Definition:Inclusion Relation on Subobject Classes

Definition
Let $\mathbf C$ be a metacategory.

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

Let $\overline{\mathbf{Sub}}_{\mathbf C} \left({C}\right)$ be the category of subobject classes of $C$.

The inclusion relation $\subseteq$ on subobject classes of $C$ is defined as follows:


 * $\left[\!\left[{m}\right]\!\right] \subseteq \left[\!\left[{m'}\right]\!\right]$ iff there exists a morphism $\left[\!\left[{f}\right]\!\right]: \left[\!\left[{m}\right]\!\right] \to \left[\!\left[{m'}\right]\!\right]$

Also see

 * Inclusion Relation on Subobjects
 * Inclusion Relation on Subobject Classes is Ordering
 * Inclusion Relation on Subobject Classes Induced by Category of Subobject Classes