A common way to describe the structure in an abelian category is to describe some class of all the "small" objects and then prove that every object in the category can be "built" from the small objects. For example
- A category is Krull-Remak-Schmidt if every object can be written uniquely as a direct sum of indecomposable objects. The category of finite abelian groups is an example.
- The category of finite groups, or more generally the category of finitely generated modules over an Artinian ring, has the property that every object can be written uniquely by iteratively taking extensions of simple objects from its composition series. I don't know if such a category has an established name.
- A category is semisimple it is Krull-Remak-Schmidt and every indecomposable object is simple. Fusion categories are semisimple.
What are other common ways that mathematicians describe all the objects in a category as being "built from smaller objects"? And what are the examples of categories in which such a characterization of the objects this way naturally comes up?