# Chain complexes

A fundamental intuitive fact reproduced in this formalism is that the boundary of a boundary is zero. A useful algebraic generalization of this idea is the chain complex, defined to be a sequence of homomorphisms of abelian groups $${\partial_{n}\colon C_{n}\to C_{n-1}}$$ with $${\partial_{n}\partial_{n+1}=0}$$. In our case the abelian groups $${C_{n}}$$ are the $${n}$$-chains, and the chain complex can be illustrated as follows: