Kerodon

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

Example 4.3.4.7. For each integer $n \geq 0$, let $| \Delta ^{n} | = \{ (u_0, \ldots , u_ n) \in \operatorname{\mathbf{R}}_{\geq 0}: u_0 + \cdots + u_ n = 1 \} $ denote the topological $n$-simplex. For $p,q \geq 0$, we have maps $| \Delta ^{p} | \xrightarrow {\iota } | \Delta ^{p+1+q} | \xleftarrow {\iota '} | \Delta ^{q} |$ given by the formulae

\[ \iota ( u_0, \ldots , u_ p) = (u_0, \ldots , u_ p, 0, \ldots , 0) \quad \quad \iota '( v_0, \ldots , v_ q) = (0, \ldots , 0, v_0, \ldots , v_ q). \]

There is a “straight-line” homotopy $h: | \Delta ^{p} | \times [0,1] \times | \Delta ^{q} | \rightarrow | \Delta ^{p+1+q} |$ from $\iota \circ \pi _{| \Delta ^{p} | }$ to $\iota ' \circ \pi _{ | \Delta ^{q} |}$, given concretely by the formula

\[ h( (u_0, \ldots , u_ p), t, (v_0, \ldots , v_ q) ) = ( (1-t) u_0, (1-t) u_1, \ldots , (1-t) u_ p, t v_0, \ldots , t v_ q ). \]

By virtue of Remark 4.3.4.3, the triple $(\iota , \iota ', h)$ can be identified with a continuous function $H_{p,q}: | \Delta ^{p} | \star | \Delta ^{q} | \rightarrow | \Delta ^{p+1+q} |$.