chain

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.

$H_n = Z_n/B_n$ | (chain-)homology | (cochain-)cohomology | $H^n = Z^n/B^n$ |
---|---|---|---|

$C_n$ | chain | cochain | $C^n$ |

$Z_n \subset C_n$ | cycle | cocycle | $Z^n \subset C^n$ |

$B_n \subset C_n$ | boundary | coboundary | $B^n \subset C^n$ |