Definition:Subobject Class/Morphism Class

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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 $C$.

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 see