Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Warning 4.3.4.14. Let $X$ and $Y$ be simplicial sets, and let $X \diamond Y$ denote the simplicial set given by the iterated coproduct

\[ X \coprod _{ (X \times \{ 0\} \times Y)} (X \times \Delta ^1 \times Y) \coprod _{ (X \times \{ 1\} \times Y) } Y. \]

Since the formation of geometric realization commutes with the formation of colimits, we have an evident comparison map of topological spaces

\[ |X \diamond Y | \rightarrow |X| \star |Y|. \]

This map is always bijective, and is a homeomorphism if either $X$ or $Y$ is finite (see Corollary 3.5.2.2). In this case, Corollary 4.3.4.13 supplies a homeomorphism of geometric realizations $| X \diamond Y | \simeq | X \star Y |$. Beware that this homeomorphism does not arise from a morphism of simplicial sets. In the case $X = \Delta ^ p$ and $Y = \Delta ^{q}$, it arises from the homotopy

\[ h: | \Delta ^{p} | \times | \Delta ^1 | \times | \Delta ^{q} | \rightarrow | \Delta ^{p+1+q} | \]
\[ h( (u_0, \ldots , u_ p), t, (v_0, \ldots , v_ q) ) = ( (1-t) u_0, (1-t) u_1, \ldots , (1-t) u_ p, t v_0, \ldots , t v_ q ). \]

appearing in Example 4.3.4.7, which is not piecewise-linear with respect to the natural triangulation of the polysimplex $| \Delta ^{p} | \times | \Delta ^1 | \times | \Delta ^ q |$.