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

In summary: 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 ...PeterUpdateComments are welcome.Update
  • #1
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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 \(\displaystyle M(1)\) ... ... "I am having trouble applying the UMP for the free monoid \(\displaystyle M(A)\) ... 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 \(\displaystyle A\) is \(\displaystyle 1\) ... but what is \(\displaystyle i, f, \overline{f}, N \) 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
 

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  • Awodey - 1 - UMP for Free Monoids ... PART 1 ... .png
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  • Awodey - 2 - UMP for Free Monoids ... PART 2 .png
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  • Awodey - 1 - Start of Section 2.1, Epis and Monos ... PART 1 .png
    Awodey - 1 - Start of Section 2.1, Epis and Monos ... PART 1 .png
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  • Awodey - 2 - Start of Section 2.1, Epis and Monos ... PART 2 .png
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  • #2
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)
 
Last edited:
  • #3
steenis said:
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
 
  • #4
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 ...
 

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  • MHB [#01.000].pdf
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  • #5
steenis said:
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
 
  • #6

Attachments

  • MHB [#01.004].pdf
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  • #7
steenis said:
Update
Comments are welcome.
Thanks Steenis ...

Working through your updated post now ...

Peter
 

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

1. What are monoids?

A monoid is a mathematical structure consisting of a set and an associative binary operation defined on that set. It also has an identity element that when combined with any element in the set using the operation, results in that same element.

2. What are monos in the context of category theory?

In category theory, monos (short for monomorphisms) are arrows between objects that preserve the structure of those objects. In other words, they are one-to-one maps that do not lose any information when mapping between objects.

3. What are injective maps?

An injective map is a function that maps distinct elements in the domain to distinct elements in the range. In other words, each element in the range is mapped to by at most one element in the domain.

4. What is Example 2.3 on page 30 in Awodey's book?

Example 2.3 in Awodey's book "Category Theory" introduces the concept of a monoid in the context of category theory. It demonstrates how the definition of a monoid can be abstracted to any category, not just the category of sets and functions.

5. How are monoids related to injective maps?

Monoids and injective maps are related in that monoids can be seen as a generalization of injective maps. In a monoid, the binary operation can be seen as a "mapping" between elements, similar to how a function maps elements in its domain to elements in its range. However, unlike injective maps, monoids also have an identity element and the operation is associative.

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