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5.2.2 Cartesian Morphisms of $\infty $-Categories

Let $q: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between ordinary categories and let $g: Y \rightarrow Z$ be a morphism in $\operatorname{\mathcal{C}}$ having image $\overline{g}: \overline{Y} \rightarrow \overline{Z}$ in $\operatorname{\mathcal{D}}$. Recall that $g$ is $q$-cartesian if, for every object $X \in \operatorname{\mathcal{C}}$ having image $\overline{X} = q(X)$ in $\operatorname{\mathcal{D}}$, the diagram of sets

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \ar [r]^-{g \circ } \ar [d] & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z) \ar [d] \\ \operatorname{Hom}_{\operatorname{\mathcal{D}}}( \overline{X}, \overline{Y} ) \ar [r]^-{\overline{g} \circ } & \operatorname{Hom}_{\operatorname{\mathcal{D}}}( \overline{X}, \overline{Z} ) } \]

is a pullback square (Definition 5.1.4.1). Our goal in this section is to give an analogous characterization of cartesian morphisms in the setting of $\infty $-categories.

We now encounter a slight complication: if $X$, $Y$, and $Z$ are objects of an $\infty $-category $\operatorname{\mathcal{C}}$ and $g: Y \rightarrow Z$ is a morphism, then the composition map $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \xrightarrow { g \circ } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)$ is only well-defined up to homotopy. We can circumvent this difficulty using the Kan complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y,Z)$ of Notation 4.6.3.1. By virtue of Corollary 4.6.3.5, the restriction map $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y,Z) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)$ is a trivial Kan fibration of simplicial sets, and therefore induces a homotopy equivalence $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y,Z) \times _{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) } \{ g\} \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$. Moreover, the “long edge” of $\Delta ^2$ determines a map of Kan complexes

\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y,Z) \times _{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) } \{ g\} \hookrightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y,Z) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z), \]

which we can regard as a surrogate for the composition map $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \xrightarrow { g \circ } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)$. This construction depends functorially on $\operatorname{\mathcal{C}}$ in the following sense: if $q: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a functor of $\infty $-categories carrying $X$ to $\overline{X} \in \operatorname{\mathcal{D}}$ and $g$ to $\overline{g}: \overline{Y} \rightarrow \overline{Z}$, then it induces a commutative diagram of Kan complexes

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y,Z) \times _{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) } \{ g\} \ar [r] \ar [d] & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z) \ar [d] \\ \operatorname{Hom}_{\operatorname{\mathcal{D}}}( \overline{X}, \overline{Y}, \overline{Z} ) \times _{ \operatorname{Hom}_{\operatorname{\mathcal{D}}}( \overline{Y}, \overline{Z} ) } \{ \overline{g} \} \ar [r] & \operatorname{Hom}_{\operatorname{\mathcal{D}}}( \overline{X}, \overline{Z} ), } \]

where the vertical maps are determined by $q$ and the horizontal maps are given by restriction. We can now state our main result, which we will prove at the end of this section:

Proposition 5.2.2.1. Let $q: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of $\infty $-categories and let $g: Y \rightarrow Z$ be a morphism in $\operatorname{\mathcal{C}}$ having image $\overline{g}: \overline{Y} \rightarrow \overline{Z}$ in $\operatorname{\mathcal{D}}$. Then $g$ is $q$-cartesian if and only if, for every object $X \in \operatorname{\mathcal{C}}$ having image $\overline{X} = q(X)$ in $\operatorname{\mathcal{D}}$, the diagram of Kan complexes

5.5
\begin{equation} \begin{gathered}\label{equation:mapping-space-cartesian-diagram} \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y,Z) \times _{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) } \{ g\} \ar [r] \ar [d] & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z) \ar [d] \\ \operatorname{Hom}_{\operatorname{\mathcal{D}}}( \overline{X}, \overline{Y}, \overline{Z} ) \times _{ \operatorname{Hom}_{\operatorname{\mathcal{D}}}( \overline{Y}, \overline{Z}) } \{ \overline{g} \} \ar [r] & \operatorname{Hom}_{\operatorname{\mathcal{D}}}( \overline{X}, \overline{Z} ) } \end{gathered} \end{equation}

is a homotopy pullback square.

Corollary 5.2.2.2. Let $q: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between categories, and let $\operatorname{N}_{\bullet }(q): \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}})$ be the induced morphism of simplicial sets. Let $g: Y \rightarrow Z$ be a morphism in the category $\operatorname{\mathcal{C}}$. Then $g$ is $q$-cartesian (in the sense of Definition 5.1.4.1) if and only if it is $\operatorname{N}_{\bullet }(q)$-cartesian (when regarded as an edge of the simplicial set $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$).

Notation 5.2.2.3. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing objects $X$, $Y$, and $Z$, so that Construction 4.6.3.9 determines a morphism

\[ \circ : \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z) \]

in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$. If $g: Y \rightarrow Z$ is a morphism in $\operatorname{\mathcal{C}}$, then the restriction of this composition law to $\{ g \} \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ can be regarded as a morphism from $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ to $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)$ in $\mathrm{h} \mathit{\operatorname{Kan}}$. We will denote this morphism by $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \xrightarrow { [g] \circ } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)$; note that it depends only on the homotopy class $[g]$ of the morphism $g$.

Corollary 5.2.2.4. Let $Q$ be a partially ordered set, let $q: \operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }(Q)$ be an inner fibration of $\infty $-categories, and let $g: Y \rightarrow Z$ be a morphism in $\operatorname{\mathcal{C}}$. Then $g$ is $q$-cartesian if and only if, for every object $X \in \operatorname{\mathcal{C}}$ satisfying $q(X) \leq q(Y)$, the map

\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \xrightarrow { [g] \circ } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z) \]

of Notation 5.2.2.3 is an isomorphism in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$.

Proof. By virtue of Proposition 5.2.2.1, the morphism $g$ is $q$-cartesian if and only if, for each object $X \in \operatorname{\mathcal{C}}$, the diagram of Kan complexes

5.6
\begin{equation} \begin{gathered}\label{diagram:trivial-cartesian-poset} \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y,Z) \times _{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) } \{ g\} \ar [r]^-{\theta _ X} \ar [d] & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z) \ar [d] \\ \operatorname{Hom}_{\operatorname{N}_{\bullet }(Q)}( q(X), q(Y), q(Z) ) \times _{ \operatorname{Hom}_{\operatorname{N}_{\bullet }(Q)}( q(Y), q(Z)) } \{ q(g) \} \ar [r] & \operatorname{Hom}_{\operatorname{N}_{\bullet }(Q)}( q(X), q(Z) ) } \end{gathered} \end{equation}

is a homotopy pullback square. If $q(X) \nleq q(Y)$, then the Kan complexes on the left side of the diagram (5.6), so this condition is vacuous. If $q(X) \leq q(Y)$, then the Kan complexes on the lower half of the diagram are isomorphic to $\Delta ^{0}$, so that (5.6) is a homotopy pullback square if and only if $\theta _{X}$ is a homotopy equivalence (Corollary 3.4.1.3). We conclude by observing that, in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$, we have a commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y,Z) \times _{\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) } \{ g\} \ar [rr] \ar [dr]^{\theta _{X}} & & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \ar [dl]_{ [g] \circ } \\ & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z), & } \]

where the horizontal map is an isomorphism (Corollary 4.6.3.5). $\square$

Corollary 5.2.2.5. Let $q: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of $\infty $-categories and let $\sigma : \Delta ^2 \rightarrow \operatorname{\mathcal{C}}$ be a $2$-simplex of $\operatorname{\mathcal{C}}$, which we will depict as a diagram

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

  • Suppose that $g$ is $q$-cartesian. Then $f$ is $q$-cartesian if and only if $h$ is $q$-cartesian.

  • Suppose that $f$ is $q$-cocartesian. Then $g$ is $q$-cocartesian if and only if $h$ is $q$-cocartesian.

Proof. We will prove the first assertion; the second follows by a similar argument. For every simplex $\tau $ of the $\infty $-category $\operatorname{\mathcal{C}}$, let $\overline{\tau }$ denote its image $q(\tau )$ in the $\infty $-category $\operatorname{\mathcal{D}}$. By virtue of Proposition 5.2.2.1, it will suffice to show that for every object $W \in \operatorname{\mathcal{C}}$, the following conditions are equivalent:

$(a)$

The commutative diagram of Kan complexes

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}( W, X, Y) \times _{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X, Y) } \{ f\} \ar [r] \ar [d] & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(W,Y) \ar [d] \\ \operatorname{Hom}_{\operatorname{\mathcal{D}}}( \overline{W}, \overline{X}, \overline{Y} ) \times _{ \operatorname{Hom}_{\operatorname{\mathcal{D}}}( \overline{X}, \overline{Y} ) } \{ \overline{f} \} \ar [r] & \operatorname{Hom}_{\operatorname{\mathcal{D}}}( \overline{W}, \overline{Y}) } \]

is a homotopy pullback square.

$(b)$

The commutative diagram of Kan complexes

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}( W, X, Z) \times _{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X, Z) } \{ h\} \ar [r] \ar [d] & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(W,Z) \ar [d] \\ \operatorname{Hom}_{\operatorname{\mathcal{D}}}( \overline{W}, \overline{X}, \overline{Z} ) \times _{ \operatorname{Hom}_{\operatorname{\mathcal{D}}}( \overline{X}, \overline{Z} ) } \{ \overline{h} \} \ar [r] & \operatorname{Hom}_{\operatorname{\mathcal{D}}}( \overline{W}, \overline{Z} ) } \]

is a homotopy pullback square.

By virtue of Corollaries 4.6.3.5 and 3.4.1.10, these conditions can be reformulated as follows:

$(a')$

The commutative diagram of Kan complexes

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}( W, X, Y, Z) \times _{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X, Y, Z) } \{ \sigma \} \ar [r] \ar [d] & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(W,Y, Z) \times _{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}( Y, Z) } \{ g \} \ar [d] \\ \operatorname{Hom}_{\operatorname{\mathcal{D}}}( \overline{W}, \overline{X}, \overline{Y}, \overline{Z} ) \times _{ \operatorname{Hom}_{\operatorname{\mathcal{D}}}( \overline{X}, \overline{Y}, \overline{Z} ) } \{ \overline{\sigma } \} \ar [r] & \operatorname{Hom}_{\operatorname{\mathcal{D}}}( \overline{W}, \overline{Y}, \overline{Z}) \times _{ \operatorname{Hom}_{\operatorname{\mathcal{D}}}( \overline{Y}, \overline{Z} ) } \{ \overline{g} \} } \]

is a homotopy pullback square.

$(b')$

The commutative diagram of Kan complexes

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}( W, X, Y, Z) \times _{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X, Y, Z) } \{ \sigma \} \ar [r] \ar [d] & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(W,Z) \ar [d] \\ \operatorname{Hom}_{\operatorname{\mathcal{D}}}( \overline{W}, \overline{X}, \overline{Y}, \overline{Z} ) \times _{ \operatorname{Hom}_{\operatorname{\mathcal{D}}}( \overline{X}, \overline{Y}, \overline{Z}) } \{ \overline{\sigma } \} \ar [r] & \operatorname{Hom}_{\operatorname{\mathcal{D}}}( \overline{W}, \overline{Z}) } \]

is a homotopy pullback square.

The equivalence of $(a')$ and $(b')$ follows by applying Proposition 3.4.1.9 to the commutative diagram of Kan complexes

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}( W, X, Y, Z) \times _{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X, Y, Z) } \{ \sigma \} \ar [r] \ar [d] & \operatorname{Hom}_{\operatorname{\mathcal{D}}}( \overline{W}, \overline{X}, \overline{Y}, \overline{Z} ) \times _{ \operatorname{Hom}_{\operatorname{\mathcal{D}}}( \overline{X}, \overline{Y}, \overline{Z}) ) } \{ \overline{\sigma } \} \ar [d] \\ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(W,Y, Z) \times _{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}( Y, Z) } \{ g \} \ar [d] \ar [r] & \operatorname{Hom}_{\operatorname{\mathcal{D}}}( \overline{W}, \overline{Y}, \overline{Z}) \times _{ \operatorname{Hom}_{\operatorname{\mathcal{D}}}( \overline{Y}, \overline{Z} ) } \{ \overline{g} \} \ar [d] \\ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(W,Z) \ar [r] & \operatorname{Hom}_{\operatorname{\mathcal{D}}}( \overline{W}, \overline{Z}), } \]

noting that the lower half of the diagram is a homotopy pullback square by virtue of our assumption that $g$ is $q$-cartesian (Proposition 5.2.2.1). $\square$

Corollary 5.2.2.6. Let $q: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of $\infty $-categories, and let $f,g: X \rightarrow Y$ be homotopic morphisms in the $\infty $-category $\operatorname{\mathcal{C}}$. Then $f$ is $q$-cartesian if and only if $g$ is $q$-cartesian. Similarly, $f$ is $q$-cocartesian if and only if $g$ is $q$-cocartesian.

Proof. Our assumption that $f$ is homotopic to $g$ guarantees that there exists a $2$-simplex of $\operatorname{\mathcal{C}}$ as depicted in the diagram

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

Since the identity morphism $\operatorname{id}_{Y}$ is $q$-cartesian (Corollary 5.2.1.8), Corollary 5.2.2.5 shows that $f$ is $q$-cartesian if and only if $g$ is $q$-cartesian. The dual assertion for $q$-cocartesian morphisms follows by a similar argument. $\square$

Using Proposition 5.2.2.1, we deduce the following stronger version of Remark 5.2.1.6:

Corollary 5.2.2.7 (Transitivity). Let $p: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $q: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be inner fibrations of simplicial sets, and let $e: Y \rightarrow Z$ be an edge of the simplicial set $\operatorname{\mathcal{C}}$.

  • Assume that $p(e)$ is a $q$-cartesian edge of $\operatorname{\mathcal{D}}$. Then $e$ is $p$-cartesian if and only if it is $(q \circ p)$-cartesian.

  • Assume that $p(e)$ is a $q$-cocartesian edge of $\operatorname{\mathcal{D}}$. Then $e$ is $p$-cocartesian if and only if it is $(q \circ p)$-cocartesian.

Proof. We will prove the first assertion; the second follows by a similar argument. Using Remark 5.2.1.10, we can reduce to the case where $\operatorname{\mathcal{E}}$ is an $\infty $-category (or even a simplex), so that $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are also $\infty $-categories (Remark 4.1.1.9). Fix an object $X \in \operatorname{\mathcal{C}}$, and set $r = q \circ p$. We have a commutative diagram of Kan complexes

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y,Z) \times _{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) } \{ e\} \ar [r] \ar [d] & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z) \ar [d] \\ \operatorname{Hom}_{\operatorname{\mathcal{D}}}( p(X), p(Y), p(Z) ) \times _{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}( p(Y), p(Z) ) } \{ q(e) \} \ar [r] \ar [d] & \operatorname{Hom}_{\operatorname{\mathcal{D}}}( p(X), p(Z) ) \ar [d] \\ \operatorname{Hom}_{\operatorname{\mathcal{E}}}( r(X), r(Y), r(Z) ) \times _{ \operatorname{Hom}_{\operatorname{\mathcal{E}}}( r(Y), r(Z) ) } \{ r(e) \} \ar [r] & \operatorname{Hom}_{\operatorname{\mathcal{E}}}( r(X), r(Z) ). } \]

If $p(e)$ is a $q$-cartesian morphism of $\operatorname{\mathcal{D}}$, then the bottom square is a homotopy pullback (Proposition 5.2.2.1). Invoking Proposition 3.4.1.9, we deduce that the upper square is a homotopy pullback if and only if the outer rectangle is a homotopy pullback. Allowing $X$ to vary and invoking Proposition 5.2.2.1, we conclude that $e$ is $p$-cartesian if and only if is $r$-cartesian. $\square$

Proof of Proposition 5.2.2.1. Let $q: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of $\infty $-categories, and let $g: Y \rightarrow Z$ be a morphism in the $\infty $-category $\operatorname{\mathcal{C}}$ having image $\overline{g}: \overline{Y} \rightarrow \overline{Z}$ in the $\infty $-category $\operatorname{\mathcal{D}}$. By virtue of Proposition 5.2.1.11, the morphism $g$ is $q$-cartesian if and only if the restriction map

\[ \theta : \operatorname{\mathcal{C}}_{/g} \rightarrow \operatorname{\mathcal{C}}_{/Z} \times _{\operatorname{\mathcal{D}}_{/\overline{Z}}} \operatorname{\mathcal{D}}_{/\overline{g}} \]

is a trivial Kan fibration of simplicial sets. Since $q$ is an inner fibration, the morphism $\theta $ is a right fibration (Proposition 4.3.6.5). For each object $X \in \operatorname{\mathcal{C}}$, $\theta $ restricts to a right fibration of simplicial sets

\[ \theta _{X}: \{ X\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/g} \rightarrow \{ X\} \times _{ \operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/Z} \times _{\operatorname{\mathcal{D}}_{/\overline{Z}}} \operatorname{\mathcal{D}}_{/\overline{g}}. \]

Note that if $\theta $ is a trivial Kan fibration, then so is $\theta _{X}$. Conversely, if each $\theta _{X}$ is a trivial Kan fibration, then every fiber of $\theta $ is a contractible Kan complex, so that $\theta $ is a trivial Kan fibration by virtue of Proposition 4.4.2.11. To complete the proof, it will suffice to show that $\theta _{X}$ is a trivial Kan fibration if and only if the diagram (5.5) appearing in the statement of Proposition 5.2.2.1 is a homotopy pullback square.

For the remainder of the proof, let us regard the object $X \in \operatorname{\mathcal{C}}$ as fixed, and set $\overline{X} = q(X)$. We then have a commutative diagram of simplicial sets

5.7
\begin{equation} \begin{gathered}\label{equation:mapping-space-cartesian-diagram2} \xymatrix@R =50pt@C=50pt{ \{ X\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/g} \ar [r] \ar [d] & \{ X\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/Z} \ar [d]^{\rho } \\ \{ \overline{X} \} \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{ / \overline{g} } \ar [r] & \{ \overline{X} \} \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{ / \overline{Z} }. } \end{gathered} \end{equation}

Corollary 4.3.6.8 guarantees that the restriction maps

\[ \operatorname{\mathcal{C}}_{/g} \rightarrow \operatorname{\mathcal{C}}_{/Z} \rightarrow \operatorname{\mathcal{C}}\quad \quad \operatorname{\mathcal{D}}_{/\overline{g}} \rightarrow \operatorname{\mathcal{D}}_{ / \overline{Z} } \rightarrow \operatorname{\mathcal{D}} \]

are right fibrations, so that each of the simplicial sets appearing in the diagram (5.7) is a Kan complex. The morphism $\rho $ is a pullback of the restriction map $\operatorname{\mathcal{C}}_{/Z} \rightarrow \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{/ \overline{Z}}$, and is therefore a right fibration by virtue of Proposition 4.3.6.5. Applying Corollary 4.4.3.7, we deduce that $\rho $ is a Kan fibration. The projection map

\[ \{ X\} \times _{ \operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/Z} \times _{\operatorname{\mathcal{D}}_{/\overline{Z}}} \operatorname{\mathcal{D}}_{/\overline{g}} \rightarrow \{ \overline{X} \} \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{ / \overline{Z} } \]

is a pullback of $\rho $, and therefore also a Kan fibration. In particular, the target of the right fibration $\theta _{X}$ is a Kan complex, so that $\theta _{X}$ is a Kan fibration (Corollary 4.4.3.7). It follows that $\theta _{X}$ is a trivial Kan fibration if and only if is a homotopy equivalence (Proposition 3.3.7.4): that is, if and only if the diagram (5.7) is a homotopy pullback square.

Let $\sigma $ be an $n$-simplex of the simplicial set $\{ X\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/g}$. Then we can identify $\sigma $ with a morphism of simplicial sets $u_{\sigma }: \Delta ^{n} \star \Delta ^1 \rightarrow \operatorname{\mathcal{C}}$ such that $u_{\sigma }|_{\Delta ^ n}$ is the constant map taking the value $X$ and $u_{\sigma }|_{\Delta ^1} = g$. Let $\pi : \Delta ^ n \times \Delta ^2 \rightarrow \Delta ^{n} \star \Delta ^1 \simeq \Delta ^{n+2}$ be the map given on vertices by the formula

\[ \pi (i,j) = \begin{cases} i & \text{ if } j = 0 \\ n+1 & \text{ if } j = 1 \\ n+2 & \text{ if } j = 2. \end{cases} \]

The composition $u_{\sigma } \circ \pi : \Delta ^ n \times \Delta ^2 \rightarrow \operatorname{\mathcal{C}}$ can then be regarded as an $n$-simplex $\sigma '$ of the simplicial set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y,Z) \times _{\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)} \{ g\} $. The construction $\sigma \mapsto \sigma '$ depends functorially on $[n] \in \operatorname{{\bf \Delta }}$, and therefore determines a morphism of Kan complexes

\[ \iota ^{\mathrm{R}}_{X,g}: \{ X\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/g} \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y,Z) \times _{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) } \{ g\} . \]

Note that the morphism $\iota _{X,g}$ fits into a commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ \{ X\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/g} \ar [d] \ar [r]^-{ \iota ^{\mathrm{R}}_{X,g} } & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y,Z) \times _{\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) } \{ g\} \ar [d] \\ \{ X\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/Y} \ar [r]^-{\iota ^{\mathrm{R}}_{X,Y}} & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y), } \]

where the left vertical map is a pullback of the restriction morphism $\operatorname{\mathcal{C}}_{/g} \rightarrow \operatorname{\mathcal{C}}_{/Y}$ (and therefore a trivial Kan fibration by virtue of Corollary 4.3.6.10), the right vertical map is a pullback of the restriction morphism $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y,Z) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)$ (and therefore a trivial Kan fibration by virtue of Corollary 4.6.3.5), and $\iota ^{\mathrm{R}}_{X,Y}: \operatorname{Hom}^{\mathrm{R}}_{\operatorname{\mathcal{C}}}(X,Y) \hookrightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ is the right-pinch inclusion map of Construction 4.6.5.12 (which is a homotopy equivalence of Kan complexes by virtue of Proposition 4.6.6.9). It follows that $\iota ^{\mathrm{R}}_{X,g}$ is also a homotopy equivalence of Kan complexes. Applying the same construction to the $\infty $-category $\operatorname{\mathcal{D}}$, we obtain a homotopy equivalence

\[ \iota ^{\mathrm{R}}_{\overline{X}, \overline{g}}: \{ \overline{X} \} \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{/ \overline{g}} \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}(\overline{X}, \overline{Y}, \overline{Z}) \times _{ \operatorname{Hom}_{\operatorname{\mathcal{D}}}( \overline{Y}, \overline{Z}) } \{ \overline{g} \} . \]

We have a commutative diagram of Kan complexes

\[ \xymatrix@R =50pt@C=10pt{ \{ X\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/g} \ar [rr] \ar [dd] \ar [dr]^{ \iota ^{\mathrm{R}}_{X,g}} & & \{ X\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/Z} \ar [dd] \ar [dr]^{ \iota _{X,Z}^{\mathrm{R}} } & \\ & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y,Z) \times _{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) } \{ g\} \ar [rr] \ar [dd] & & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z) \ar [dd] \\ \{ \overline{X} \} \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{ / \overline{g} } \ar [rr] \ar [dr]^{ \iota ^{\mathrm{R}}_{\overline{X}, \overline{g} } } & & \{ \overline{X} \} \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{/ \overline{Z} } \ar [dr]^{ \iota ^{\mathrm{R}}_{ \overline{X}, \overline{Z} }} & \\ & \operatorname{Hom}_{\operatorname{\mathcal{D}}}( \overline{X}, \overline{Y}, \overline{Z} ) \times _{ \operatorname{Hom}_{\operatorname{\mathcal{D}}}( \overline{Y}, \overline{Z} ) } \{ \overline{g} \} \ar [rr] & & \operatorname{Hom}_{\operatorname{\mathcal{D}}}( \overline{X}, \overline{Z} ), } \]

where the right-pinch inclusion maps $\iota ^{\mathrm{R}}_{X,Z}$ and $\iota ^{\mathrm{R}}_{\overline{X}, \overline{Z}}$ are homotopy equivalences (Proposition 4.6.6.9). Applying Corollary 3.4.1.10, we conclude that the front face (5.5) is a homotopy pullback square if and only if the back face (5.7) is a homotopy pullback square: that is, if and only if $\theta _{X}$ is a trivial Kan fibration. $\square$