definition of homotopy equivalence
Topics
About: category
About: topological space
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 hTop category.
- The reader knows a definition of %category name% isomorphism.
Target Context
- The reader will have a definition of homotopy equivalence.
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:
\( T_1\): \(\in \{\text{ the topological spaces }\}\)
\( T_2\): \(\in \{\text{ the topological spaces }\}\)
\(*f\): \(: T_1 \to T_2\), \(\in \{\text{ the continuous maps }\}\)
\( [f]\): \(= \text{ the equivalence class of } f \text{ by being homotopic }\)
//
Conditions:
\([f] \in \{\text{ the hTop isomorphisms }\}\)
//
2: Natural Language Description
For any topological spaces, \(T_1, T_2\), any continuous map, \(f: T_1 \to T_2\), such that the equivalence class of \(f\) by being homotopic, \([f]\), is an hTop isomorphism.
3: Note
In other words, there is a continuous map, \(f': T_2 \to T_1\), such that \(f' \circ f \simeq id_{T_1}\) and \(f \circ f' \simeq id_{T_2}\), which indeed equals the definition by this article, because that equals \([f'] \circ [f] = [id_{T_1}]\) and \([f] \circ [f'] = [id_{T_2}]\), which is nothing but that \([f]\) is a \(hTop\) isomorphism.
The definition by this article seems better in emphasizing the significance of the concept: it is significant because it is an isomorphism in a category.
