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 monomorphisms

:"u": "S → A" and:"v": "T → A".

We say "u" "factors through" "v" and write

:"u" ≤ "v"

when "u" = "v∘u′" 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 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, we call the category "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.

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 monomorphisms.

ee also

*Subobject classifier