Definition:Category of Subobject Classes

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Let $\mathbf C$ be a metacategory.

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

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

The category of subobject classes of $C$, denoted $\overline{\mathbf{Sub}}_{\mathbf C} \left({C}\right)$, is defined as follows:

Objects:         Subobject classes $\left[\!\left[{m}\right]\!\right]$ of $C$
Morphisms: Morphism classes $\left[\!\left[{f}\right]\!\right]: \left[\!\left[{m}\right]\!\right] \to \left[\!\left[{m'}\right]\!\right]$
Composition: Inherited from $\mathbf{Sub}_{\mathbf C} \left({C}\right)$, i.e. $\left[\!\left[{g}\right]\!\right] \circ \left[\!\left[{f}\right]\!\right] := \left[\!\left[{g \circ f}\right]\!\right]$
Identity morphisms: $\operatorname{id}_{\left[\!\left[{m}\right]\!\right]} := \left[\!\left[{\operatorname{id}_m}\right]\!\right]$, the morphism class of the identity morphism of $m$ in $\mathbf{Sub}_{\mathbf C} \left({C}\right)$

Also denoted as

Most authors don't care to distinguish the category of subobject classes symbolically from the category of subobjects $\mathbf{Sub}_{\mathbf C} \left({C}\right)$.

Also see