Chain

The same as a totally ordered set.

A formal linear combination of simplices (of a triangulation, of a simplicial set and, in particular, of singular simplices of a topological space) or of cells. In the most general sense it is an element of the group of chains of an arbitrary (as a rule, free) chain complex. A chain with coefficients in a group $G$ is an element of the tensor product of a chain complex by the group $G$.

References

[1]	N.E. Steenrod, S. Eilenberg, "Foundations of algebraic topology" , Princeton Univ. Press (1966)
[2]	P.J. Hilton, S. Wylie, "Homology theory. An introduction to algebraic topology" , Cambridge Univ. Press (1960)