Math Amateur
Gold Member
MHB
- 3,920
- 48
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
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
Last edited: