Can a subcategory be an ideal?

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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 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 know nothing more about category theory than the ability to phrase the above question.

Does anyone know what to call it?
 

Answers and Replies

  • #2
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You could call it a normal subcategory, but I do not think there is a special name for it. The usual properties for subcategories are "full" or "regular". I wouldn't bet that "normal" isn't occupied either. Have a look:
https://ncatlab.org/nlab/show/full+subcategory
 

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