# Kerodon

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Warning 7.6.3.3. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $\sigma : \Delta ^1 \times \Delta ^1 \rightarrow \operatorname{\mathcal{C}}$ be a morphism, which we depict as a diagram

$\xymatrix@C =50pt@R=50pt{ X_{01} \ar [r]^-{g_0} \ar [d]^-{g_1} & X_0 \ar [d]^-{f_0} \\ X_1 \ar [r]^-{f_1} & X. }$

Beware that, if $\sigma$ is a pullback square in the $\infty$-category $\operatorname{\mathcal{C}}$, then the associated diagram

$\xymatrix@C =50pt@R=50pt{ X_{01} \ar [r]^-{[g_0]} \ar [d]^-{[g_1]} & X_0 \ar [d]^-{[f_0]} \\ X_1 \ar [r]^-{[f_1]} & X }$

need not be a pullback square in the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ (see Example 7.6.3.4 and Exercise 7.6.3.5). If $Y$ is an object of $\operatorname{\mathcal{C}}$, then the map of sets

$\operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}( Y, X_{01} ) \xrightarrow { ( [g_0], [g_1]) \circ } \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{\mathcal{C}}}}( Y, X_0) \times _{ \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}( Y, X ) } \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(Y, X_1 )$

is surjective, but need not be injective. Given a commutative diagram

7.56
$$\begin{gathered}\label{equation:pullback-not-preserved-by-homotopy} \xymatrix@C =50pt@R=50pt{ Y \ar [r]^-{[g_0]} \ar [d]^-{[g_1]} & X_0 \ar [d]^-{[f_0]} \\ X_1 \ar [r]^-{[f_1]} & X } \end{gathered}$$

in the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$, we can always find a morphism $g_{01}: Y \rightarrow X_{01}$ satisfying $g_{01}: Y \rightarrow X_{01}$ satisfying $[g_0] = [f'_0] \circ [g_{01}]$ and $[ g_1 ] = [ f'_1 ] \circ [ g_{01} ]$. However, the homotopy class $[g_{01}]$ is not uniquely determined: roughly speaking, to construct $g_{01}$, we need to lift (7.56) to a commutative diagram in the $\infty$-category $\operatorname{\mathcal{C}}$. Such a lift always exists (Exercise 1.4.2.10), but is not unique (even up to homotopy).