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

Properties

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

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

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

References

• Power object at the nLab
• Johnstone, Peter T. (2002). Sketches of an elephant: a topos theory compendium. Vol. 1. Clarendon Press. ISBN 9780198534259.