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 $\chi$.

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 $A\times {\mathrm{\Omega }}^A\to \mathrm{\Omega }$.

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