4.5 On the definition of equivalences

We have shown that all three definitions of equivalence satisfy the three desirable properties and are pairwise equivalent:

$$\mathrm{𝗂𝗌𝖢𝗈𝗇𝗍𝗋}(f)\simeq \mathrm{𝗂𝗌𝗁𝖺𝖾}(f)\simeq \mathrm{𝖻𝗂𝗂𝗇𝗏}(f).$$

(There are yet more possible definitions of equivalence, but we will stop with these three.See http://planetmath.org/node/87824Exercise 3.11 and the exercises in this chapter for some more.)Thus, we may choose any one of them as “the” definition of $\mathrm{𝗂𝗌𝖾𝗊𝗎𝗂𝗏}(f)$.For definiteness, we choose to define

$$\mathrm{𝗂𝗌𝖾𝗊𝗎𝗂𝗏}(f):\equiv \mathrm{𝗂𝗌𝗁𝖺𝖾}(f).$$

This choice is advantageous for formalization, since $\mathrm{𝗂𝗌𝗁𝖺𝖾}(f)$ contains the most directly useful data.On the other hand, for other purposes, $\mathrm{𝖻𝗂𝗂𝗇𝗏}(f)$ is often easier to deal with, since it contains no 2-dimensional paths and its two symmetrical halves can be treated independently.However, for purposes of this book, the specific choice will make little difference.

In the rest of this chapter, we study some other properties and characterizations of equivalences.