Monoids as Categories .... Awodey Section 1.4, Example 13 ....

[Math Amateur]

I am reading Steve Awodey's book: Category Theory (Second Edition) and am focused on Section 1.4 Examples of Categories ...

I need some help in order to fully understand some aspects of Section 1.4 Example 13 ...

Section 1.4 Example 13 reads as follows:https://www.physicsforums.com/attachments/8343
View attachment 8344In the above text by Awodey we read the following:

" ... ... But also for any set \(\displaystyle X\) the set of functions from \(\displaystyle X\) to \(\displaystyle X\) , written as

\(\displaystyle \text{HOM}_\text{Sets} (X, X)\)

is a monoid under the operation of composition. More generally, for any object \(\displaystyle C\) in any Category \(\displaystyle C\), the set of arrows from \(\displaystyle C\) to \(\displaystyle C\), written as

\(\displaystyle \text{HOM}_C (C, C)\)

is a monoid under the composition operation of \(\displaystyle C\). ... ... "
I am slightly unsure regarding how to interpret the objects and arrows of \(\displaystyle \text{HOM}_\text{Sets} (X, X)\) and \(\displaystyle \text{HOM}_C (C, C)\) when viewed as categories ... ... ?My interpretation of \(\displaystyle \text{HOM}_\text{Sets} (X, X)\) is that the single object is \(\displaystyle X\) and the arrows are the functions for \(\displaystyle X\) to \(\displaystyle X\) ... ... is that correct?My interpretation of \(\displaystyle \text{HOM}_C (C, C)\) is that that the single object is \(\displaystyle C\) and the arrows are the arrows from \(\displaystyle C\) to \(\displaystyle C\) ... ... is that correct?Help will be appreciated ...

Peter

 

[Evgeny.Makarov]

I am slightly unsure regarding how to interpret the objects and arrows of \(\displaystyle \text{HOM}_\text{Sets} (X, X)\) and \(\displaystyle \text{HOM}_C (C, C)\) when viewed as categories ... ... ?

I think the author suggests viewing \(\displaystyle \text{HOM}_\text{Sets} (X, X)\) as a monoid in the usual sense of an algebraic structure rather than as a category. The fact that it is an algebraic monoid is obvious since function from $X$ to $X$ are elements and composition is the operation. But yes, since every monoid is a category, \(\displaystyle \text{HOM}_\text{Sets} (X, X)\) can be viewed as a category as well.

My interpretation of \(\displaystyle \text{HOM}_\text{Sets} (X, X)\) is that the single object is \(\displaystyle X\) and the arrows are the functions for \(\displaystyle X\) to \(\displaystyle X\) ... ... is that correct?

We can make anything to be the single object, for example, the symbol $*$: it does not matter. The arrows are indeed functions from $X$ to $X$, viewed as arrows from and to that single object.

My interpretation of \(\displaystyle \text{HOM}_C (C, C)\) is that that the single object is \(\displaystyle C\) and the arrows are the arrows from \(\displaystyle C\) to \(\displaystyle C\) ... ... is that correct?

It can be viewed this way.

 

[Math Amateur]

I think the author suggests viewing \(\displaystyle \text{HOM}_\text{Sets} (X, X)\) as a monoid in the usual sense of an algebraic structure rather than as a category. The fact that it is an algebraic monoid is obvious since function from $X$ to $X$ are elements and composition is the operation. But yes, since every monoid is a category, \(\displaystyle \text{HOM}_\text{Sets} (X, X)\) can be viewed as a category as well.

We can make anything to be the single object, for example, the symbol $*$: it does not matter. The arrows are indeed functions from $X$ to $X$, viewed as arrows from and to that single object.

It can be viewed this way.

Thanks Evgeny ... Appreciate your help...

Peter

 

[steenis]

Evgeny.Makarov is right. $Hom(X,X)$ is not viewed as a category, here.

See Example 1.5.1 of Simmons for a better explanation of how to view a monoid as a category.

 

[Math Amateur]

Evgeny.Makarov is right. $Hom(X,X)$ is not viewed as a category, here.

See Example 1.5.1 of Simmons for a better explanation of how to view a monoid as a category.

Thanks Steenis ... will check Simmons ...

Peter