615: 'Finite Simplicial Complexes - Simplicial Maps' Category to Top Functor

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definition of 'finite simplicial complexes - simplicial maps' category to Top functor

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Topics

About: category

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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 'finite simplicial complexes - simplicial maps' category.
• The reader knows a definition of Top category.
• The reader knows a definition of covariant functor.

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Target Context

• The reader will have a definition of 'finite simplicial complexes - simplicial maps' category to Top functor.

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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:
$K$: $=\text{ the 'finite simplicial complexes - simplicial maps' category }$
$Top$: $=\text{ the Top category }$
$\ast |\bullet |$: $:Obj(K)\to Obj(Top),Mor(K)\to Mor(Top)$, $\in \{\text{ the covariant functors }\}$
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Conditions:
$\mathrm{\forall }C\in Obj(K)(|C|=|C|)$, where the right hand side is the underlying space
$\wedge$
$\mathrm{\forall }f\in Mor(C_1,C_2)(\mathrm{\forall }S\in C_1(|f{|}_{{|}_S}=\text{ the affine map }))$
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2: Natural Language Description

For the 'finite simplicial complexes - simplicial maps' category, $K$, and the $Top$ category, the covariant functor, $|\bullet |$, $:Obj(K)\to Obj(Top),Mor(K)\to Mor(Top)$, such that for each $C\in Obj(K)$, $|C|=|C|$, where the right hand side is the underlying space, for each $f\in Mor(C_1,C_2)$, for each $S\in C_1$, $|f{|}_{{|}_S}=\text{ the affine map }$

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3: Note

$|\bullet |$ is indeed a covariant functor: $|C|$ is indeed a topological space; 1) $|f_1|\in Mor(|C_1|,|C_2|)$, because $|f|$ is indeed a continuous map from $|C_1|$ into $|C_2|$, because each simplex in $C_1$ is closed on $|C_1|$, by the proposition that any affine simplex on any finite-dimensional real vectors space is closed and compact on the canonical topological superspace and each affine map is continuous, by the proposition that any affine map from the affine or convex set spanned by any possibly-non-affine-independent set of base points on any finite-dimensional real vectors space into any finite-dimensional real vectors space is continuous with respect to the canonical topologies, and the proposition that any map between topological spaces is continuous if the domain restriction of the map to each closed set of a finite closed cover is continuous can be applied; 2) $|i{d}_{C_1}|=i{d}_{|O_1|}$, obviously; 3) $|f_2\circ f_1|=|f_2|\circ |f_1|$, because $|f_2\circ f_1{|}_{{|}_S}$ is $\sum _{j\in \{0,...,n\}}{t}^j{p}_j\mapsto \sum _{j\in \{0,...,n\}}{t}^j{f}_2\circ f_1(p_j)$ while $|f_2|\circ |f_1{|}_{{|}_S}$ is $\sum _{j\in \{0,...,n\}}{t}^j{p}_j\mapsto \sum _{j\in \{0,...,n\}}{t}^j{f}_1(p_j)\mapsto \sum _{j\in \{0,...,n\}}{t}^j{f}_2\circ f_1(p_j)$, by the proposition that any affine map from the affine or convex set spanned by any possibly-non-affine-independent set of base points on any real vectors space is linear

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References

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