Subquotient

A subquotient of an object A is any object that can be obtained as a quotient of a subobject of A. More concretely, if N is a subobject of A and Q is a quotient object of N, then Q is a subquotient of A. Equivalently, in many algebraic settings a subquotient of A is isomorphic to a quotient H/K where K is a subobject of a subobject H of A (K ≤ H ≤ A in the appropriate sense).

In familiar categories, subquotients take concrete forms. For groups, a subquotient has the form H/K where K ≤ H ≤ G, with K normal in H if one wants a group quotient. For modules over a ring, a subquotient is (N/L) with L ≤ N ≤ M, where N is a submodule of M and L is a submodule of N. Subquotients include all subobjects of A (via the choice L = 0) and all quotients of A (via choosing N = A and L appropriate), up to isomorphism.

Subquotients are often used to study the structure of a given object by looking at its “pieces.” In a composition series of a module or group, the successive quotients are subquotients of the original object and are called composition factors. The notion extends to category theory: a subquotient of an object X is any object isomorphic to a quotient of a subobject of X, providing a flexible way to discuss factor-like structures without fixing a particular subobject.

Overall, the concept emphasizes the idea that both subobjects and their quotients contribute to the building blocks of an algebraic object, with subquotients capturing all pieces obtainable through this two-step process.