# Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Warning 3.4.0.1. Suppose we are given a commutative diagram of simplicial sets

$\xymatrix@R =50pt@C=50pt{ X_0 \ar [d] \ar [r] & X \ar [d] & X_1 \ar [l] \ar [d] \\ Y_0 \ar [r] & Y & Y_1 \ar [l] }$

for which the vertical maps are weak homotopy equivalences. Then the induced map

$X_0 \times _{X} X_1 \rightarrow Y_0 \times _{Y} Y_1$

need not be a weak homotopy equivalence. For example, the pullback of the upper half of the diagram

$\xymatrix@R =50pt@C=50pt{ \{ 0\} \ar [r] \ar@ {=}[d] & \Delta ^1 \ar [d] & \{ 1\} \ar@ {=}[d] \ar [l] \\ \{ 0\} \ar [r]^-{\sim } & \Delta ^0 & \{ 1\} , \ar [l]_{\sim } }$

is empty, while the pullback of the lower half is isomorphic to $\Delta ^0$.