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Math Help - Can a subcategory be called an ideal?

  1. #1
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    Can a subcategory be called an ideal?

    Consider a category C with objects ob(C) and morphisms hom(C). Suppose there is a subcategory D such that ob(D)=ob(C) but hom(D) is a subset of hom(C), with the property that the product of two compatible morphisms in hom(C), f*g, is an element of hom(D) if either f or g is in hom(D).

    This subcategory is basically acting like an "ideal" in algebra, but I'm not sure what this thing is called in the context of categories. I'm a physicist and know nothing more about category theory than the ability to phrase the above question.

    Does anyone know what to call it?
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  2. #2
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    Re: Can a subcategory be called an ideal?

    Quote Originally Posted by tmatrix View Post
    Consider a category C with objects ob(C) and morphisms hom(C). Suppose there is a subcategory D such that ob(D)=ob(C) but hom(D) is a subset of hom(C), with the property that the product of two compatible morphisms in hom(C), f*g, is an element of hom(D) if either f or g is in hom(D).

    This subcategory is basically acting like an "ideal" in algebra, but I'm not sure what this thing is called in the context of categories. I'm a physicist and know nothing more about category theory than the ability to phrase the above question.

    Does anyone know what to call it?
    is that even possible? choose a,b \in \text{ob}(C)=\text{ob}(D) and f \in \text{hom}_C(a,b) \setminus \text{hom}_D(a,b). then f = f \circ \text{id}_a and we know \text{id}_a \in \text{hom}(D). so, according to your condition, we must have f \in \text{hom}(D), which is nonsense!
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    Re: Can a subcategory be called an ideal?

    Quote Originally Posted by NonCommAlg View Post
    is that even possible? choose a,b \in \text{ob}(C)=\text{ob}(D) and f \in \text{hom}_C(a,b) \setminus \text{hom}_D(a,b). then f = f \circ \text{id}_a and we know \text{id}_a \in \text{hom}(D). so, according to your condition, we must have f \in \text{hom}(D), which is nonsense!
    Thank you, I had neglected to notice that my "subcategory" D does not have an identity. Therefore I guess it does not count as a category. With this qualification my question still stands, if there is an answer.
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    MHF Contributor Drexel28's Avatar
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    Re: Can a subcategory be called an ideal?

    This was already asked on mathoverflow already.
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    Re: Can a subcategory be called an ideal?

    Thank you, the discussion seems relevant to the notion I am considering. In the paper I am writing I guess it would be understood if I called the subcategory D an ideal, unless there are objections?
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    MHF Contributor Drexel28's Avatar
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    Re: Can a subcategory be called an ideal?

    Quote Originally Posted by tmatrix View Post
    Thank you, the discussion seems relevant to the notion I am considering. In the paper I am writing I guess it would be understood if I called the subcategory D an ideal, unless there are objections?
    Perhaps, but I think, especially considering the uncertainty of the definition, it would be better to just define it.
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