Subobject

In category theory, a branch of mathematics, a subobject is, roughly speaking, an object that sits inside another object in the same category. The notion is a generalization of concepts such as subsets from set theory, subgroups from group theory, and subspaces from topology. 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 . This generalizes concepts such as quotient sets, quotient groups, quotient spaces, quotient graphs, etc.
Definitions.
In detail, let "formula_1" be an object of some category. Given two monomorphisms
with codomain "formula_1", we write formula_4 if formula_5 factors through "formula_6"—that is, if there exists formula_7 such that formula_8. The binary relation formula_9 defined by
is an equivalence relation on the monomorphisms with codomain "formula_1", and the corresponding equivalence classes of these monomorphisms are the subobjects of "formula_1". (Equivalently, one can define the equivalence relation by formula_13 if and only if there exists an isomorphism formula_7 with formula_8.)
The relation ≤ induces a partial order on the collection of subobjects of formula_1.
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."