differential on total chain complex

Question

There is the definition of (second) total chain complex of double complex of chains from GTM 004.He says $(\partial b)_{p,q}=\partial'b_{p+1,q}+\partial''b_{p,q+1}$,but I don't have any clues what $b_{p+1,q},b_{p,q+1}$ are.

Answer 1

An element of the total complex consists of a choice of an element $b_{p,q}\in B_{p,q}$ for each pair of integers $p$ and $q$. We need to define what its boundary is. The definition is that is consists of the elements $c_{p,q}$ determined by the formula $$c_{p,q}=\partial'b_{p+1,q}+\partial''b_{p,q+1}.$$ As $b_{p+1,q}\in B_{p+1,q}$ then $\partial'b_{p+1,q}\in B_{p,q}$ etc., so that $c_{p,q}\in B_{p,q}$.