definition of barycentric subdivision of simplicial complex
Topics
About: vectors space
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Natural Language Description
- 3: Note
Starting Context
- The reader knows a definition of simplicial complex.
- The reader knows a definition of ascending sequence of barycenters of faces of affine simplex.
Target Context
- The reader will have a definition of barycentric subdivision of simplicial complex.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\( V\): \(\in \{\text{ the real vectors spaces }\}\)
\( C\): \(\in \{\text{ the simplicial complexes on } V\}\)
\( Vert Sd C\): \(= \{bary (S) \vert S \in C\}\)
\(*Sd C\): \(= \{[S] \vert S \subseteq Vert Sd C \text{ such that } S \in \{\text{ the linearly-ordered subsets of } Vert Sd C\}\}\)
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Conditions:
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"\(S \in \{\text{ the linearly-ordered subsets of } Vert Sd C\}\)" is based on the partial order described in Note of the definition of ascending sequence of barycenters of faces of affine simplex: \(Vert Sd C\) is partially-ordered, and a subset may be linearly-ordered or not; being linearly-ordered means that the subset is of a subsequence of an ascending sequence of barycenters of faces of an affine simplex.
2: Natural Language Description
For any real vectors space, \(V\), and any simplicial complex, \(C\), on \(V\), \( Vert Sd C := \{bary (S) \vert S \in C\}\) and \(Sd C := \{[S] \vert S \subseteq Vert Sd C \text{ such that } S \in \{\text{ the linearly-ordered subsets of } Vert Sd C\}\}\)
3: Note
\(Sd C\) is indeed a simplicial complex: by the proposition that for any affine simplex and its any ascending sequence of faces, the set of the barycenters of the faces is affine-independent, \([S]\) is indeed an affine simplex; for each \([S]\), its each face is contained in \(Sd C\), because each subset of \(S\) is linearly-ordered; for each \([S_1], [S_2]\), \([S_1] \cap [S_2]\) is \([S_1 \cap S_2]\), by the proposition that for any simplicial complex, the nonempty intersection of the 2 affine simplexes determined by any subsequences of any ascending sequences of barycenters of faces of any elements of the complex is the affine simplex determined by the intersection of the subsequences, and so, is a face of \([S_1]\) and a face of \([S_2]\).
