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Math Help - Forgetful Functor

  1. #1
    Senior Member slevvio's Avatar
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    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 F: \textbf{Gp} \rightarrow \textbf{Set}. I was wondering if the structure \tilde{\textbf{Gp}} where the collection of objects is \{ F(G) | G \in \text{Ob}(\textbf{Gp})\} and the collection of morphisms between two objects F(G) and F(H) is \{ F(\phi) | \phi \in \text{Hom}_{\textbf{Gp}}(G,H))\} forms a category?

    The reason I ask is because I read that \textbf{Gp} was a subcategory of \textbf{Set} but the objects of groups arent really contained in the objects of Sets, but I can see it if this 'image' of \text{Gp} is a category. Thanks very much for any advice.
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  2. #2
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    Re: Forgetful Functor

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

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

    The reason I ask is because I read that \textbf{Gp} was a subcategory of \textbf{Set} but the objects of groups arent really contained in the objects of Sets, but I can see it if this 'image' of \text{Gp} is a category. Thanks very much for any advice.
    yes it is a category because F is one-to-one on the class of objects.
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  3. #3
    Senior Member slevvio's Avatar
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    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?
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    Re: Forgetful Functor

    Quote Originally Posted by slevvio View Post
    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.
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  5. #5
    Senior Member slevvio's Avatar
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    Re: Forgetful Functor

    Ah ok, thanks, but that is not useful here unfortunately because the functor is not injective on the class of objects
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    Re: Forgetful Functor

    the only problem that occurs, if F is not injective, is that we might have morphisms \phi:G_1 \longrightarrow G_2 and \psi : G_3 \longrightarrow G_4 with F(G_2)=F(G_3) but G_2 \neq G_3. then we won't necessarily have a morphism from G_1 to G_4 but we do have a morphism F(\phi)F(\psi) : F(G_1) \longrightarrow F(G_4).
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