Subobject
of an object in a category
A concept analogous to the concept of a substructure of a mathematical structure. Let be any category and let
be a fixed object in
. In the class of all monomorphisms of
with codomain (target)
, one may define a pre-order relation (the relation of divisibility from the right):
precedes
, or
, if
for some
. In fact, the morphism
is uniquely determined because
is a monomorphism. The pre-order relation induces an equivalence relation between the monomorphisms with codomain
: The monomorphisms
and
are equivalent if and only if
and
. An equivalence class of monomorphisms is called a subobject of the object
. A subobject with representative
is sometimes denoted by
or by
. It is also possible to use Hilbert's
-symbol to select representatives of subobjects of
and consider these representatives as subobjects. In the categories of sets, groups, Abelian groups, and vector spaces, a subobject of any object is defined by the imbedding of a subset (subgroup, subspace) in the ambient set (group, space). However, in the category of topological spaces, the concept of a subobject is wider than that of a subset with the induced topology.
The pre-order relation between the monomorphisms with codomain induces a partial order relation between the subobjects of
:
if
. This relation is analogous to the inclusion relation for subsets of a given set.
If the monomorphism is regular (cf. Normal monomorphism), then any monomorphism equivalent to it is also regular. One can therefore speak of the regular subobjects of any object
. In particular, the subobject represented by
is regular. In categories with zero morphisms one similarly introduces normal subobjects. If
possesses a bicategory structure
, then a subobject
of an object
is called admissible (with respect to this bicategory structure) if
.
Comments
The notation used in this article (and elsewhere in this Encyclopaedia) is not standard. Most authors do not bother to distinguish notationally between a subobject and a monomorphism which represents it.
For references see Category.
Subobject. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Subobject&oldid=34635