# Forgetful Functor

• Sep 8th 2011, 01:25 PM
slevvio
Forgetful Functor
Hello, I am reading some notes on category theory, and I just had a question about the forgetful functor?

Consider the forgetful functor $\displaystyle F: \textbf{Gp} \rightarrow \textbf{Set}$. I was wondering if the structure $\displaystyle \tilde{\textbf{Gp}}$ where the collection of objects is $\displaystyle \{ F(G) | G \in \text{Ob}(\textbf{Gp})\}$ and the collection of morphisms between two objects $\displaystyle F(G)$ and $\displaystyle F(H)$ is $\displaystyle \{ F(\phi) | \phi \in \text{Hom}_{\textbf{Gp}}(G,H))\}$ forms a category?

The reason I ask is because I read that $\displaystyle \textbf{Gp}$ was a subcategory of $\displaystyle \textbf{Set}$ but the objects of groups arent really contained in the objects of Sets, but I can see it if this 'image' of $\displaystyle \text{Gp}$ is a category. Thanks very much for any advice.
• Sep 8th 2011, 03:49 PM
NonCommAlg
Re: Forgetful Functor
Quote:

Originally Posted by slevvio
Hello, I am reading some notes on category theory, and I just had a question about the forgetful functor?

Consider the forgetful functor $\displaystyle F: \textbf{Gp} \rightarrow \textbf{Set}$. I was wondering if the structure $\displaystyle \tilde{\textbf{Gp}}$ where the collection of objects is $\displaystyle \{ F(G) | G \in \text{Ob}(\textbf{Gp})\}$ and the collection of morphisms between two objects $\displaystyle F(G)$ and $\displaystyle F(H)$ is $\displaystyle \{ F(\phi) | \phi \in \text{Hom}_{\textbf{Gp}}(G,H))\}$ forms a category?

The reason I ask is because I read that $\displaystyle \textbf{Gp}$ was a subcategory of $\displaystyle \textbf{Set}$ but the objects of groups arent really contained in the objects of Sets, but I can see it if this 'image' of $\displaystyle \text{Gp}$ is a category. Thanks very much for any advice.

yes it is a category because $\displaystyle F$ is one-to-one on the class of objects.
• Sep 8th 2011, 03:51 PM
slevvio
Re: Forgetful Functor
what about two groups (G,*) and (G,#) which are non isomorphic, but F(G,*) = F(G,#) = G? Is all we require injectivity on the class of objects?
• Sep 8th 2011, 04:00 PM
NonCommAlg
Re: Forgetful Functor
Quote:

Originally Posted by slevvio
what about two groups (G,*) and (G,#) which are non isomorphic, but F(G,*) = F(G,#) = G? Is all we require injectivity on the class of objects?

injectivity on the objects is a sufficient condition for the image of a category under a functor to be a category.
• Sep 8th 2011, 04:04 PM
slevvio
Re: Forgetful Functor
Ah ok, thanks, but that is not useful here unfortunately because the functor is not injective on the class of objects :(
• Sep 8th 2011, 04:23 PM
NonCommAlg
Re: Forgetful Functor
the only problem that occurs, if $\displaystyle F$ is not injective, is that we might have morphisms $\displaystyle \phi:G_1 \longrightarrow G_2$ and $\displaystyle \psi : G_3 \longrightarrow G_4$ with $\displaystyle F(G_2)=F(G_3)$ but $\displaystyle G_2 \neq G_3$. then we won't necessarily have a morphism from $\displaystyle G_1$ to $\displaystyle G_4$ but we do have a morphism $\displaystyle F(\phi)F(\psi) : F(G_1) \longrightarrow F(G_4).$