Monoids .... Monos and Injective Maps .... Awodey, Example 2.3, Page 30 ....

[Math Amateur]

I am reading Steve Awodey's book: Category Theory (Second Edition) and am focused on Chapter 2: Abstract Structures ... ...

I need some help in order to fully understand Awodey Example 2.3, Chapter 2 ... ... Awodey Example 2.3, Chapter 2 reads as follows:View attachment 8398
In the Example above Awodey writes the following:

" ... By the UMP of the free monoid $$M(1)$$ ... ... "I am having trouble applying the UMP for the free monoid $$M(A)$$ ... see text below from Awodey pages 18-19 ...

Can someone please help ... in particular I am having trouble identifying the various functions, sets and structures in the UMP on pages 18-19 with the Example ... I can see the set $$A$$ is $$1$$ ... but what is $$i, f, \overline{f}, N $$ in terms of Example 2.3 ... specifically how is Awodey applying the UMP to the Example ... can someone clearly explain the situation ...
Hope that someone can help ...

Peter
=========================================================================================The above post mentions the UMP for free monoids ... so I am providing the text of the same ... ... as follows:
View attachment 8399
View attachment 8400It may also help readers of the above post to have access to the start of Awodey's section on Epis and Monos (up to Example 2.3) ... so I am providing the same ... as follows ...
View attachment 8401
View attachment 8402Hope that helps ...

Peter

 

[steenis]

I gave it some time but I don't see it.

I think the trick is to see that a monoid $M$ equals the free monoid $M(1) = M( \star )$. In that case you can apply the universal mapping property of $M(1)$. For now, I give up, maybe someone else can give it a try.

You can go on reading Awodey, you do not have to understand everything. Just skip everything that references this example.Try to do example 2.4 (easy) and the remark between example 2.4 and example 2.5 (a little bit harder)

 

[Math Amateur]

I gave it some time but I don't see it.

I think the trick is to see that a monoid $M$ equals the free monoid $M(1) = M( \star )$. In that case you can apply the universal mapping property of $M(1)$. For now, I give up, maybe someone else can give it a try.

You can go on reading Awodey, you do not have to understand everything. Just skip everything that references this example.Try to do example 2.4 (easy) and the remark between example 2.4 and example 2.5 (a little bit harder)

Can someone please please help Steenis and me to find an answer to Post #1 above ...Help will be much appreciated ...Peter

 

[steenis]

I found this solution, but it is not quite ready. I need help to resolve the "why's" and to find errors.

View attachment 8421

Awodey uses a nuclear bomb to kill a fly. This proof can much easier. Well, it is an exercise ...

 

[Math Amateur]

I found this solution, but it is not quite ready. I need help to resolve the "why's" and to find errors.
Awodey uses a nuclear bomb to kill a fly. This proof can much easier. Well, it is an exercise ...

Thanks so so much for your help Steenis ...

I have left the state of Tasmania for a trip to the mainland ... will be staying with a friend outside Melbourne in Victoria... will be in touch shortly ..

... will work through your solution as soon as possible ...

It is great that you have an answer/clarification to Awodey's example ...

Peter

 

[steenis]

Update

View attachment 8422

Comments are welcome.

 

[Math Amateur]

Update
Comments are welcome.

Thanks Steenis ...

Working through your updated post now ...

Peter