Definition
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 u = v ∘ w. 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. 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.
Read more about this topic: Subobject
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