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

Let ${\displaystyle {\mathcal {C}}}$ be a finitely complete category. A power object of ${\displaystyle A\in {\mathcal {C}}}$ is an object ${\displaystyle {\mathcal {P}}(A)}$ together with a subobject ${\displaystyle (\in )\hookrightarrow A\times {\mathcal {P}}(A)}$ satisfying the following universal property: for every other object ${\displaystyle B\in {\mathcal {C}}}$ and subobject ${\displaystyle R\hookrightarrow B\times A}$, there exists a unique morphism ${\displaystyle \chi :B\to {\mathcal {P}}(A)}$ such that ${\displaystyle R\hookrightarrow B\times A}$ is the pullback of ${\displaystyle (\in )\hookrightarrow A\times {\mathcal {P}}(A)}$ along ${\displaystyle \chi }$.^[1]

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

More generally, in any elementary topos, the power object of ${\displaystyle A}$ can be constructed as ${\displaystyle {\mathcal {P}}(A):=\Omega ^{A}}$ (where ${\displaystyle \Omega }$ is the subobject classifier), with ${\displaystyle (\in )\hookrightarrow A\times \Omega ^{A}}$ being the subobject classified by the evaluation map ${\displaystyle A\times \Omega ^{A}\to \Omega }$.^[2]

Conversely, every finitely complete category with power objects is an elementary topos.^[3] Thus, power objects provide a possible simplification of the definition of an elementary topos.

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