Power object

In category theory, a branch of mathematics, a power object in a category is an analogue of a powerset in the category of sets.

Definition

Let $𝒞$ be a finitely complete category. A power object of $A\in 𝒞$ is an object $𝒫(A)$ together with a subobject $(\in )\hookrightarrow A\times 𝒫(A)$ satisfying the following universal property: for every other object $B\in 𝒞$ and subobject $R\hookrightarrow B\times A$, there exists a unique morphism $\chi :B\to 𝒫(A)$ such that $R\hookrightarrow B\times A$ is the pullback of $(\in )\hookrightarrow A\times 𝒫(A)$ along $\chi$.Template:Sfn

Properties

In the category of sets, power objects exist: $𝒫(A)$ is the usual power set of $A$, and $(\in )\hookrightarrow A\times 𝒫(A)$ is the set membership relation.

More generally, in any elementary topos, the power object of $A$ can be constructed as $𝒫(A):={\mathrm{\Omega }}^A$ (where $\mathrm{\Omega }$ is the subobject classifier), with $(\in )\hookrightarrow A\times {\mathrm{\Omega }}^A$ being the subobject classified by the evaluation map $A\times {\mathrm{\Omega }}^A\to \mathrm{\Omega }$.Template:Sfn

Conversely, every finitely complete category with power objects is an elementary topos.Template:Sfn Thus, power objects provide a possible simplification of the definition of an elementary topos.

References

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