Definition 34.2:

The subdivision map $B:A_p(\Delta_n)\longrightarrow A_p(\Delta_n)$ is defined inductively as follows:

(i) For a zero simplex $\sigma$ we define $B\sigma = \sigma$.

(ii) For $p \geq 1$, we assume that $B$ is defined on $A_{k}(\Delta_n)$ for each $k\leq p-1$. For a $p-$simplex $\sigma$ define

$$B\sigma = K_{\bf b}(B(\partial \sigma)), \eqno(34.5)$$

the cone over the chain $B(\partial \sigma)$ with apex ${\bf b}$ as the barycenter of $\sigma$.