Category:Definitions/Categories of Subobjects

This category contains definitions related to Categories of Subobjects.
Related results can be found in Category:Categories of Subobjects.

Let $\mathbf C$ be a metacategory.

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

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

Objects:		Subobjects $m: B \to C$ of $C$
Morphisms:		Morphisms $f: \operatorname{dom} m \to \operatorname{dom} m'$ of $\mathbf C$ such that $m' \circ f = m$ in $\mathbf C$
Composition:		Inherited from $\mathbf C$
Identity morphisms:		$\operatorname{id}_m := \operatorname{id}_{\operatorname{dom} m}$, the identity morphism in $\mathbf C$ of the domain of $m$

The behaviour of the morphisms is shown in the following commutative diagram in $\mathbf C$:

$\begin{xy}\xymatrix@+1em{

B
 \ar[r]^*+{f}
 \ar[rd]_*+{m}

&

B'
 \ar[d]^*+{m'}

\\ &

}\end{xy}$

Pages in category "Definitions/Categories of Subobjects"

The following 3 pages are in this category, out of 3 total.