612: Simplicial Map

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definition of simplicial map

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Topics

About: vectors space

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Structured Description
• 2: Natural Language Description
• 3: Note

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Starting Context

• The reader knows a definition of simplicial complex.
• The reader knows a definition of convex set spanned by possibly-non-affine-independent set of base points on real vectors space.
• The reader knows a definition of map.

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Target Context

• The reader will have a definition of simplicial map.

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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.

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Main Body

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1: Structured Description

Here is the rules of Structured Description.

Entities:
$V_1$: $\in \{\text{ the real vectors spaces }\}$
$V_2$: $\in \{\text{ the real vectors spaces }\}$
$C_1$: $\in \{\text{ the simplicial complexes on }{V}_1\}$
$C_2$: $\in \{\text{ the simplicial complexes on }{V}_2\}$
$\ast f$: $:Vert{C}_1\to Vert{C}_2$
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Conditions:
$\mathrm{\forall }\{p_0,...,p_n\}\subseteq Vert{C}_1\text{ such that }\{p_0,...,p_n\}\text{ spans a simplex in }{C}_1(\{f(p_0),...,f(p_n)\}\text{ spans a simplex in }{C}_2)$.
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2: Natural Language Description

For any real vectors spaces, $V_1,V_2$, and any simplicial complexes, $C_1$, on $V_1$ and $C_2$, on $V_2$, any map, $f:Vert{C}_1\to Vert{C}_2$, such that for each $\{p_0,...,p_n\}\subseteq Vert{C}_1$ that spans a simplex in $C_1$, $\{f(p_0),...,f(p_n)\}$ spans a simplex in $C_2$

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3: Note

"$\{p_0,...,p_n\}$ spans a simplex in $C_1$" is not equal to '$[p_0,...,p_n]$ is a simplex in $C_1$' when $\{p_0,...,p_n\}$ contains some duplications, but supposing that the duplications have been removed, they are equal by the proposition that for any simplicial complex, each vertex of each simplex that is on any another simplex is a vertex of the latter simplex: each $p_j$ is on the spanned simplex, so, $p_j$ is a vertex of the simplex, and the spanned simplex is nothing but $[p_0,...,p_n]$. "$\{f(p_0),...,f(p_n)\}$ spans a simplex in $C_2$" is likewise.

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References

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