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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 5.2.4.15). 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 Corollary 4.3.3.24:

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.

Lemma 5.2.4.12. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{E}}'$ be an inner fibration of simplicial sets. Then the induced map

\[ \Delta ^1 \times \operatorname{\mathcal{E}}= \operatorname{\mathcal{E}}\star _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{E}}\star _{\operatorname{\mathcal{E}}'} \operatorname{\mathcal{E}} \]

is also an inner fibration of simplicial sets.

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

5.18
\begin{equation} \begin{gathered}\label{equation:inner-fibration-silliness} \xymatrix@R =50pt@C=50pt{ \Lambda ^{n}_{i} \ar [r]^-{ \sigma _0 } \ar [d] & \Delta ^1 \times \operatorname{\mathcal{E}}\ar [d] \\ \Delta ^ n \ar@ {-->}[ur]^{\sigma } \ar [r]^-{ \overline{\sigma } } & \operatorname{\mathcal{E}}\star _{\operatorname{\mathcal{E}}'} \operatorname{\mathcal{E}}} \end{gathered} \end{equation}

admits a solution. Let $\alpha $ denote the composite map

\[ \Delta ^ n \xrightarrow { \overline{\sigma } } \operatorname{\mathcal{E}}\star _{\operatorname{\mathcal{E}}'} \operatorname{\mathcal{E}}\rightarrow \Delta ^0 \star _{\Delta ^0} \Delta ^0 \simeq \Delta ^1. \]

If $\alpha $ is a constant morphism, then the existence of $\sigma $ is immediate. We may therefore assume without loss of generality that $\alpha $ is not constant. Write $\sigma _0 = (\alpha _0, \tau _0)$, where $\alpha _0 = \alpha |_{ \Lambda ^{n}_{i} }$ and $\tau _0: \Lambda ^{n}_{i} \rightarrow \operatorname{\mathcal{E}}$ is a morphism of simplicial sets, and let $\overline{\tau }$ denote the composite map $\Delta ^ n \xrightarrow { \overline{\sigma } } \operatorname{\mathcal{E}}\star _{\operatorname{\mathcal{E}}'} \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{E}}'$. Since $F$ is an inner fibration, the lifting problem

\[ \xymatrix { \Lambda ^{n}_{i} \ar [r]^{ \tau _0 } \ar [d] & \operatorname{\mathcal{E}}\ar [d]^{U} \\ \Delta ^ n \ar [r]^{ \overline{\tau } } \ar@ {-->}[ur]^{ \tau } & \operatorname{\mathcal{E}}' } \]

admits a solution. We now observe that the pair $\sigma = (\alpha , \tau )$ can be regarded as an $n$-simplex of $\Delta ^1 \times \operatorname{\mathcal{E}}$ which solves the lifting problem (5.18). $\square$

Lemma 5.2.4.13. Suppose we are given a commutative diagram of simplicial sets

\[ \xymatrix { \operatorname{\mathcal{C}}\ar [r] \ar [d]^{U} & \operatorname{\mathcal{E}}\ar [d]^{W} & \operatorname{\mathcal{D}}\ar [d]^{V} \ar [l] \\ \operatorname{\mathcal{C}}' \ar [r] & \operatorname{\mathcal{E}}' & \operatorname{\mathcal{D}}' \ar [l] } \]

in which the vertical morphisms are inner fibrations. Then the induced map

\[ F: \operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}' \star _{\operatorname{\mathcal{E}}'} \operatorname{\mathcal{D}}' \]

is also an inner fibration.

Proof. Unwinding the definitions, we see that $F$ factors as a composition

\[ \operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}\xrightarrow {G} \operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{E}}'} \operatorname{\mathcal{D}}\xrightarrow {H} \operatorname{\mathcal{C}}' \star _{\operatorname{\mathcal{E}}'} \operatorname{\mathcal{D}}', \]

where $G$ is a pullback of the inner fibration $\operatorname{\mathcal{E}}\star _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{E}}\star _{\operatorname{\mathcal{E}}'} \operatorname{\mathcal{E}}$ of Lemma 5.2.4.12 and $H$ is a pullback of the inner fibration $\operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}' \star \operatorname{\mathcal{D}}'$ of Proposition 4.3.3.23. $\square$

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. Applying Lemma 5.2.4.13, we see that the natural map

\[ \operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}\rightarrow \Delta ^0 \star _{\Delta ^0} \Delta ^0 \simeq \Delta ^1 \]

is an inner fibration of simplicial sets. Since $\Delta ^1$ is an $\infty $-category, 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. Our goal is to prove the following:

Proposition 5.2.4.15. 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

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

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

The proof of Proposition 5.2.4.15 will require some preliminaries.

Lemma 5.2.4.16. Suppose we are given a commutative diagram of simplicial sets

\[ \xymatrix { \operatorname{\mathcal{C}}\ar [r]^{F} \ar [d]^{U} & \operatorname{\mathcal{D}}\ar [d]^{V} \\ \operatorname{\mathcal{C}}' \ar [r]^{F'} & \operatorname{\mathcal{D}}', } \]

so that $U$ and $V$ induce a morphism $W: \operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}' \star _{\operatorname{\mathcal{D}}'} \operatorname{\mathcal{D}}'$. Let $e$ be an edge of the simplicial set $\operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$ satisfying the following conditions:

$(1)$

If $e$ is contained in $\operatorname{\mathcal{C}}$, then it is $U$-cocartesian when viewed as an edge of $\operatorname{\mathcal{C}}$.

$(2)$

The image of $e$ under the map

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

is $V$-cocartesian.

Then $e$ is $W$-cocartesian.

Proof. Let $n \geq 2$ be an integer and suppose we are given a lifting problem

5.23
\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]^-{W} \\ \Delta ^{n} \ar@ {-->}[ur]^{\sigma } \ar [r]^{ \sigma ' } & \operatorname{\mathcal{C}}' \star _{\operatorname{\mathcal{D}}'} \operatorname{\mathcal{D}}',} \end{gathered} \end{equation}

where $\sigma _0$ carries $\operatorname{N}_{\bullet }( \{ 0 < 1 \} ) \subseteq \Lambda ^{n}_{0}$ to the edge $e$. If $\sigma '$ is contained in the simplicial subset $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}' \star _{\operatorname{\mathcal{D}}'} \operatorname{\mathcal{D}}'$, then the lifting problem (5.23) admits a solution by virtue of assumption $(1)$. Let us therefore assume that $\sigma '$ is not contained in $\operatorname{\mathcal{C}}'$. Let $\rho : \operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{D}}$ be as in $(2)$, and define $\rho ': \operatorname{\mathcal{C}}' \star _{\operatorname{\mathcal{D}}'} \operatorname{\mathcal{D}}' \rightarrow \operatorname{\mathcal{D}}'$ similarly. Unwinding the definitions, we can rewrite (5.23) as a lifting problem

\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{n}_{0} \ar [r]^-{\rho \circ \sigma _0} \ar [d] & \operatorname{\mathcal{D}}\ar [d]^-{V} \\ \Delta ^{n} \ar@ {-->}[ur] \ar [r]^{\rho ' \circ \sigma ' } & \operatorname{\mathcal{D}}',} \]

which admits a solution by virtue of assumption $(2)$. $\square$

Lemma 5.2.4.17. Suppose we are given a commutative diagram of simplicial sets

\[ \xymatrix { \operatorname{\mathcal{C}}\ar [r]^{F} \ar [d]^{U} & \operatorname{\mathcal{D}}\ar [d]^{V} \\ \operatorname{\mathcal{C}}' \ar [r]^{F'} & \operatorname{\mathcal{D}}'. } \]

Suppose that $U$ and $V$ are cocartesian fibrations, and that the morphism $F$ carries $U$-cocartesian edges of $\operatorname{\mathcal{C}}$ to $V$-cocartesian edges of $\operatorname{\mathcal{D}}$. Then the induced map $W: \operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}' \star _{\operatorname{\mathcal{D}}'} \operatorname{\mathcal{D}}'$ is also a cocartesian fibration. Moreover, an edge $e$ of $\operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$ is $W$-cocartesian if and only if it satisfies conditions $(1)$ and $(2)$ of Lemma 5.2.4.16.

Proof. It follows from Lemma 5.2.4.12 that $W$ is an inner fibration of simplicial sets. Let us say that an edge of $\operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$ is special if it satisfies conditions $(1)$ and $(2)$ of Lemma 5.2.4.16, so that every special edge of $\operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$ is $W$-cocartesian. We consider three cases:

  • Suppose that $X$ belongs to $\operatorname{\mathcal{C}}$ and $\overline{Y}$ belongs to $\operatorname{\mathcal{C}}'$. In this case, our assumption that $U$ is a cocartesian fibration guarantees that we can lift $\overline{e}$ to $U$-cocartesian edge $e: X \rightarrow Y$ of $\operatorname{\mathcal{C}}\subseteq \operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$. Since $F(e)$ is a $V$-cocartesian edge of $\operatorname{\mathcal{D}}$, the edge $e$ is special.

  • Suppose that $X$ belongs to $\operatorname{\mathcal{C}}$ and $\overline{Y}$ belongs to $\operatorname{\mathcal{D}}'$. In this case, we can identify $\overline{e}$ with an edge $\overline{e}_0: V( F(X) ) \rightarrow \overline{Y}$ of the simplicial set $\operatorname{\mathcal{D}}'$. Since $V$ is a cocartesian fibration, we can lift $\overline{e}_0$ to a $V$-cocartesian morphism $e_0: F(X) \rightarrow Y$ of $\operatorname{\mathcal{D}}$, which we can identify with a special edge $e: X \rightarrow Y$ of the simplicial set $\operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$ satisfying $W(e) = \overline{e}$.

  • Suppose that $X$ belongs to $\operatorname{\mathcal{D}}$ and $\overline{Y}$ belongs to $\operatorname{\mathcal{D}}'$. In this case, our assumption that $V$ is a cocartesian fibration guarantees that we can lift $\overline{e}$ to $V$-cocartesian edge $e: X \rightarrow Y$ of $\operatorname{\mathcal{D}}\subseteq \operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$, which is then special when regarded as an edge of $\operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$.

To complete the proof, it will suffice to show that every $W$-cocartesian edge $e: X \rightarrow Y$ of $\operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$ is special. Applying the preceding argument, we can choose a special edge $e': X \rightarrow Y'$ satisfying $W(e') = W(e)$. Set $\overline{Y} = W(Y) = W(Y')$. Since $e$ and $e'$ are both $W$-cocartesian, Remark 5.1.3.8 supplies a $2$-simplex $\sigma $ of the simplicial set $\operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$ with boundary given by

\[ \xymatrix { & Y \ar [dr]^{u} & \\ X \ar [ur]^{e} \ar [rr]^{e'} & & Y', } \]

where $u$ is an isomorphism in the $\infty $-category $\{ \overline{Y} \} \times _{ ( \operatorname{\mathcal{C}}' \star _{\operatorname{\mathcal{D}}'} \operatorname{\mathcal{D}}' ) } (\operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}})$. Applying Remark 5.1.3.8 to the cocartesian fibrations $U$ and $V$, we deduce that the edge $e$ is also special. $\square$

Example 5.2.4.18. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Applying Lemma 5.2.4.17 to the diagram

\[ \xymatrix { \operatorname{\mathcal{C}}\ar [r]^{F} \ar [d] & \operatorname{\mathcal{D}}\ar [d] \\ \Delta ^{0} \ar@ {=}[r] & \Delta ^{0}, } \]

we deduce that the projection map

\[ \pi : \operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}\rightarrow \Delta ^{0} \star _{\Delta ^{0} } \Delta ^{0} \simeq \Delta ^1 \]

is a cocartesian fibration. Moreover, a morphism $e: X \rightarrow Y$ of the $\infty $-category $\operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$ is $\pi $-cocartesian if and only if it satisfies one of the following three conditions:

  • The objects $X$ and $Y$ belong to $\operatorname{\mathcal{C}}$ and $e$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{C}}$.

  • The objects $X$ and $Y$ belong to $\operatorname{\mathcal{D}}$ and $e$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{D}}$.

  • The object $X$ belongs to $\operatorname{\mathcal{C}}$, the object $Y$ belongs to $\operatorname{\mathcal{D}}$, and $e$ corresponds to an isomorphism $e_0: F(X) \rightarrow Y$ in the $\infty $-category $\operatorname{\mathcal{D}}$ (under the identification of Remark 5.2.4.14).

Proof of Proposition 5.2.4.15. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories, so that the projection map $\pi : \operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}\rightarrow \Delta ^{1}$ of Example 5.2.4.18 is a cocartesian fibration. Note that the morphism

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

satisfies $H|_{ \{ 0\} \times \operatorname{\mathcal{C}}} = \operatorname{id}_{\operatorname{\mathcal{C}}}$ and $H|_{ \{ 1\} \times \operatorname{\mathcal{C}}} = F$. To complete the proof, it will suffice to show that for every object $C \in \operatorname{\mathcal{C}}$, the restriction $H|_{ \Delta ^1 \times \{ C\} }$ is a $\pi $-cocartesian morphism $e: X \rightarrow F(X)$ in the $\infty $-category $\operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$. This follows from the criterion of Example 5.2.4.18, since $e$ corresponds to the identity morphism $\operatorname{id}_{ F(X) }: F(X) \rightarrow F(X)$ under the identification of Remark 5.2.4.14. $\square$

Passing to opposite $\infty $-categories, we obtain a dual version of Proposition 5.2.4.15:

Variant 5.2.4.19. 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 5.2.4.20. 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.