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.