Category Theory .... Groups and Isomorphisms .... Awodey, Section 1.5 .... ....

[Math Amateur]

I am reading Steve Awodey's book: Category Theory (Second Edition) and am focused on Section 1.5 Isomorphisms ...

I need some further help in order to fully understand some aspects of Definition 1.4, Page 12 ... ...

The start of Section 1.5, including Definition 1.4 ... reads as follows:https://www.physicsforums.com/attachments/8355In the above text from Awodey, in Definition 1.4, we read the following:

" ... ... Thus \(\displaystyle G\) is a category with one object, in which every arrow is an isomorphism. ... ... "Can someone please demonstrate a proof that in the category \(\displaystyle G\) every arrow is an isomorphism ... ?Hope someone can help ...

Peter

 

[steenis]

Recall the definition of a group.
In a group $G$, what is the definition of the inverse $g^{-1}$ of an element $g \in G$ ?

Just like we viewed monoids as categories. we now can view the group $G$ as a category. Can you describe how? You can follow the definition of monoids viewed as categories in section 1.5.1 of Simmons.

But you have to make an additional rule for the inverse $g^{-1}$ of an element $g \in G$

If you have correctly defined inverses in a group viewed as a category, you can compare this definition with the definition of an isomorphism in a general category $C$. What do you observe ?

 

[Math Amateur]

Recall the definition of a group.
In a group $G$, what is the definition of the inverse $g^{-1}$ of an element $g \in G$ ?

Just like we viewed monoids as categories. we now can view the group $G$ as a category. Can you describe how? You can follow the definition of monoids viewed as categories in section 1.5.1 of Simmons.

But you have to make an additional rule for the inverse $g^{-1}$ of an element $g \in G$

If you have correctly defined inverses in a group viewed as a category, you can compare this definition with the definition of an isomorphism in a general category $C$. What do you observe ?

Hi Steenis ... thanks for the help ...

Well ...

... a group \(\displaystyle G\) is a monoid in which every element \(\displaystyle x \in G\) has an inverse: that is, for each \(\displaystyle x \in G\) there exists an element \(\displaystyle x^{-1}\), called the inverse of \(\displaystyle x\) which is such that \(\displaystyle x \bullet x^{-1} = x^{-1} \bullet x = 1_G\) ... ...

Now ... a group \(\displaystyle G\) viewed as a category is a monoid: that is a category with one element \(\displaystyle \star\) ... but as a group viewed as a category we have the extra condition that there exist inverses for every element (arrow).

So ... with a group \(\displaystyle G\) as with a monoid, the elements are arrows ... but the existence of inverses in the case of a group means that for each arrow \(\displaystyle x \in G\) there exists an arrow \(\displaystyle x^{-1}\) such that \(\displaystyle x \circ x^{-1} = x^{-1} \circ x = 1_G\) where \(\displaystyle \circ\) is the composition of arrows ... BUT ... this is the condition for the arrow \(\displaystyle x : \star \to \star\) to be an isomorphism ... ... so then because this is true for every arrow in \(\displaystyle G\), we have that every arrow is an isomorphism ...Is the above correct?

Peter

 

[steenis]

Ok, Peter, correct