# Math Help - Can a subcategory be called an ideal?

1. ## 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?

2. ## Re: Can a subcategory be called an ideal?

Originally Posted by tmatrix
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!

3. ## Re: Can a subcategory be called an ideal?

Originally Posted by NonCommAlg
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.

5. ## 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?

6. ## Re: Can a subcategory be called an ideal?

Originally Posted by tmatrix
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.