definition of homotopy equivalence relation on collection of topological spaces
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of homotopy equivalence.
- The reader knows a definition of equivalence relation on collection.
Target Context
- The reader will have a definition of homotopy equivalence relation on collection of topological spaces.
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:
\( C\): \(= \{\text{ the topological spaces }\}\)
\(*\sim\): \(\in \{\text{ the equivalence relations on } C\}\), such that \(T_1 \sim T_2 \iff \exists f: T_1 \to T_2 \in \{\text{ the homotopy equivalences }\}\)
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Conditions:
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2: Note
As for the distinction between 'collection' and 'set', refer to Are You Talking About Sets or About Collections?.
Let us see that \(\sim\) is indeed an equivalence relation.
1) \(\forall T \in C (T \sim T)\): reflexivity: \(id: T \to T\) is a homotopy equivalence, because \([id] \circ [id] = [id \circ id] = [id]\).
2) \(\forall T_1, T_2 \in C (T_1 \sim T_2 \implies T_2 \sim T_1)\): symmetry: there are an \(f_1: T_1 \to T_2\) and an \(\widetilde{f_2}: T_2 \to T_1\) such that \([\widetilde{f_2}] \circ [f_1] = [id]\) and \([f_1] \circ [\widetilde{f_2}] = [id]\), which implies that \(T_2 \sim T_1\).
3) \(\forall T_1, T_2, T_3 \in C ((T_1 \sim T_2 \land T_2 \sim T_3) \implies T_1 \sim T_3)\): transitivity: there are an \(f_1: T_1 \to T_2\) and an \(\widetilde{f_2}: T_2 \to T_1\) such that \([\widetilde{f_2}] \circ [f_1] = [id]\) and \([f_1] \circ [\widetilde{f_2}] = [id]\) and there are an \(f_2: T_2 \to T_3\) and an \(\widetilde{f_3}: T_3 \to T_2\) such that \([\widetilde{f_3}] \circ [f_2] = [id]\) and \([f_2] \circ [\widetilde{f_3}] = [id]\), and \([\widetilde{f_2}] \circ [\widetilde{f_3}] \circ [f_2] \circ [f_1] = [\widetilde{f_2}] \circ ([\widetilde{f_3}] \circ [f_2]) \circ [f_1] = [\widetilde{f_2}] \circ [id] \circ [f_1] = [\widetilde{f_2}] \circ [f_1] = [id]\) and \([f_2] \circ [f_1] \circ [\widetilde{f_2}] \circ [\widetilde{f_3}] = [f_2] \circ ([f_1] \circ [\widetilde{f_2}]) \circ [\widetilde{f_3}] = [f_2] \circ [id] \circ [\widetilde{f_3}] = [f_2] \circ [\widetilde{f_3}] = [id]\), which implies that \(T_1 \sim T_3\).
When \(T_1 \sim T_2\), \(T_1\) and \(T_2\) are called "homotopy equivalent", \(T_1\) is called "homotopy equivalent to \(T_2\)", or \(T_1\) and \(T_2\) are called to "have the same homotopy type".
In fact, as \(\sim\) is an equivalence relation, there is the quotient collection, \(C / \sim\), and each element of \(C / \sim\) is called "homotopy type".
