Definition:Subobject Class

Definition

Let $\mathbf C$ be a metacategory.

Let $C$ be an object of $\mathbf C$.

Let $\map {\mathbf{Sub}_{\mathbf C} } C$ be the category of subobjects of $C$.

A subobject class of $C$ is an equivalence class of subobjects of $C$ under the equivalence of subobjects.

If $m$ is a subobject, its associated subobject class may be denoted by $\overline m$ or $\eqclass m {}$.

Define the equivalence $\sim$ on the morphisms of $\map {\mathbf{Sub}_{\mathbf C} } C$ as follows.

For morphisms $f: m \to n$ and $g: m' \to n'$ of $\map {\mathbf{Sub}_{\mathbf C} } C$:

$f \sim g$ if and only if $m \sim m'$ and $n \sim n'$

where $m \sim m'$ signifies equivalence of subobjects.

A morphism class is an equivalence class $\eqclass f {}$ under $\sim$ of a morphism $f: m \to m'$.

The domain and codomain of $\eqclass f {}$ are taken to be $\eqclass m {}$ and $\eqclass {m'} {}$, respectively.

Many authors like to abuse language and call this a subobject as well.