Subobject

In category theory, there is a general definition of subobject extending the idea of subset and subgroup.

In detail, suppose we are given some category C and monics

u: S → A and

v: T → A.

We say u factors through v and write

u ≤ v

when u = vu′ for some morphism u′  : S → T. We also write

u ≡ v

to denote that both

u ≤ v and v ≤ u.

This defines an equivalence relation ≡ on the collection of monics with codomain A, and the corresponding equivalence classes of these monics are the subobjects of A. The collection of monics with codomain A under the relation ≤ forms a preorder, but the definition of a subobject ensures that the collection of subobjects of A is a partial order. (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, we call the category well-powered.)

The dual concept to a subobject is a quotient object; that is, to define quotient object replace monic by epic above and reverse arrows.

Examples

In the category Sets, a subobject of A corresponds to a subset B of A, or rather the collection of all maps from sets equipotent to B with image exactly B. The subobject partial order of a set in Sets is just its subset lattice. Similar results hold in Groups, and some other categories.

Given a partially ordered class P, we can form a category with P's elements as objects and a single arrow going from one object (element) to another if the first is less than or equal to the second. If P has a greatest element, the subobject partial order of this greatest element will be P itself. This is in part because all arrows in such a category will be monic.