# Standard simplex

The simplex of dimension in the space with vertices at the points , (the stands in the -th place), i.e. For any topological space , the continuous mappings are the singular simplices of (see Singular homology).

The simplicial complex whose vertices are the points , , while the simplices are arbitrary non-empty subsets of vertices. The geometric realization of this simplicial complex coincides with the standard simplex in the sense of 1).

The simplicial set , obtained by applying the functor to the simplicial scheme in 2), which is a contra-variant functor on the category (see Simplicial object in a category), for which Thus, non-decreasing sequences of numbers from are -dimensional simplices of the simplicial set , while the face operators and the degeneracy operators of this simplicial set are defined by the formulas  where the sign signifies that the symbol beneath it is deleted. The simplicial set is also called a simplicial segment. The simplex (the unique non-degenerate -dimensional simplex of ) is called the fundamental simplex of . The smallest simplicial subset of containing all simplices of the form with is denoted by and is called the -th standard horn.

For any simplicial set and an arbitrary -dimensional simplex of , there is a unique simplicial mapping for which . This mapping is said to be characteristic for .

The fundamental simplex of a simplicial set as in 3), which in this instance is denoted by .