Chain complex

 A cochain complex $\mathcal{C}$ is a collection of vector spaces ${C^k}_{k\in\mathcal{Z}}$ together with sequence of linear maps $d_k:C^k \rightarrow C^{k+1}$
$$\begin{equation}   \cdots \rightarrow C^{-1} \xrightarrow{d_{-1}} C^{0} \xrightarrow{d_{0}} C^1 \xrightarrow{d_{1}} C^2\xrightarrow{d_{2}}\cdots \end{equation}$$
with
$$\begin{equation}   d_k \circ d_{k-1} = 0 \end{equation}$$
${d_k}$ are collection of linear maps known as ``differentials'' of the cochain complex.
One relevant example of Cochain complex is the vector space $\Omega^{*}(M)$ of differential forms on Manifold together with exterior derivative.
$$\begin{equation}   \cdots \rightarrow \Omega^{-1}(M) \xrightarrow{d_{-1}} \Omega^{0}(M) \xrightarrow{d_{0}} \Omega^1(M) \xrightarrow{d_{1}} \Omega^2(M)\xrightarrow{d_{2}}\cdots,\;d\circ d = 0 \end{equation}$$
Above cochain complex is known as deRahm complex.