# Category:Categories of Subobjects

Jump to navigation
Jump to search

This category contains results about Categories of Subobjects.

Definitions specific to this category can be found in Definitions/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}\[email protected]+1em{ B \ar[r]^*+{f} \ar[rd]_*+{m} & B' \ar[d]^*+{m'} \\ & C }\end{xy}$

## Pages in category "Categories of Subobjects"

The following 6 pages are in this category, out of 6 total.