In category theory, a branch of mathematics, a subobject is, roughly speaking, an object which sits inside another object in the same category. The notion is a generalization of the older concepts of subset from set theory and subgroup from group theory. Since the actual structure of objects is immaterial in category theory, the definition of subobject relies on a morphism which describes how one object sits inside another, rather than relying on the use of elements.
In detail, let A be an object of some category. Given two monomorphisms
- u: S → A and
- v: T → A
with codomain A, say that u ≤ v if u factors through v — that is, if there exists w: S → T such that . The binary relation ≡ defined by
- u ≡ v if and only if u ≤ v and v ≤ u
is an equivalence relation on the monomorphisms with codomain A, and the corresponding equivalence classes of these monomorphisms are the subobjects of A. If two monomorphisms represent the same subobject of A, then their domains are isomorphic. The collection of monomorphisms 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, the category is well-powered.)
The dual concept to a subobject is a quotient object; that is, to define quotient object replace monomorphism by epimorphism above and reverse arrows.
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 monomorphisms.
- Mac Lane, p. 126