901: Homotopy Equivalence

<The previous article in this series | The table of contents of this series | The next article in this series>

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:
$T_1$: $\in \{\text{ the topological spaces }\}$
$T_2$: $\in \{\text{ the topological spaces }\}$
$\ast f$: $:T_1\to T_2$, $\in \{\text{ the continuous maps }\}$
$[f]$: $=\text{ the equivalence class of }f\text{ by being homotopic }$
//

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, $f^{\prime }:T_2\to T_1$, such that $f^{\prime }\circ f\simeq i{d}_{T_1}$ and $f\circ f^{\prime }\simeq i{d}_{T_2}$, which indeed equals the definition by this article, because that equals $[f^{\prime }]\circ [f]=[i{d}_{T_1}]$ and $[f]\circ [f^{\prime }]=[i{d}_{T_2}]$, 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.

---

References

<The previous article in this series | The table of contents of this series | The next article in this series>