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5.2.4 Digression: The Relative Join

Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Our goal in this section is to show that $F$ is given by covariant transport, in the sense of Definition 5.2.2.1. More precisely, we will show that there exists a cocartesian fibration of $\infty $-categories $\operatorname{\mathcal{M}}\rightarrow \Delta ^1$ equipped with identifications $\operatorname{\mathcal{C}}\simeq \{ 0\} \times _{\Delta ^1} \operatorname{\mathcal{M}}$ and $\operatorname{\mathcal{D}}= \{ 1\} \times _{\Delta ^1} \operatorname{\mathcal{M}}$ carry $F$ to a functor

\[ \{ 0\} \times _{\Delta ^1} \operatorname{\mathcal{M}}\rightarrow \{ 1\} \times _{\Delta ^1} \operatorname{\mathcal{M}} \]

given by covariant transport along the nondegenerate edge of $\Delta ^1$ (Proposition 5.2.4.16). We will prove this by an explicit construction, using a generalization of the join operation studied in §4.3 (in §5.2.5, we will show that the $\infty $-category $\operatorname{\mathcal{M}}$ is determined up to equivalence by the functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$; see Corollary 5.2.5.2 and Remark 5.2.5.3).

Construction 5.2.4.1 (The Relative Join). Let $\operatorname{\mathcal{E}}$ be a simplicial set. By virtue of Remark 4.3.3.20, there is a unique morphism of simplicial sets $\rho : \Delta ^1 \times \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{E}}\star \operatorname{\mathcal{E}}$ for which the diagram

\[ \xymatrix@R =50pt@C=50pt{ \{ 0\} \times \operatorname{\mathcal{E}}\ar [r] \ar [d]^-{\operatorname{id}_{\operatorname{\mathcal{E}}}} & \Delta ^1 \times \operatorname{\mathcal{E}}\ar [d]^-{\rho } & \{ 1\} \times \operatorname{\mathcal{E}}\ar [l] \ar [d]^-{\operatorname{id}_{\operatorname{\mathcal{E}}}} \\ \operatorname{\mathcal{E}}\star \, \, \emptyset \ar [r] & \operatorname{\mathcal{E}}\star \operatorname{\mathcal{E}}& \emptyset \star \operatorname{\mathcal{E}}\ar [l] } \]

is commutative.

Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be morphisms of simplicial sets. We let $\operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}$ denote the fiber product $(\operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}}) \times _{ (\operatorname{\mathcal{E}}\star \operatorname{\mathcal{E}})} (\Delta ^1 \times \operatorname{\mathcal{E}})$, so that we have a pullback diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}\ar [r] \ar [d] & \operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}}\ar [d] \\ \Delta ^1 \times \operatorname{\mathcal{E}}\ar [r]^-{\rho } & \operatorname{\mathcal{E}}\star \operatorname{\mathcal{E}}. } \]

We will refer to $\operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}$ as the join of $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ relative to $\operatorname{\mathcal{E}}$.

Remark 5.2.4.2. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be morphisms of simplicial sets, and let $K$ be a simplicial set. By virtue of Remark 4.3.3.20, morphisms from $K$ to the relative join $\operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}$ are given by maps $\pi : K \rightarrow \Delta ^1$ together with commutative diagrams

\[ \xymatrix@R =50pt@C=50pt{ \{ 0\} \times _{\Delta ^1} K \ar [d] \ar [r] & K \ar [d] & \{ 1\} \times _{\Delta ^1} K \ar [d] \ar [l] \\ \operatorname{\mathcal{C}}\ar [r]^-{F} & \operatorname{\mathcal{E}}& \operatorname{\mathcal{D}}. \ar [l]_-{G} } \]

Remark 5.2.4.3. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be morphisms of simplicial sets. Then the inclusion maps $\operatorname{\mathcal{C}}\hookrightarrow \operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}}\hookleftarrow \operatorname{\mathcal{D}}$ lift uniquely to monomorphisms

\[ \iota _{\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\hookrightarrow \operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}\quad \quad \iota _{\operatorname{\mathcal{D}}}: \operatorname{\mathcal{D}}\hookrightarrow \operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}, \]

which fit into a commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [r]^-{\iota _{\operatorname{\mathcal{C}}}} \ar [d] & \operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}\ar [d] & \operatorname{\mathcal{D}}\ar [l]_-{\iota _{\operatorname{\mathcal{D}}}} \ar [d] \\ \{ 0\} \ar [r] & \Delta ^1 & \{ 1\} \ar [l] } \]

in which both squares are pullbacks. In the future, we will often abuse notation by identifying $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ with their images under the monomorphisms $\iota _{\operatorname{\mathcal{C}}}$ and $\iota _{\operatorname{\mathcal{D}}}$, respectively (which are full simplicial subsets of the relative join $\operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}$).

Example 5.2.4.4. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be morphisms of simplicial sets. If $\operatorname{\mathcal{D}}$ is empty, then the inclusion map $\iota _{\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\hookrightarrow \operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}$ is an isomorphism of simplicial sets. If $\operatorname{\mathcal{C}}$ is empty, then the inclusion map $\iota _{\operatorname{\mathcal{D}}}: \operatorname{\mathcal{D}}\hookrightarrow \operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}$ is an isomorphism of simplicial sets.

Example 5.2.4.5. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be simplicial sets, so that we have unique morphisms $F: \operatorname{\mathcal{C}}\rightarrow \Delta ^{0}$ and $G: \operatorname{\mathcal{D}}\rightarrow \Delta ^{0}$. Then the relative join $\operatorname{\mathcal{C}}\star _{\Delta ^{0}} \operatorname{\mathcal{D}}$ agrees with the join $\operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}}$ introduced in Construction 4.3.3.13.

Example 5.2.4.6. Let $\operatorname{\mathcal{E}}$ be a simplicial set. Then the relative join $\operatorname{\mathcal{E}}\star _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{E}}$ is isomorphic to $\Delta ^1 \times \operatorname{\mathcal{E}}$.

Example 5.2.4.7. Let $\operatorname{\mathcal{E}}$ be a simplicial set equipped with a morphism $\pi : \operatorname{\mathcal{E}}\rightarrow \Delta ^1$, and set $\operatorname{\mathcal{C}}= \{ 0\} \times _{\Delta ^1} \operatorname{\mathcal{E}}$ and $\operatorname{\mathcal{D}}= \{ 1\} \times _{\Delta ^1} \operatorname{\mathcal{E}}$. Then the relative join $\operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}$ is isomorphic to $\operatorname{\mathcal{E}}$.

Example 5.2.4.8. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be functors between categories. Then the relative join $\operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}) \star _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{E}}) } \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}})$ can be identified with the nerve of the category

\[ \operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}= (\operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}}) \times _{ ( \operatorname{\mathcal{E}}\star \operatorname{\mathcal{E}}) } ( [1] \times \operatorname{\mathcal{E}}), \]

which can be described more concretely as follows:

  • The set of objects $\operatorname{Ob}( \operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}})$ is the disjoint union of $\operatorname{Ob}(\operatorname{\mathcal{C}})$ with $\operatorname{Ob}(\operatorname{\mathcal{D}})$.

  • For every pair of objects $X,Y \in \operatorname{Ob}( \operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}})$, we have

    \[ \operatorname{Hom}_{\operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}}( X, Y) = \begin{cases} \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) & \textnormal{ if $X,Y \in \operatorname{Ob}(\operatorname{\mathcal{C}})$} \\ \operatorname{Hom}_{\operatorname{\mathcal{D}}}( X,Y) & \textnormal{ if $X,Y \in \operatorname{Ob}(\operatorname{\mathcal{D}})$} \\ \operatorname{Hom}_{\operatorname{\mathcal{E}}}( F(X), G(Y) ) & \textnormal{ if $X \in \operatorname{Ob}(\operatorname{\mathcal{C}})$, $Y \in \operatorname{Ob}(\operatorname{\mathcal{D}})$} \\ \emptyset & \textnormal{ if $X \in \operatorname{Ob}(\operatorname{\mathcal{D}})$, $Y \in \operatorname{Ob}(\operatorname{\mathcal{C}}).$} \end{cases} \]

Remark 5.2.4.9 (Base Change). Suppose we are given a commutative diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}' \ar [r] \ar [d] & \operatorname{\mathcal{E}}' \ar [d] & \operatorname{\mathcal{D}}' \ar [l] \ar [d] \\ \operatorname{\mathcal{C}}\ar [r] & \operatorname{\mathcal{E}}& \operatorname{\mathcal{D}}, \ar [l] } \]

where both squares are pullbacks. Then the induced diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}' \star _{\operatorname{\mathcal{E}}'} \operatorname{\mathcal{D}}' \ar [r] \ar [d] & \operatorname{\mathcal{E}}' \ar [d] \\ \operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}\ar [r] & \operatorname{\mathcal{E}}} \]

is also a pullback square.

Remark 5.2.4.10. Let $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be a fixed morphism of simplicial sets. Then the construction

\[ (F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}) \mapsto \operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}} \]

carries colimits in the category $(\operatorname{Set_{\Delta }})_{/\operatorname{\mathcal{E}}}$ to colimits in the category $(\operatorname{Set_{\Delta }})_{\operatorname{\mathcal{D}}/}$. In particular, the construction $\operatorname{\mathcal{C}}\mapsto (\operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}})$ commutes with filtered colimits and carries pushout diagrams to pushout diagrams.

The relative join $\operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}$ of Construction 5.2.4.1 is defined for arbitrary diagrams of simplicial sets $\operatorname{\mathcal{C}}\xrightarrow {F} \operatorname{\mathcal{E}}\xleftarrow {G} \operatorname{\mathcal{D}}$. However, as our notation suggests, we will be primarily interested in the special case where $\operatorname{\mathcal{C}}$, $\operatorname{\mathcal{D}}$, and $\operatorname{\mathcal{E}}$ are $\infty $-categories. In this case, we have the following generalization of Proposition 4.3.3.23:

Proposition 5.2.4.11. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be morphisms of $\infty $-categories. Then the relative join $\operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}$ is an $\infty $-category.

We will deduce Proposition 5.2.4.11 from the following:

Lemma 5.2.4.12. Let $\operatorname{\mathcal{E}}$ be an $\infty $-category. Then the morphism $\rho : \Delta ^1 \times \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{E}}\star \operatorname{\mathcal{E}}$ of Construction 5.2.4.1 is an inner fibration of simplicial sets.

Proof. Suppose we are given integers $0 < i < n$; we wish to show that every lifting problem

\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{n}_{i} \ar [r]^-{ \sigma _0 } \ar [d] & \Delta ^1 \times \operatorname{\mathcal{E}}\ar [d]^-{\rho } \\ \Delta ^ n \ar@ {-->}[ur]^{\sigma } \ar [r]^-{ \overline{\sigma } } & \operatorname{\mathcal{E}}\star \operatorname{\mathcal{E}}} \]

admits a solution. Let us identify $\sigma _0$ with a pair of maps

\[ \sigma '_0: \Lambda ^{n}_{i} \rightarrow \Delta ^1 \quad \quad \sigma ''_0: \Lambda ^{n}_{i} \rightarrow \operatorname{\mathcal{E}}. \]

By virtue of Proposition 1.2.3.1, we can extend $\sigma '_{0}$ uniquely to a morphism $\sigma ': \Delta ^ n \rightarrow \Delta ^1$. If $\sigma '$ is constant, then $\overline{\sigma }$ factors through one of the inclusion maps

\[ \operatorname{\mathcal{E}}\star \emptyset \hookrightarrow \operatorname{\mathcal{E}}\star \operatorname{\mathcal{E}}\hookleftarrow \emptyset \star \operatorname{\mathcal{E}}, \]

in which case the desired extension exists and is unique. If $\sigma '$ is not constant, then our assumption that $\operatorname{\mathcal{E}}$ is an $\infty $-category guarantees that we can extend $\sigma ''_0$ to an $n$-simplex $\sigma '': \Delta ^ n \rightarrow \operatorname{\mathcal{E}}$, and we can take $\sigma $ to be the pair $(\sigma ', \sigma '')$. $\square$

Lemma 5.2.4.13. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be morphisms of simplicial sets. If $\operatorname{\mathcal{E}}$ is an $\infty $-category, then the natural map $\operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}}$ is an inner fibration of simplicial sets.

Proof. By construction, we have a pullback diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}\ar [r] \ar [d] & \operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}}\ar [d] \\ \Delta ^1 \times \operatorname{\mathcal{E}}\ar [r]^-{\rho } & \operatorname{\mathcal{E}}\star \operatorname{\mathcal{E}}, } \]

where $\rho $ is an inner fibration by virtue of Lemma 5.2.4.12. $\square$

Proof of Proposition 5.2.4.11. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be functors of $\infty $-categories. By virtue of Lemma 5.2.4.13, the tautological map $\operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}}$ is an inner fibration of simplicial sets. Since $\operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}}$ is an $\infty $-category (Proposition 4.3.3.23), it follows that $\operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}$ is also an $\infty $-category (Remark 4.1.1.9). $\square$

Remark 5.2.4.14 (Morphism Spaces in the Relative Join). Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be morphisms of simplicial sets. If $X$ and $Y$ are vertices of the relative join $\operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}$, then we have canonical isomorphisms of simplicial sets

\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}}( X, Y) \simeq \begin{cases} \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) & \textnormal{ if $X,Y \in \operatorname{\mathcal{C}}$} \\ \operatorname{Hom}_{\operatorname{\mathcal{D}}}( X,Y) & \textnormal{ if $X,Y \in \operatorname{\mathcal{D}}$} \\ \operatorname{Hom}_{\operatorname{\mathcal{E}}}( F(X), G(Y) ) & \textnormal{ if $X \in \operatorname{\mathcal{C}}$, $Y \in \operatorname{\mathcal{D}}$} \\ \emptyset & \textnormal{ if $X \in \operatorname{\mathcal{D}}$, $Y \in \operatorname{\mathcal{C}}.$} \end{cases} \]

The pinched morphism spaces $\operatorname{Hom}^{\mathrm{L}}_{\operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}}( X, Y)$ and $\operatorname{Hom}^{\mathrm{R}}_{\operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}}( X, Y)$ admit similar descriptions.

We now specialize Construction 5.2.4.1 to the case where $\operatorname{\mathcal{D}}= \operatorname{\mathcal{E}}$ and the morphism $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ is the identity.

Lemma 5.2.4.15. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories and let $\pi : \operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}\rightarrow \Delta ^1$ be the projection map. Let $X$ be an object of $\operatorname{\mathcal{C}}$, let $Y$ be an object of $\operatorname{\mathcal{D}}$, and let $f: X \rightarrow Y$ be a morphism in the relative join $\operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$, which we identify with a morphism $e: F(X) \rightarrow Y$ in the $\infty $-category $\operatorname{\mathcal{D}}$ (see Remark 5.2.4.14). Then $f$ is $\pi $-cocartesian if and only if $e$ is an isomorphism in $\operatorname{\mathcal{D}}$.

Proof. Let $n \geq 2$ be an integer. Unwinding the definitions, we see that the datum of a lifting problem

5.19
\begin{equation} \begin{gathered}\label{equation:cocartesian-in-relative-join} \xymatrix@R =50pt@C=50pt{ \Lambda ^{n}_{0} \ar [r]^-{\sigma _0} \ar [d] & \operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}\ar [d]^-{\pi } \\ \Delta ^{n} \ar@ {-->}[ur]^{\sigma } \ar [r] & \Delta ^1 } \end{gathered} \end{equation}

satisfying $\sigma _0|_{ \operatorname{N}_{\bullet }( \{ 0 < 1 \} ) } = f$ is equivalent to the datum of a diagram $\tau _0: \Lambda ^{n}_{0} \rightarrow \operatorname{\mathcal{D}}$ satisfying $\tau _0|_{ \operatorname{N}_{\bullet }( \{ 0 < 1 \} )} = e$ (see Remark 5.2.4.2). Moreover, the lifting problem (5.19) admits a solution if and only if the corresponding map $\tau _0: \Lambda ^{n}_{0} \rightarrow \operatorname{\mathcal{D}}$ can be extended to an $n$-simplex of $\operatorname{\mathcal{D}}$. The desired equivalence now follows from the characterization of isomorphisms given in Theorem 4.4.2.6 $\square$

Proposition 5.2.4.16. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Then:

  • The projection map $\pi : \operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}\rightarrow \Delta ^1$ is a cocartesian fibration of $\infty $-categories.

  • The map

    \[ h: \Delta ^1 \times \operatorname{\mathcal{C}}\simeq ( \operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}} \]

    witnesses the functor $F$ as given by covariant transport along the nondegenerate edge of $\Delta ^1$.

Proof. Proposition 5.2.4.11 guarantees that the relative join $\operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$ is an $\infty $-category, so that the projection map $\pi : \operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}\rightarrow \Delta ^1$ is automatically an inner fibration (Proposition 4.1.1.10). To prove that it is a cocartesian fibration, we must show that for every object $X \in \operatorname{\mathcal{C}}$ there exists an object $Y \in \operatorname{\mathcal{D}}$ and a $\pi $-cocartesian morphism $f: X \rightarrow Y$ in the $\infty $-category $\operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$. By virtue of Lemma 5.2.4.15, this is equivalent to the statement that there exists an object $Y \in \operatorname{\mathcal{D}}$ equipped with an isomorphism $e: F(X) \rightarrow Y$, which is clear (we can take $Y = F(X)$ and $e$ to be the identity morphism). This proves assertion $(1)$. To prove assertion $(2)$, we first observe that the restriction $h|_{ \{ 1\} \times \operatorname{\mathcal{C}}}$ coincides with the functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$. It will therefore suffice to show that, for each object $X \in \operatorname{\mathcal{C}}$, the edge $h|_{ \Delta ^1 \times \{ X\} }$ is $\pi $-cocartesian. Note that the correspondence of Remark 5.2.4.14 carries $h|_{ \Delta ^1 \times \{ X\} }$ to the identity morphism $\operatorname{id}_{ F(X)}$ in the $\infty $-category $\operatorname{\mathcal{D}}$, so the desired result follows from the criterion of Lemma 5.2.4.15. $\square$

Passing to opposite $\infty $-categories, we obtain a dual version of Proposition 5.2.4.16:

Variant 5.2.4.17. Let $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a functor of $\infty $-categories. Then:

  • The projection map $\pi : \operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}\rightarrow \Delta ^1$ is a cartesian fibration of $\infty $-categories.

  • The map

    \[ h: \Delta ^1 \times \operatorname{\mathcal{D}}\simeq ( \operatorname{\mathcal{D}}\star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}} \]

    witnesses the functor $G$ as given by contravariant transport along the nondegenerate edge of $\Delta ^1$.

Exercise 5.2.4.18. Suppose we are given a finite sequence of $\infty $-categories $\{ \operatorname{\mathcal{E}}(m) \} _{0 \leq im \leq n}$ and functors

\[ \operatorname{\mathcal{E}}(0) \xrightarrow { F(1) } \operatorname{\mathcal{E}}(1) \xrightarrow { F(2) } \operatorname{\mathcal{E}}(2) \rightarrow \cdots \xrightarrow {F(n)} \operatorname{\mathcal{E}}(n). \]

Let $\operatorname{\mathcal{E}}$ denote the iterated relative join

\[ ((( \operatorname{\mathcal{E}}(0) \star _{\operatorname{\mathcal{E}}(1)} \operatorname{\mathcal{E}}(1) ) \star _{\operatorname{\mathcal{E}}(2)} \operatorname{\mathcal{E}}(2)) \star \cdots ) \star _{\operatorname{\mathcal{E}}(n)} \operatorname{\mathcal{E}}(n). \]

Show that the associated projection map $U: \operatorname{\mathcal{E}}\rightarrow \Delta ^ n$ is a cocartesian fibration whose homotopy transport representation $\operatorname{hTr}_{U}: [n] \rightarrow \mathrm{h} \mathit{\operatorname{QCat}}$ is the diagram

\[ \operatorname{\mathcal{E}}(0) \xrightarrow { [F(1)] } \operatorname{\mathcal{E}}(1) \xrightarrow { [F(2)] } \operatorname{\mathcal{E}}(2) \rightarrow \cdots \xrightarrow {[F(n)]} \operatorname{\mathcal{E}}(n). \]

For a more general statement, see Proposition 5.5.3.13 and Remark 5.5.3.15.