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# Exercise 18 - Chapter 3

edited June 2018

Write down all the morphisms in the category D.

• Options
1.
edited May 2018

Apparently there is a difference between paths and morphisms. I believe the morphisms form an equivalence group over the paths.

For example, to which morphisms do the following paths belong?

$$id_z$$ $$s$$ $$s.s$$ $$s.id_z.s.s$$

Comment Source:Apparently there is a difference between paths and morphisms. I believe the morphisms form an equivalence group over the paths. For example, to which morphisms do the following paths belong? $$id_z$$ $$s$$ $$s.s$$ $$s.id_z.s.s$$
• Options
2.
edited July 2018

Since the set of paths is isomorphic to $$\mathbb{N}$$, let's see what the equation means in $$\mathbb{N}$$. $$s^4=s^2$$ corresponds to $$n\equiv n-2$$ for $$n\geq4$$. This means that every even number except 0 is equivalent to 2, and every odd number except 1 is equivalent to 3. So, there are 4 morphisms in $$\mathcal{D}$$: $$\mathrm{id}_z,s,s^2,$$ and $$s^3$$.

Fredrick Eisele wrote in #1:

I believe the morphisms form an equivalence group over the paths.

They certainly form equivalence classes. They don't quite form a group, though the paths of length at least 2 have a $$\mathbb{Z}/2\mathbb{Z}$$-like structure (with composition corresponding to addition modulo 2, and the even numbers corresponding to the identity).

Comment Source:Since the set of paths is isomorphic to \$$\mathbb{N}\$$, let's see what the equation means in \$$\mathbb{N}\$$. \$$s^4=s^2\$$ corresponds to \$$n\equiv n-2\$$ for \$$n\geq4\$$. This means that every even number except 0 is equivalent to 2, and every odd number except 1 is equivalent to 3. So, there are 4 morphisms in \$$\mathcal{D}\$$: \$$\mathrm{id}_z,s,s^2,\$$ and \$$s^3\$$. Fredrick Eisele wrote in [#1](https://forum.azimuthproject.org/discussion/comment/18569/#Comment_18569): >I believe the morphisms form an equivalence group over the paths. They certainly form equivalence classes. They don't quite form a group, though the paths of length at least 2 have a \$$\mathbb{Z}/2\mathbb{Z}\$$-like structure (with composition corresponding to addition modulo 2, and the even numbers corresponding to the identity).