372: Functionally Structured Topological Spaces Category Morphisms Are Morphisms

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A description/proof of that functionally structured topological spaces category morphisms are morphisms

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Topics

About: topological space

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Description
• 2: Proof
• 3: Note

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Starting Context

• The reader knows a definition of functional structured topological spaces category.

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Target Context

• The reader will have a description and a proof of the proposition that the functionally structured topologically spaces category morphisms are morphisms.

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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: Description

The definition of morphisms for the functionally structured topological spaces category, $C$, is legitimate. This is the definition: for any functionally structured topological spaces, $(T_1,F_{T_1})$ and $(T_2,F_{T_2})$, the morphisms, $C((T_1,F_{T_1}),(T_2,F_{T_2}))$, are all the continuous maps, $\{\varphi :T_1\to T_2\}$ such that for any $f\in F_{T_2}(U_2)$, $f\circ \varphi {|}_{{\varphi }^{-1}(U_2)}\in F_{T_1}({\varphi }^{-1}(U_2))$; the composition of morphisms is the composition of the continuous maps; any identity morphism is the identity map.

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2: Proof

Let us check the transitivity of morphisms. For any morphisms, ${\varphi }_{1,2}:(T_1,F_{T_1})\to (T_2,F_{T_2})$ and ${\varphi }_{2,3}:(T_2,F_{T_2})\to (T_3,F_{T_3})$, is ${\varphi }_{2,3}\circ {\varphi }_{1,2}$ a morphism, $(T_1,F_{T_1})\to (T_3,F_{T_3})$? For any $f\in F_{T_3}(U_3)$, $f\circ {\varphi }_{2,3}\circ {\varphi }_{1,2}{|}_{({\varphi }_{2,3}\circ {\varphi }_{1,2}{)}^{-1}(U_3)}\in F_{T_1}(({\varphi }_{2,3}\circ {\varphi }_{1,2}{)}^{-1}(U_3))$? $f\circ {\varphi }_{2,3}{|}_{{{\varphi }_{2,3}}^{-1}(U_3)}\in F_{T_2}({{\varphi }_{2,3}}^{-1}(U_3))$. $f\circ {\varphi }_{2,3}\circ {\varphi }_{1,2}{|}_{{{\varphi }_{1,2}}^{-1}({{\varphi }_{2,3}}^{-1}(U_3))}=(f\circ {\varphi }_{2,3})\circ {\varphi }_{1,2}{|}_{{{\varphi }_{1,2}}^{-1}({{\varphi }_{2,3}}^{-1}(U_3))}\in F_{T_1}({{\varphi }_{1,2}}^{-1}({{\varphi }_{2,3}}^{-1}(U_3)))=F_{T_1}(({\varphi }_{2,3}\circ {\varphi }_{1,2}{)}^{-1}(U_3))$.

Let us check that the identity map is a morphism. For any object, $(T_1,F_{T_1})$, and the identity map, ${\varphi }_{1,1}:T_1\to T_1$, for any $f\in F_{T_1}(U_1)$, $f\circ {\varphi }_{1,1}{|}_{{{\varphi }_{1,1}}^{-1}(U_1)}\in F_{T_1}({{\varphi }_{1,1}}^{-1}(U_1))$? As ${\varphi }_{1,1}$ is the identity map, $f\circ {\varphi }_{1,1}{|}_{{{\varphi }_{1,1}}^{-1}(U_1)}=f\circ {\varphi }_{1,1}{|}_{U_1}=f\in F_{T_1}(U_1)$.

The associativity is because maps compositions are associative.

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3: Note

Just calling some things "morphisms" does not make them morphisms; they have to satisfy the conditions to be morphisms.

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References

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