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.