# Definition:Subobject Class

## 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 $C$.

A **subobject class of $C$** is an equivalence class of subobjects of $C$ under the equivalence of subobjects.

If $m$ is a subobject, its associated **subobject class** may be denoted by $\overline m$ or $\eqclass m {}$.

### Morphism Class

Define the equivalence $\sim$ on the morphisms of $\map {\mathbf{Sub}_{\mathbf C} } C$ as follows.

For morphisms $f: m \to n$ and $g: m' \to n'$ of $\map {\mathbf{Sub}_{\mathbf C} } C$:

- $f \sim g$ if and only if $m \sim m'$ and $n \sim n'$

where $m \sim m'$ signifies equivalence of subobjects.

That $\sim$ in fact is an equivalence is shown on Morphism Class Equivalence is Equivalence.

A **morphism class** is an equivalence class $\eqclass f {}$ under $\sim$ of a morphism $f: m \to m'$.

The domain and codomain of $\eqclass f {}$ are taken to be $\eqclass m {}$ and $\eqclass {m'} {}$, respectively.

## Also known as

Many authors like to abuse language and call this a subobject as well.

## Also see

## Sources

- 2010: Steve Awodey:
*Category Theory*(2nd ed.) ... (previous) ... (next): $\S 5.1$: Remark $5.2$