613: 'Finite Simplicial Complexes - Simplicial Maps' Category

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definition of 'finite simplicial complexes - simplicial maps' category

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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 category.
• The reader knows a definition of simplicial complex.
• The reader knows a definition of simplicial map.

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Target Context

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

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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:
$\ast K$: $\in \{\text{ the categories }\}$
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Conditions:
$Obj(K)=\{\text{ the finite simplicial complexes on finite-dimensional real vectors spaces }\}$.
$\wedge$
$\mathrm{\forall }{O}_1,O_2\in Obj(K)(Mor(O_1,O_2)=\{f:O_1\to O_2|f\in \{\text{ the simplicial maps }\}\})$.
$\wedge$
$\mathrm{\forall }{O}_1,O_2,O_3\in Obj(K),\mathrm{\forall }{f}_1\in Mor(O_1,O_2),\mathrm{\forall }{f}_2\in Mor(O_2,O_3)(f_2\circ f_1=f_2\circ f_1)$.
//

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2: Natural Language Description

The category, $K$, such that $Obj(K)=\{\text{ the finite simplicial complexes on finite-dimensional real vectors spaces }\}$, $\mathrm{\forall }{O}_1,O_2\in Obj(K)(Mor(O_1,O_2)=\{f:O_1\to O_2|f\in \{\text{ the simplicial maps }\}\})$, and $\mathrm{\forall }{O}_1,O_2,O_3\in Obj(K),\mathrm{\forall }{f}_1\in Mor(O_1,O_2),\mathrm{\forall }{f}_2\in Mor(O_2,O_3)(f_2\circ f_1=f_2\circ f_1)$.

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3: Note

"$f_2\circ f_1=f_2\circ f_1$" may seem trivial, but that is indeed meaningful, because the $\circ$ in the left hand side is the composition of the morphisms while the $\circ$ in the right hand side is the composition of the maps, so, what that means is that composition of morphisms is defined to be composition of maps, which is not trivial.

It is indeed a category: for any $f_1\in Mor(O_1,O_2)$, $f_2\in Mor(O_2,O_3)$, and $f_3\in Mor(O_3,O_4)$, 1) $f_2\circ f_1\in Mor(O_1,O_3)$, because when $\{p_0,...,p_n\}\subseteq Vert{O}_1$ spans a simplex in $O_1$, $\{f_1(p_0),...,f_1(p_n)\}\subseteq Vert{O}_2$ spans a simplex in $O_2$, and $\{f_2\circ f_1(p_0),...,f_2\circ f_1(p_n)\}\subseteq Vert{O}_3$ spans a simplex in $O_3$; 2) for each object, $O$, there is the identity morphism, $i{d}_O\in Mor(O,O)$ as the identity map, $i{d}_0:VertO\to VertO$ ; 3) $f_3\circ (f_2\circ f_1)=(f_3\circ f_2)\circ f_1$.

This definition requires that the simplicial complexes are finite; the category without the requirement is possible, but for our immediate purposes, the requirement does not cause any trouble while avoids some troubles.

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References

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