Sajun.org
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
monics
:''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
monics with codomain ''A'', and the corresponding equivalence classes of these monics are the '''subobjects''' of ''A''. The collection of monics 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 ''monic'' by ''
epic'' 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 monic.
