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'' = ''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 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.

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Examples

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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.

See also

★ Subobject classifier