609: Top Category

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definition of Top 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 topological space.
• The reader knows a definition of continuous map.

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Target Context

• The reader will have a definition of Top 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:
$\ast Top$: $\in \{\text{ the categories }\}$
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Conditions:
$Obj(Top)=\{\text{ the topological spaces }\}$.
$\wedge$
$\mathrm{\forall }{O}_1,O_2\in Obj(Top)(Mor(O_1,O_2)=\{f:O_1\to O_2|f\in \text{ the continuous maps }\})$.
$\wedge$
$\mathrm{\forall }{O}_1,O_2,O_3\in Obj(Top),\mathrm{\forall }{f}_1\in Mor(O_1,O_2),\mathrm{\forall }{f}_2\in Mor(O_2,O_3)(f_2\circ f_1=f_2\circ f_1)$.
//

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2: Natural Language Description

The category, $Top$, such that $Obj(Top)=\{\text{ the topological spaces }\}$, $\mathrm{\forall }{O}_1,O_2\in Obj(Top)(Mor(O_1,O_2)=\{f:O_1\to O_2|f\in \text{ the continuous maps }\})$, and $\mathrm{\forall }{O}_1,O_2,O_3\in Obj(Top),\mathrm{\forall }{f}_1\in Mor(O_1,O_2),\mathrm{\forall }{f}_2\in Mor(O_2,O_3)(f_2\circ f_1=f_2\circ f_1)$

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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.

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References

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