Subobject Class in Category of Sets

Theorem

Let $\mathbf {Set}$ be the category of sets.

Let $S$ be a set.

Let $\map {\overline {\mathbf {Sub} }_{\mathbf {Set} } } S$ be the category of subobject classes of $S$.

Let $\powerset S$ be the order category on the power set of $S$ induced by Subset Relation on Power Set is Partial Ordering.

Then $\map {\overline {\mathbf {Sub} }_{\mathbf {Set} } } S \cong \powerset S$.

Proof

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Sources

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