nLab chain

Idea

In homology theory

In homology/homological algebra, 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 submanifolds? with boundary.

In order theory

In order theory, a chain is a totally ordered subset of a given poset (or proset). See also antichain

For applications of this concept see for instance

• Zorn's Lemma

• shuffle

• …

Related concepts

$H_n = Z_n/B_n$	(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	$B^n \subset C^n$