A description/proof of that functionally structured topological spaces category morphisms are morphisms
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of functional structured topological spaces category.
Target Context
- The reader will have a description and a proof of the proposition that the functionally structured topologically spaces category morphisms are morphisms.
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.
Main Body
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, \(\{\phi: T_1 \to T_2\}\) such that for any \(f \in F_{T_2} (U_2)\), \(f \circ \phi \vert_{\phi^{-1} (U_2)} \in F_{T_1} (\phi^{-1} (U_2))\); the composition of morphisms is the composition of the continuous maps; any identity morphism is the identity map.
2: Proof
Let us check the transitivity of morphisms. For any morphisms, \(\phi_{1, 2}: (T_1, F_{T_1}) \to (T_2, F_{T_2})\) and \(\phi_{2, 3}: (T_2, F_{T_2}) \to (T_3, F_{T_3})\), is \(\phi_{2, 3} \circ \phi_{1, 2}\) a morphism, \((T_1, F_{T_1}) \to (T_3, F_{T_3})\)? For any \(f \in F_{T_3} (U_3)\), \(f \circ \phi_{2, 3} \circ \phi_{1, 2} \vert_{(\phi_{2, 3} \circ \phi_{1, 2})^{-1} (U_3)} \in F_{T_1} ((\phi_{2, 3} \circ \phi_{1, 2})^{-1} (U_3))\)? \(f \circ \phi_{2, 3} \vert_{{\phi_{2, 3}}^{-1} (U_3)} \in F_{T_2} ({\phi_{2, 3}}^{-1} (U_3))\). \(f \circ \phi_{2, 3} \circ \phi_{1, 2} \vert_{{\phi_{1, 2}}^{-1} ({\phi_{2, 3}}^{-1} (U_3))} = (f \circ \phi_{2, 3}) \circ \phi_{1, 2} \vert_{{\phi_{1, 2}}^{-1} ({\phi_{2, 3}}^{-1} (U_3))} \in F_{T_1} ({\phi_{1, 2}}^{-1} ({\phi_{2, 3}}^{-1} (U_3))) = F_{T_1} ((\phi_{2, 3} \circ \phi_{1, 2})^{-1} (U_3))\).
Let us check that the identity map is a morphism. For any object, \((T_1, F_{T_1})\), and the identity map, \(\phi_{1, 1}: T_1 \to T_1\), for any \(f \in F_{T_1} (U_1)\), \(f \circ \phi_{1, 1} \vert_{{\phi_{1, 1}}^{-1} (U_1)} \in F_{T_1} ({\phi_{1, 1}}^{-1} (U_1))\)? As \(\phi_{1, 1}\) is the identity map, \(f \circ \phi_{1, 1} \vert_{{\phi_{1, 1}}^{-1} (U_1)} = f \circ \phi_{1, 1} \vert_{U_1} = f \in F_{T_1} (U_1)\).
The associativity is because maps compositions are associative.
3: Note
Just calling some things "morphisms" does not make them morphisms; they have to satisfy the conditions to be morphisms.
