Power object ($PA={\Omega }^A$)

A power object $PA$ represents parameterized subobjects of $A$: $\mathcal{E}(X,PA)\cong \mathrm{Sub}(X\times A)$. In an elementary topos, it is ${\Omega }^A$, with the membership subobject supplied by evaluation at the subobject classifier.

If a Cartesian category has power objects, its slice categories do too. In $\mathcal{E}/A$, the constant family $A\times X\to A$ has power object $A\times PX\to A$, since its product with a family $Y\to A$ is $X\times Y$. An arbitrary family $p:X\to A$ is a subobject of the constant family through its graph $(p,1_X)$, so the power object of a subobject construction applies in the slice.

For a monomorphism $Y\hookrightarrow X$ in a Cartesian category, intersect the universal membership relation on $X\times PX$ with $Y\times PX$. The resulting subobject is classified by $r:PX\to PX$, and $r^2=r$. The equalizer of $r$ and the identity represents precisely families of subobjects of $X$ contained in $Y$, hence is a power object for $Y$. This construction only requires the power object of $X$ and finite limits.

An arrow $f:A\to B$ induces $Pf:PB\to PA$ by inverse image of predicates. Transposition of predicates on $A\times B$ gives $\mathcal{E}(A,PB)\cong \mathcal{E}(B,PA)$ and the adjunction $P^{\mathrm{op}}⊣P$. The induced monad on the topos is double power-object formation; it is distinct from the covariant powerset monad on sets.

The contravariant power-object functor of an elementary topos is monadic. Its double-power unit ${\eta }_A(a)(S)=(a\in S)$ is monic, which helps prove that $P$ reflects isomorphisms. Finite equalizers and the fact that power objects turn coreflexive equalizers into coequalizers give the remaining hypotheses of the crude monadicity theorem.

For an elementary topos, existence of products in a category indexed by a fixed set is equivalent to existence of the corresponding coproducts in a category. The power-object monadicity equivalence ${\mathcal{E}}^{\mathrm{op}}\simeq {\mathcal{E}}^{PP}$ gives the forward direction because Eilenberg-Moore categories create limits. For the converse, the diagonal into the pointwise product category is a logical functor. A coproduct is its left adjoint; the adjoint lifting theorem for monad algebra functors, applied through power-object monadicity, then gives its right adjoint, namely the product. The required reflexive coequalizers on the algebra side are opposites of finite equalizers in the topos.

For a coreflexive pair $f,g:B\rightrightarrows A$ with equalizer $e:E\hookrightarrow B$, direct image along the mono $f$ satisfies $Pf{\exists }_f=1$ and $Pg{\exists }_f={\exists }_{e}Pe$. Thus any map $h:PB\to Z$ equalizing $Pf,Pg$ obeys $h=h{\exists }_{e}Pe$. Since $Pe{\exists }_e=1$, the map $Pe$ is their coequalizer. These identities hold for parameterized subobjects in any elementary topos.