The homotopy category of chain complexes can also be interpreted along these lines.

The basic object is the category of chain complexes in .

The boundary operation on a simplex induces a singular chain complex.

This allows the entire chain complex to be treated as a functor.

A chain complex may then be defined as follows.

The boundary operator on a chain complex in homological algebra.

The chain complex is the central notion of homological algebra.

Dually, the boundary operators in a chain complex are sometimes called codifferentials.

The set of all k-chains forms a group and the sequence of these groups is called a chain complex.

See Homotopy category of chain complexes for further information.