MHB 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 $$G$$ is a category with one object, in which every arrow is an isomorphism. ... ... "Can someone please demonstrate a proof that in the category $$G$$ 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 $$G$$ is a monoid in which every element $$x \in G$$ has an inverse: that is, for each $$x \in G$$ there exists an element $$x^{-1}$$, called the inverse of $$x$$ which is such that $$x \bullet x^{-1} = x^{-1} \bullet x = 1_G$$ ... ...

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

Peter

 

[steenis]

Ok, Peter, correct