Is There a Name for a Subcategory Acting Like an Ideal in Category Theory?

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SUMMARY

The discussion centers on identifying a specific term for a subcategory D within category theory that behaves like an ideal in algebra. In this context, D shares the same objects as category C (ob(D) = ob(C)), but its morphisms (hom(D)) are a subset of those in C, with the property that the product of two morphisms in C remains in D if at least one of the morphisms is in D. While the term "normal subcategory" is suggested, it lacks a definitive established name in category theory, with existing classifications including "full" and "regular" subcategories.

PREREQUISITES
  • Understanding of basic category theory concepts, including objects and morphisms.
  • Familiarity with the definitions of full and regular subcategories.
  • Knowledge of algebraic structures, particularly ideals.
  • Ability to interpret mathematical notation and properties of morphisms.
NEXT STEPS
  • Research the concept of "full subcategories" in category theory.
  • Explore the properties and examples of "regular subcategories".
  • Investigate the relationship between ideals in algebra and subcategories in category theory.
  • Examine the definitions and implications of morphism products in categorical contexts.
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Mathematicians, category theorists, and algebraists interested in the structural properties of subcategories and their parallels to algebraic ideals.

tmatrix
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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?
 
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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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