definition of hTop 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 homotopic maps.
- The reader admits the proposition that on the set of the continuous maps between any topological spaces, being homotopic is an equivalence relation.
- The reader admits the proposition that for any homotopic maps from any 1st topological space into any 2nd topological space and any homotopic maps from the 2nd topological space into any 3rd topological space, the compositions of the homotopic maps are homotopic.
Target Context
- The reader will have a definition of hTop 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:
\(*hTop\): \(\in \{\text{ the categories }\}\)
//
Conditions:
\(Obj (Top) = \{\text{ the topological spaces }\}\).
\(\land\)
\(\forall O_1, O_2 \in Obj (Top) (Hom (O_1, O_2) = \{[f: O_1 \to O_2] \vert [f] \in \{\text{ the equivalence classes of homotopic maps }\}\})\).
\(\land\)
\(\forall O_1, O_2, O_3 \in Obj (Top), \forall [f_1] \in Hom (O_1, O_2), \forall [f_2] \in Hom (O_2, O_3) ([f_2] \circ [f_1] = [f_2 \circ f_1])\).
//
2: Natural Language Description
The category, \(hTop\), such that \(Obj (Top) = \{\text{ the topological spaces }\}\), \(\forall O_1, O_2 \in Obj (Top) (Hom (O_1, O_2) = \{[f: O_1 \to O_2] \vert [f] \in \{\text{ the equivalence classes of homotopic maps }\}\})\), and \(\forall O_1, O_2, O_3 \in Obj (Top), \forall [f_1] \in Hom (O_1, O_2), \forall [f_2] \in Hom (O_2, O_3) ([f_2] \circ [f_1] = [f_2 \circ f_1])\)
3: Note
The definition is possible, by the proposition that on the set of the continuous maps between any topological spaces, being homotopic is an equivalence relation and the proposition that for any homotopic maps from any 1st topological space into any 2nd topological space and any homotopic maps from the 2nd topological space into any 3rd topological space, the compositions of the homotopic maps are homotopic.
Maps' being homotopic is a congruence on the \(Top\) category and \(hTop\) is the quotient category of \(Top\) by the congruence.
