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 $$X$$ the set of functions from $$X$$ to $$X$$ , written as

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

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

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

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

Peter

 

[Evgeny.Makarov]

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

I think the author suggests viewing $$\text{HOM}_\text{Sets} (X, X)$$ 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, $$\text{HOM}_\text{Sets} (X, X)$$ can be viewed as a category as well.

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

It can be viewed this way.

 

[Math Amateur]

I think the author suggests viewing $$\text{HOM}_\text{Sets} (X, X)$$ 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, $$\text{HOM}_\text{Sets} (X, X)$$ 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