Subobject - Citizendia

In category theory, there is a general definition of subobject extending the idea of subset and subgroup. In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets In Group theory, given a group G under a Binary operation * we say that some Subset H of G is a subgroup of

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

u: S → A and

v: T → A. In the context of Abstract algebra or Universal algebra, a monomorphism is simply an Injective Homomorphism.

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. In Mathematics, especially in Order theory, preorders are Binary relations that satisfy certain conditions In Mathematics, especially Order theory, a partially ordered set (or poset) formalizes the intuitive concept of an ordering sequencing or arrangement (The collection of subobjects of an object may in fact be a proper class; this means that the discussion given is somewhat loose. In Set theory and its applications throughout Mathematics, a class is a collection of sets (or sometimes other mathematical objects that can be unambiguously 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. In Category theory, an abstract branch of Mathematics, the dual category or opposite category C op of a category C is the

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.

Examples

See also