Generally, a chain is an element of a chain complex. Specifically for the complex computing the singular homology of a topological space, a singular chain is a formal linear combination of simplices in that space. In de Rham cohomology, a de Rham chain? is a formal linear combination of parametrized submanifold?s with boundary.

In order theory, the term has another meaning: a totally ordered subset of a given poset (or proset). See Zorn's Lemma for an application of this concept; see also antichain.

H n=Z n/B nH_n = Z_n/B_n(chain-)homology(cochain-)cohomologyH n=Z n/B nH^n = Z^n/B^n
C nC_nchaincochainC nC^n
Z nC nZ_n \subset C_ncyclecocycleZ nC nZ^n \subset C^n
B nC nB_n \subset C_nboundarycoboundaryB nC nB^n \subset C^n