Encyclopedia

Power object

Wikipedia contributorsAdapted for Fathomly · View authors

1 min read
Source updated July 27, 2026

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

01Definition

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 $\chi$ .

02Properties

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.

Watch videos about Power objectExplainers and documentaries on YouTube (opens in a new tab)

Sources and credits

This article is adapted from the Wikipedia article “Power object”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

Fathomly is not affiliated with or endorsed by the Wikimedia Foundation. Spotted a problem? Tell us.