In category theory, a branch of mathematics, a subobject is, roughly speaking, an object that sits inside another object in the same category.
Since the detailed structure of objects is immaterial in category theory, the definition of subobject relies on a morphism that describes how one object sits inside another, rather than relying on the use of elements.
The dual concept to a subobject is a quotient object.
An appropriate categorical definition of "subobject" may vary with context, depending on the goal.
The relation ≤ induces a partial order on the collection of subobjects of
The collection of subobjects of an object may in fact be a proper class; this means that the discussion given is somewhat loose.
If the subobject-collection of every object is a set, the category is called well-powered or, rarely, locally small (this clashes with a different usage of the term locally small, namely that there is a set of morphisms between any two objects).
To get the dual concept of quotient object, replace "monomorphism" by "epimorphism" above and reverse arrows.
A quotient object of A is then an equivalence class of epimorphisms with domain A.
However, in some contexts these definitions are inadequate as they do not concord with well-established notions of subobject or quotient object.
In the category of topological spaces, monomorphisms are precisely the injective continuous functions; but not all injective continuous functions are subspace embeddings.
To get maps which truly behave like subobject embeddings or quotients, rather than as arbitrary injective functions or maps with dense image, one must restrict to monomorphisms and epimorphisms satisfying additional hypotheses.
Therefore, one might define a "subobject" to be an equivalence class of so-called "regular monomorphisms" (monomorphisms which can be expressed as an equalizer of two morphisms) and a "quotient object" to be any equivalence class of "regular epimorphisms" (morphisms which can be expressed as a coequalizer of two morphisms) This definition corresponds to the ordinary understanding of a subobject outside category theory.
In Grp, the category of groups, the subobjects of A correspond to the subgroups of A.
Given a partially ordered class P = (P, ≤), we can form a category with the elements of P as objects, and a single arrow from p to q iff p ≤ q.