nLab
chains and cochains - table

$H_n=Z_n/B_n$	(chain-)homology	(cochain-)cohomology	$H^n=Z^n/B^n$
$C_n$	chain	cochain	$C^n$
$Z_n\subset C_n$	cycle	cocycle	$Z^n\subset C^n$
$B_n\subset C_n$	boundary	coboundary	$B^n\subset C^n$