# Can a subcategory be an ideal?

• tmatrix
In summary, the conversation discusses the existence of a subcategory D within a category C, where the objects in D are the same as those in C but the morphisms are a subset of those in C. This subcategory behaves similarly to an ideal in algebra and could potentially be called a "normal subcategory" or "full subcategory." However, there is no specific term for it in the context of categories.
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 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?

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

## 1. Can a subcategory be an ideal?

Yes, it is possible for a subcategory to be an ideal. In category theory, an ideal is a special type of subobject that behaves like a kernel in algebra. A subcategory can be considered as a subobject in the category of categories, and therefore can also have properties of an ideal.

## 2. What is the difference between a subcategory and an ideal?

A subcategory is a subset of objects and morphisms in a larger category, while an ideal is a special type of subobject with specific properties. The main difference is that an ideal has additional algebraic properties, such as being closed under composition and containing the identity morphism.

## 3. Are all subcategories also ideals?

No, not all subcategories are also ideals. A subcategory can only be considered an ideal if it satisfies the necessary algebraic properties, such as being closed under composition and containing the identity morphism. If these conditions are not met, then the subcategory cannot be considered an ideal.

## 4. Can a subcategory be an ideal in any category?

No, a subcategory cannot be considered an ideal in any category. As mentioned before, an ideal must satisfy certain algebraic properties, which may not always hold in every category. Additionally, the concept of ideals is specific to category theory and may not be applicable in other fields of science.

## 5. How are subcategories and ideals related in category theory?

In category theory, an ideal is a special type of subobject that behaves like a kernel in algebra. This means that a subcategory can be considered as an ideal if it satisfies the necessary algebraic properties. Additionally, ideals can be used to define subcategories, as a subcategory can be seen as a subobject in the category of categories.

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