.3 factors through the subcategory $\mathrm{h} \mathit{\operatorname{Kan}}^{\simeq } \subseteq \mathrm{h} \mathit{\operatorname{Cat}_{\infty }}$.

  • The morphism $q$ is a cocartesian fibration and the homotopy transport representation $\operatorname{hTr}'_{q}: \mathrm{h} \mathit{\operatorname{\mathcal{D}}} \rightarrow \mathrm{h} \mathit{\operatorname{Cat}_{\infty }}$ of Construction 5.3.3.7 factors through the subcategory $\mathrm{h} \mathit{\operatorname{Kan}}^{\simeq } \subseteq \mathrm{h} \mathit{\operatorname{Cat}_{\infty }}$.

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

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

where $\iota $ is the isomorphism which carries each morphism in $\mathrm{h} \mathit{\operatorname{Kan}}^{\simeq }$ to its inverse (see Warning 5.1.2.16).