Difference between revisions of "Standard simplex"
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| + | $#C+1 = 46 : ~/encyclopedia/old_files/data/S087/S.0807170 Standard simplex | ||
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| − | + | The [[Simplex|simplex]] $ \Delta ^ {n} $ | |
| + | of dimension $ n $ | ||
| + | in the space $ \mathbf R ^ {n+1} $ | ||
| + | with vertices at the points $ e _ {i} = ( 0 \dots 1 \dots 0) $, | ||
| + | $ i = 0 \dots n $( | ||
| + | the $ 1 $ | ||
| + | stands in the $ i $- | ||
| + | th place), i.e. | ||
| − | + | $$ | |
| + | \Delta ^ {n} = \{ {( t _ {0} \dots t _ {n+1} ) } : {0 \leq t _ {i} \leq 1, \sum t _ {i} = 1 } \} | ||
| + | \subset \mathbf R ^ {n+1} . | ||
| + | $$ | ||
| − | + | For any topological space $ X $, | |
| + | the continuous mappings $ \sigma : \Delta ^ {n} \rightarrow X $ | ||
| + | are the singular simplices of $ X $( | ||
| + | see [[Singular homology|Singular homology]]). | ||
| − | + | The [[Simplicial complex|simplicial complex]] $ \Delta ^ {n} $ | |
| + | whose vertices are the points $ l _ {i} $, | ||
| + | $ 0 \leq i \leq n $, | ||
| + | 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|simplicial set]] $ \Delta ^ {n} $, | |
| + | obtained by applying the functor $ O ^ {+} $ | ||
| + | to the simplicial scheme in 2), which is a contra-variant functor on the category $ \Delta $( | ||
| + | see [[Simplicial object in a category|Simplicial object in a category]]), for which | ||
| − | + | $$ | |
| + | \Delta ^ {n} ([ m]) = \Delta ([ m], [ n]),\ \ | ||
| + | \Delta ^ {n} ( \lambda )( \mu ) = \mu \lambda . | ||
| + | $$ | ||
| − | + | Thus, non-decreasing sequences $ ( a _ {0} \dots a _ {m} ) $ | |
| + | of numbers from $ [ n] $ | ||
| + | are $ m $- | ||
| + | dimensional simplices of the simplicial set $ \Delta ^ {n} $, | ||
| + | while the face operators $ d _ {i} $ | ||
| + | and the degeneracy operators $ s _ {i} $ | ||
| + | of this simplicial set are defined by the formulas | ||
| − | + | $$ | |
| + | d _ {i} ( a _ {0} \dots a _ {m} ) = ( a _ {0} \dots a _ {i-1} , \widehat{a} _ {i} , a _ {i+1} \dots a _ {m} ), | ||
| + | $$ | ||
| − | + | $$ | |
| + | s _ {i} ( a _ {0} \dots a _ {m} ) = ( a _ {0} \dots a _ {i} , a _ {i} , a _ {i+1} \dots a _ {m} ), | ||
| + | $$ | ||
| − | The | + | where the sign $ \widehat{ {}} $ |
| + | signifies that the symbol beneath it is deleted. The simplicial set $ \Delta ^ {1} $ | ||
| + | is also called a simplicial segment. The simplex $ \iota _ {n} = ( 0, 1 \dots n) $( | ||
| + | the unique non-degenerate $ n $- | ||
| + | dimensional simplex of $ \Delta ^ {n} $) | ||
| + | is called the fundamental simplex of $ \Delta ^ {n} $. | ||
| + | The smallest simplicial subset of $ \Delta ^ {n+1} $ | ||
| + | containing all simplices of the form $ d _ {i} \iota _ {n+1} $ | ||
| + | with $ i \neq k $ | ||
| + | is denoted by $ \Delta _ {k} ^ {n} $ | ||
| + | and is called the $ k $- | ||
| + | th standard horn. | ||
| + | For any simplicial set $ K $ | ||
| + | and an arbitrary $ n $- | ||
| + | dimensional simplex $ \sigma $ | ||
| + | of $ K $, | ||
| + | there is a unique simplicial mapping $ \chi _ \sigma : \Delta ^ {n} \rightarrow K $ | ||
| + | for which $ \chi ( \iota _ {n} ) = \sigma $. | ||
| + | This mapping is said to be characteristic for $ \sigma $. | ||
| + | The fundamental simplex $ \iota _ {n} $ | ||
| + | of a simplicial set as in 3), which in this instance is denoted by $ \Delta _ {n} $. | ||
====Comments==== | ====Comments==== | ||
For references see [[Simplicial set|Simplicial set]]. | For references see [[Simplicial set|Simplicial set]]. | ||
Latest revision as of 16:52, 20 January 2024
The simplex $ \Delta ^ {n} $
of dimension $ n $
in the space $ \mathbf R ^ {n+1} $
with vertices at the points $ e _ {i} = ( 0 \dots 1 \dots 0) $,
$ i = 0 \dots n $(
the $ 1 $
stands in the $ i $-
th place), i.e.
$$ \Delta ^ {n} = \{ {( t _ {0} \dots t _ {n+1} ) } : {0 \leq t _ {i} \leq 1, \sum t _ {i} = 1 } \} \subset \mathbf R ^ {n+1} . $$
For any topological space $ X $, the continuous mappings $ \sigma : \Delta ^ {n} \rightarrow X $ are the singular simplices of $ X $( see Singular homology).
The simplicial complex $ \Delta ^ {n} $ whose vertices are the points $ l _ {i} $, $ 0 \leq i \leq n $, 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 $ \Delta ^ {n} $, obtained by applying the functor $ O ^ {+} $ to the simplicial scheme in 2), which is a contra-variant functor on the category $ \Delta $( see Simplicial object in a category), for which
$$ \Delta ^ {n} ([ m]) = \Delta ([ m], [ n]),\ \ \Delta ^ {n} ( \lambda )( \mu ) = \mu \lambda . $$
Thus, non-decreasing sequences $ ( a _ {0} \dots a _ {m} ) $ of numbers from $ [ n] $ are $ m $- dimensional simplices of the simplicial set $ \Delta ^ {n} $, while the face operators $ d _ {i} $ and the degeneracy operators $ s _ {i} $ of this simplicial set are defined by the formulas
$$ d _ {i} ( a _ {0} \dots a _ {m} ) = ( a _ {0} \dots a _ {i-1} , \widehat{a} _ {i} , a _ {i+1} \dots a _ {m} ), $$
$$ s _ {i} ( a _ {0} \dots a _ {m} ) = ( a _ {0} \dots a _ {i} , a _ {i} , a _ {i+1} \dots a _ {m} ), $$
where the sign $ \widehat{ {}} $ signifies that the symbol beneath it is deleted. The simplicial set $ \Delta ^ {1} $ is also called a simplicial segment. The simplex $ \iota _ {n} = ( 0, 1 \dots n) $( the unique non-degenerate $ n $- dimensional simplex of $ \Delta ^ {n} $) is called the fundamental simplex of $ \Delta ^ {n} $. The smallest simplicial subset of $ \Delta ^ {n+1} $ containing all simplices of the form $ d _ {i} \iota _ {n+1} $ with $ i \neq k $ is denoted by $ \Delta _ {k} ^ {n} $ and is called the $ k $- th standard horn.
For any simplicial set $ K $ and an arbitrary $ n $- dimensional simplex $ \sigma $ of $ K $, there is a unique simplicial mapping $ \chi _ \sigma : \Delta ^ {n} \rightarrow K $ for which $ \chi ( \iota _ {n} ) = \sigma $. This mapping is said to be characteristic for $ \sigma $.
The fundamental simplex $ \iota _ {n} $ of a simplicial set as in 3), which in this instance is denoted by $ \Delta _ {n} $.
Comments
For references see Simplicial set.
How to Cite This Entry:
Standard simplex. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Standard_simplex&oldid=17060
Standard simplex. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Standard_simplex&oldid=17060
This article was adapted from an original article by S.N. MalyginM.M. Postnikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article