nLab
power object

Idea

The notion of power object generalizes the notion of power set from the category Set to an arbitrary category with finite limits.

Definition

Let $C$ be a category with finite limits. A power object of an object $c\in C$ is

• an object ${\Omega }^c$

• a monomorphism ${\in }_c\hookrightarrow c\times {\Omega }^c$

such that

• for every other object $d$ and every monomorphism $r\hookrightarrow c\times d$ there is a unique morphism ${\chi }_r:d\to {\Omega }^c$ such that $r$ is the pullback

$$\begin{array}{ccc}r& \to & {\in }_c\\ \downarrow & & \downarrow \\ c\times d& \stackrel{{\mathrm{Id}}_c\times {\chi }_r}{\to }& c\times {\Omega }^c\end{array}$$\array{ r &\to& \in_c \\ \downarrow && \downarrow \\ c \times d &\stackrel{Id_c \times \chi_r}{\to}& c \times \Omega^c }

If $C$ may lack some finite limits, then we may weaken that condition as follows:

• If $C$ has all pullbacks (but may lack products), then equip each of ${\in }_c$ and $r$ with a jointly monic pair of morphisms, one to $c$ and one to ${\Omega }^c$ or $d$, in place of the single monomorphism to the product of these targets; $r$ must then be the joint pullback

  $$\begin{array}{ccc}r& \to & d\\ \downarrow & \searrow & & {\searrow }^{{\chi }_r}\\ c& & {\in }_c& \to & {\Omega }^c\\ & {\searrow }^{{\mathrm{Id}}_c}& \downarrow \\ & & c\end{array}$$\array { r & \rightarrow & d \\ \downarrow & \searrow & & \searrow^{\chi_r} \\ c & & \in_c & \rightarrow & \Omega^c \\ & \searrow^{Id_c} & \downarrow \\ & & c }

• If $C$ may lack some pullbacks, then we simply require that the pullback that $r$ is to equal must exist. But arguably we should require, if ${\Omega }^c$ is to be a power object, that this pullback exists for any given map $\chi :d\to {\Omega }^c$.

Examples

• If $1$ is a terminal object, then ${\Omega }^1$ is precisely a subobject classifier.

• A power object in Set is precisely a power set.

• A category with finite limits and power objects for all objects is precisely a topos. The power object $PA$ of any object $A$ in the topos is the exponential object $PA={\Omega }^A$ into the subobject classifier.

• See Trimble on ETCS I for the axiom of power sets in the elementary theory of the category of sets.