Subobject

of an object in a category

A concept analogous to the concept of a substructure of a mathematical structure. Let $ \mathfrak K $ be any category and let $ A $ be a fixed object in $ \mathfrak K $. In the class of all monomorphisms of $ \mathfrak K $ with codomain (target) $ A $, one may define a pre-order relation (the relation of divisibility from the right): $ \mu : X \rightarrow A $ precedes $ \sigma : Y \rightarrow A $, or $ \mu \prec \sigma $, if $ \mu = \mu ^ \prime \sigma $ for some $ \mu ^ \prime : X \rightarrow Y $. In fact, the morphism $ \mu ^ \prime $ is uniquely determined because $ \sigma $ is a monomorphism. The pre-order relation induces an equivalence relation between the monomorphisms with codomain $ A $: The monomorphisms $ \mu : X \rightarrow A $ and $ \sigma : Y \rightarrow A $ are equivalent if and only if $ \mu \prec \sigma $ and $ \sigma \prec \mu $. An equivalence class of monomorphisms is called a subobject of the object $ A $. A subobject with representative $ \mu : X \rightarrow A $ is sometimes denoted by $ ( \mu : X \rightarrow A ] $ or by $ ( \mu ] $. It is also possible to use Hilbert's $ \tau $- symbol to select representatives of subobjects of $ A $ 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 $ A $ induces a partial order relation between the subobjects of $ A $: $ ( \mu ] \leq ( \sigma ] $ if $ \mu \prec \sigma $. This relation is analogous to the inclusion relation for subsets of a given set.

If the monomorphism $ \mu $ 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 $ A $. In particular, the subobject represented by $ 1 _ {A} $ is regular. In categories with zero morphisms one similarly introduces normal subobjects. If $ \mathfrak K $ possesses a bicategory structure $ ( \mathfrak K , \mathfrak L , \mathfrak M ) $, then a subobject $ ( \mu : X \rightarrow A ] $ of an object $ A $ is called admissible (with respect to this bicategory structure) if $ \mu \in \mathfrak M $.

Comments

The notation $ ( \mu ] $ 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.