definition of Top category
Topics
About: category
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Natural Language Description
- 3: Note
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.
Target Context
- The reader will have a definition of Top category.
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: Structured Description
Here is the rules of Structured Description.
Entities:
\(*Top\): \(\in \{\text{ the categories }\}\)
//
Conditions:
\(Obj (Top) = \{\text{ the topological spaces }\}\).
\(\land\)
\(\forall O_1, O_2 \in Obj (Top) (Mor (O_1, O_2) = \{f: O_1 \to O_2 \vert f \in \text{ the continuous maps }\})\).
\(\land\)
\(\forall O_1, O_2, O_3 \in Obj (Top), \forall f_1 \in Mor (O_1, O_2), \forall f_2 \in Mor (O_2, O_3) (f_2 \circ f_1 = f_2 \circ f_1)\).
//
2: Natural Language Description
The category, \(Top\), such that \(Obj (Top) = \{\text{ the topological spaces }\}\), \(\forall O_1, O_2 \in Obj (Top) (Mor (O_1, O_2) = \{f: O_1 \to O_2 \vert f \in \text{ the continuous maps }\})\), and \(\forall O_1, O_2, O_3 \in Obj (Top), \forall f_1 \in Mor (O_1, O_2), \forall f_2 \in Mor (O_2, O_3) (f_2 \circ f_1 = f_2 \circ f_1)\)
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.
