Theorem 34.3:

For each topological space $X$, there exists a unique chain map ${\cal B}_{X}:S_p(X)\longrightarrow S_p(X)$ and a chain homotopy ${\cal J}_{X}:S_p(X)\longrightarrow S_{p+1}(X)$ between ${\cal B}$ and the identity map, which satisfies the following two conditions.

(i)

For a continuous map $f : X \longrightarrow Y$ between topological spaces $X$ and $Y$,

$${\cal B}\circ f_{\sharp} = f_{\sharp}\circ {\cal B},\quad {\cal J}\circ f_{\sharp} = f_{\sharp}\circ {\cal J}.$$

(ii)

${\cal B}$ and ${\cal J}$ when restricted to the affine simplicies reduce to $B$ and $J$ respectively.