Standard simplex
The simplex 
of dimension 
in the space 
with vertices at the points 
, 
(the 
stands in the 
-th place), i.e.
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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
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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 
.
Comments
For references see Simplicial set.
Standard simplex. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Standard_simplex&oldid=48801