Definition:Category of Subobject Classes
Jump to navigation
Jump to search
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 $C$.
The category of subobject classes of $C$, denoted $\map {\overline {\mathbf{Sub}}_{\mathbf C} } C$, is defined as follows:
Objects: | Subobject classes $\eqclass m {}$ of $C$ | |
Morphisms: | Morphism classes $\eqclass f {}: \eqclass m {} \to \eqclass {m'} {}$ | |
Composition: | Inherited from $\map {\mathbf{Sub}_{\mathbf C} } C$: $\eqclass g {} \circ \eqclass f {} := \eqclass {g \circ f} {}$ | |
Identity morphisms: | $\operatorname{id}_{\eqclass m {} } := \eqclass {\operatorname{id}_m} {}$, the morphism class of the identity morphism of $m$ in $\map {\mathbf{Sub}_{\mathbf C} } C$ |
Also denoted as
Most authors don't care to distinguish the category of subobject classes symbolically from the category of subobjects $\map {\mathbf{Sub}_{\mathbf C} } C$.
Also see
- Category of Subobject Classes is Category
- Category of Subobject Classes is Order Category
- Category of Subobjects
Sources
- 2010: Steve Awodey: Category Theory (2nd ed.) ... (previous) ... (next): $\S 5.1$: Remark $5.2$