A simplicial complex is a space with a triangulation. Formally, a simplicial complex in is a collection of simplices
in
such that

1. Every face of a simplex of is in , and

2. The intersection of any two simplices of is a face of each of them

(Munkres 1993, p. 7).

Objects in the space made up of only the simplices in the triangulation of the space are called simplicial subcomplexes. When
only simplicial complexes and simplicial subcomplexes
are considered, defining homology is particularly easy
(and, in fact, combinatorial because of its finite/counting nature). This kind of
homology is called simplicial homology.

Harary, F. Graph Theory. Reading, MA: Addison-Wesley, p. 7, 1994.Hatcher,
A. Algebraic
Topology. Cambridge, England: Cambridge University Press, 2002.Munkres,
J. R. "Simplicial Complexes and Simplicial Maps." §1.2 in Elements
of Algebraic Topology. New York: Perseus Books Pub.,pp. 7-14, 1993.