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Corollary 5.1.2.4. 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 5.1.2.1, 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 4.6.7.5 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 5.1.2.1). $\square$