Cochain complex ($(C^{\bullet },d)$)

A cochain complex is a sequence of abelian groups or modules and maps $d^n:C^n\to C^{n+1}$ satisfying $d^{n+1}{d}^n=0$. Its cohomology is $H^n(C)=\ker d^n/\mathrm{im}{d}^{n-1}$.

A coboundary map is the degree-one differential $d^n:C^n\to C^{n+1}$ of a cochain complex.